The 4th Annual Meeting of SIAM Central States Section

University of OklahomaOctober 5 - 7, 2018

Summary

1 SIAM Central States Section ii

2 Schedule at a glance 1

3 Parallel Sessions 4

MS01. Partial Differential Equations in Mathematical Fluid Mechanics . . . . . . . . . . . . . . . . . . . . 4

MS02. Numerical Methods and Stability of Nonlinear Dynamical System . . . . . . . . . . . . . . . . . . 5

MS03. Recent Advances in Modeling, Numerics, and Analysis of Electrodiffusion Phenomena . . . . . . . 7

MS04. Recent Developments of High-order Discontinuous Galerkin Methods . . . . . . . . . . . . . . . . 8

MS05. Interactions among Analysis, Optimization and Network Science . . . . . . . . . . . . . . . . . . . 9

MS06. Recent Development in Numerical PDEs and Applications . . . . . . . . . . . . . . . . . . . . . . 13

MS07. Applied and Computational Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

MS08. Recent Advances of Numerical Methods in Fluid Mechanics with Applications . . . . . . . . . . . 18

MS09. Modeling and Analysis in Ecology and Epidemiology . . . . . . . . . . . . . . . . . . . . . . . . . 21

MS10. Complex Patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

MS11. Partial Differential Equations: Analysis, Modeling, and Applications . . . . . . . . . . . . . . . . . 23

MS12. Novel Mathematical Methods with Applications to Continuum Mechanics . . . . . . . . . . . . . . 25

MS14. Recent Advances in Numerical PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

MS15. Inverse Problems in Tomography and Wave Propagation . . . . . . . . . . . . . . . . . . . . . . . 29

Contributed Talks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Posters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

List of Speakers 35

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1 SIAM Central States Section

Welcome to the University of Oklahoma and the 4th annual meeting of the SIAM Central States Section.

About SIAM Central States Section

The SIAM Central States Section was formed in 2014 to serve SIAM members in Arkansas, Colorado, Iowa,

Kansas, Mississippi, Missouri, Nebraska, and Oklahoma.

Section Purpose

The purpose of this section is to enhance the communication among the section members, promote the collabora-

tion for both basic research and applications of mathematics to industry and science, represent applied and computa-

tional mathematics in the entire proposed central region, and support the SIAM mission in the central region of the

USA.

Activities

The activities for the SIAM Central Section include annual section meetings, seminars and workshops for advanced

topics of common interests of the section members, encouraging new SIAM student chapters and facilitating all the

SIAM student chapters in the central region to connect together, promoting collaboration in applied mathematics and

its applications to industry and science, and expanding the influence of SIAM in the central states. Participation in

SIAM Central States Section activities will be open to all institutions and industries in the region with an interest in

applied and computational mathematics.

Leadership

President: Jiangguo Liu Colorado State University

Vice President: Anna Zemlyanova Kansas State University

Secretary: Ying Wang University of Oklahoma

Treasurer: Stephen Pankavich Colorado School of Mines

Department of Mathematics, University of Oklahoma

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2 Schedule at a glance

All talks are on Saturday (Oct.6) and Sunday (Oct.7)

60 minutes for a plenary talk (all in Physical Sciences Building (PHSC) 201)

20 minutes for a mini-symposium talk in all parallel sessions

Friday, October 5

05:30 pm - 07:00 pm Welcome Reception Physical Sciences Building 209(not a dinner)

05:30 pm - 07:00 pm Registration table opens Physical Sciences Building 209

Saturday, October 6

08:45 am Opening remarks by OU officials and announcements by SIAM CSS

09:00 am Plenary talk by Hong Qian, University of Washington

10:00 am Photo session and coffee break

10:40 am Parallel Session I

12:00 pm Lunch at OU Couch Restaurants

02:00 pm Plenary talk by Eitan Tadmor, University of Maryland

03:00 pm Transition to parallel sessions 2

03:10 pm Parallel Session II

04:30 pm Coffee break

05:00 pm Parallel Session III

06:45 pm Conference dinner at Molly Shi Boren ballroom in the OU Union

Sunday, October 7

09:00 am Plenary talk by Peter Kuchment, Texas A&M University

10:00 am Coffee break

10:30 am Parallel Session IV

11:50 am Adjourn

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The 4th Annual Meeting of SIAM Central States SectionPhysical Science Center, The University of Oklahoma

Planery Talks - PHSC 201Peter Kuchment, Texas A&M UniversityHong Qian, University of WashingtonEitan Tadmor, University of Maryland

Mini-Symposia:PHSC 114 PHSC 115 PHSC 116 PHSC 117 PHSC 212 PHSC 222 PHSC 313 PHSC 314 PHSC 316 PHSC 317 PHSC 321 PHSC 323

Session ISat, Oct. 610:40am-12:00pm

MS15K.ChenP.ChenS.Acosta

MS10B.HashmiD.HandwerkI.Oprea

MS05C.ScoglioT.McAllisterS.Eriksson- BiqueD.ZossoG.David

MS02A.LuoS.XingY.XuC.Guo

Contrib.D.BhattaH.KhudhairJ.Hassan

MS07E.GasparovicF.MottaA.ThomasM.K.Chung

MS06W.GengZ.WangJ.Tausch

MS14P.CazeauxQ.GuanX.TuH.GaoG.Harper

MS03H.MofidiP.LiuW.LiuZ.Wen

MS01L.HoangM.ChenW.YangX.Yu

MS08H.LeeY.EpshteynS.LiX.Zhao

Session IISat, Oct. 603:10pm-04:30pm

MS15B.YuanY.Yang (MSU)G.Ambart.

MS09S.B.,H.C.,T.G.J.LyuR.ParshadA.Basheer

MS05N.AlbinG.SpeightJ.MartinM.HigginsB.Alali

MS02Y.GuoY.YuanY.LiuB.YuJ-S.Pei

Contrib.U.FarooqZ.ChenC.Zhang

MS07K.XiaM.McGuirlN.GiansiracusaD.Guralnik

MS06S.LiW.WangX.ZhangY.Zhong

MS14J.KuJ.LiuS.LuoW.Maimaitiyiming

MS11S.DaiJ.MurphyT.Kaman

MS03Z.XuJ.ChenD.ChenM.Zhang

MS01Z.BradshawW.HuH.QiuK.Yamazaki

MS08C.WangX.YangX.He

Session IIISat, Oct. 605:00pm-06:20pm

MS12J.MearsM.EspanolY.Godin

MS09P.ShipmanA.Santhank.W.FitzgibbonF.Agusto

MS05D.CarageaJ.GillP.HornJ.LindH.Shakeri

Contrib.S.YuanA.BakerY.Wu

MS07J.SegertR.BeltonS.ChowdhuryJ.Bush

MS06H.FengH.LiR.GuoP.Yin

MS14A.MiedlarJ.MohebujjamanC.QiuJ.RossmanithD.Han

MS11W.HuB.XiaoD.HanE.Carlson

MS04Y.Yang (MTU)N.ChuenjarernY.LiuW.GuoQ.Zhuang

MS01Y.DaiN.JuY.Cao

MS08Z.WangN.JiangT.HoangJ.Singler

Session IVSun, Oct. 710:30am-11:50am

MS12A.ZemlyanovaA.Y.WuL.Greenleaf

MS09B.C-CharpentierM.Beauregard

MS05P.Poggi- CorradiniH.HakobyanJ.ClemensL.GeyerK.Kottegoda

MS07R.KomendarczykN.SandersonJ.DoverD.Heisterkamp

MS14Z.WangN.MalluwawaduD.ErkmenZ.Yang

MS11Y.PeiA.KolasinskiH.YeY.Yu

MS08J.ZhaoY.Zhang

Plenary Speakers

Peter Kuchment is an University Distinguished Professor in the Mathematics Department of Texas A&MUniversity. He is a Fellow of the AMS, SIAM, Institute of Physics, and AAAS (Amer. Soc. Adv. Sci.),as well as a Senior member of IEEE. His work is focused in the areas of partial differential equations,mathematical physics, spectral theory, medical imaging, material science, and enhancement of K-12 matheducation. He has authored nearly 200 publications in mathematics and its applications, including threeresearch monographs: Floquet Theory for Partial Differential Equations, Birkhauser 1993, Introductionto Quantum Graphs (joint with G. Berkolaiko), AMS 2013, and The Radon Transform and MedicalImaging, SIAM 2014.

Title: Mathematics of some novel imaging techniquesAbstract: The talk will provide a excursion into the mathematics of hybrid imaging, reconstructions with internalinformation, and time permitting (unlikely) Compton camera imaging.

Hong Qian received B.A. in astrophysics from Peking University and Ph.D. in molecular biology fromWashington University (St. Louis). His research interests turned to theoretical biophysical chemistry andmathematical biology when he was a Post-Doctoral Fellow with the University of Oregon and the Califor-nia Institute of Technology, where he studied protein folding. He was with the Department of Biomathe-matics, UCLA School of Medicine, from 1994 to 1997, when he worked on the theory of motor proteinsand single-molecule biophysics. He joined the University of Washington (Seattle) in 1997 and is currentlythe Olga Jung Wan Endowed Professor of Applied Mathematics. He has coauthored two books: Chem-ical Biophysics: Quantitative Analysis of Cellular Systems (Cambridge University Press, 2008) with D.A. Beard, and in Chinese, Mathematical Kinetic Models: Applications in Biophysics and Biochemistry(Peking Univ. Press, 2017) with H. Ge. His current research interest focuses on stochastic dynamics andits applications to biological systems. He is a fellow of the American Physical Society, and served andserves on the editorial board of journals on biophysical chemistry and quantitative/computational/systemsbiology.

Title: Mathematicothermodynamics: Stochastic Laws and Emergent Behavior of Population Kinetic SystemsAbstract: There is a growing awareness toward a slow shifting in the foundation of the thermodynamic laws, fromseveral macroscopic, empirical postulates concerning heat as a form of random mechanical motions, to derivablemathematical theorems based on stochastic dynamics of mesoscopic systems. It becomes increasingly clear that astochastic dynamic description of the Nature is a very effective mathematical representation of the Reality. In thistalk, I shall first introduce mathematicothermodynamics as a set of mathematical results in stochastic processes. Theresult is then applied to complex chemical kinetic systems. Via the limit by merely taking the molecular numbers tobe infinite, we are able to derive J. W. Gibbs’ macroscopic isothermal equilibrium chemical thermodynamics, as wellas generalize it to mesoscopic nonequilibrium systems such as biological cells.

Eitan Tadmor is a Distinguished University Professor at the University of Maryland (UMd) CollegePark. Earlier he was a Bateman Instructor in CalTech, 1980-1982 and on the faculty of his alma mater,Tel Aviv University. In 1995 Tadmor joined the UCLA Math Department where he was the foundingco-director of the NSF Institute for Pure and Applied Math (IPAM), 1999-2001. In 2002 he was recruitedto lead the UMd Center for Scientific Computation and Math Modeling (CSCAMM), and served as itsfirst Director, 2002-2016. In 2016-2017 he was a Senior Fellow at the Institute for Theoretical Studiesat ETH-Zurich. Tadmor was an ICM invited speaker (Beijing 2002), and delivered plenary addresses inthe international conferences on hyperbolic problems (Zurich 1990 and Beijing 1998), the SIAM invitedaddress at the 2014 Joint Math Meeting, and the 2016 Lecons Jacques-Louis Lions at the UniversitePierre et Marie Curie. In 2012 he was in the inaugural class of AMS Fellows. In 2015, he was awardedthe SIAM-ETH Henrici prize. Tadmor is the Principal Investigator of an NSF Research Network “KineticDescription of Emerging Challenges in Natural Sciences” (Ki-Net, 2012-2019).

Title: Short-Range Interactions and the Emergence of Higher-Order PatternsAbstract: A fascinating aspect in collective dynamics is self-organization: ants form colonies, birds flock, mobilenetworks coordinate a rendezvous and human crowds reach a consensus. We discuss the large-time, large-crowdflocking behavior of different models for collective dynamics driven by alignment. In particular, we address thecentral question how short-range interactions lead, over time, to the emergence of long-range, higher-order patterns inone- and multi-species dynamics.

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3 Parallel Sessions

MS01. Partial Differential Equations inMathematical Fluid MechanicsOrganizer: Ning Ju, ning.ju@okstate.edu, Okla-homa State UniversityCo-organizer: Jiahong Wu, jiahong.wu@okstate.

edu, Oklahoma State University

Schedule: PHSC 321 (for all three sessions)

Session I Sat, Oct. 6, 10:40am - 12:00pm

Developments in Asymptotic Expansions for Solutionsof Navier-Stokes EquationsLuan Hoang, Texas Tech UniversityAbstract: We study the long-time dynamics of theNavier-Stokes equations in the three-dimensional periodicdomains with a decaying outer body force. Consider theforce has a large-time asymptotic expansion in a very gen-eral system of decaying functions. We prove that anyLeray-Hopf weak solutions admits an asymptotic expan-sion of the same kind. This expansion is uniquely deter-mined by the force, and independent of the solutions. Ourgeneral theorem not only recovers the previous results ob-tained for power decaying functions, but also establishesthe expansions for the case of log(t), log(log(t)), etc. de-caying forces. This is joint work with Dat Cao (Texas techUniversity).

On classical solutions to the Cauchy problem of the 2Dcompressible non-resistive MHD equations with vac-uumMingtao Chen, Oklahoma State UniversityAbstract: In this paper, we investigate the Cauchy prob-lem of the compressible non-resistive MHD on mr2 withvacuum as far field density. We prove that the 2D Cauchyproblem has a unique local strong solution provided theinitial density and magnetic field decay not too slow at in-finity. Furthermore, if the initial data satisfies some addi-tional regularity and compatibility conditions, the strongsolution becomes a classical one. Additionally, we es-tablish a blowup criterion for the 2D compressible non-resistive MHD depending solely on the density and mag-netic fields.

3D incompressible magnetohydrodynamic equationswith fractional partial dissipationWanrong Yang, North Minzu UniversityAbstract: In this talk, we will mainly present recent re-sults about the global existence and uniqueness of H1-solutions to 3D incompressible MHD equations with onlydirectional velocity dissipation and horizontal magneticdiffusion −(−∆h)

54 b, where ∆h = p21 + p22.

New Prodi-Serrin Type Regularity Criteria for the 3D

Navier-Stokes EquationsXinwei Yu, University of AlbertaAbstract: We prove that a Leray-Hopf weak solutionfor the 3D incompressible Navier-Stokes equations wouldstay regular if a Prodi-Serrin type integrability conditionin Lorentz spaces is satisfied by div(u/|u|), or by the thirdcomponent u3, where u is the velocity function. Thekey to our proofs is a nonlinear Gronwall type inequal-ity. Our method can be easily adapted to ”upgrade” manyother Prodi-Serrin type conditions from Lebesgue spacesto Lorentz spaces. This is joint work with Mr. BenjaminPineau.

Session II Sat, Oct. 6, 03:10pm - 04:30pm

Properties of infinite energy solutions to the Navier-Stokes equationsZachary Bradshaw, University of ArkansasAbstract: Since their introduction by Lemarie-Rieusset,local Leray solutions to the Navier-Stokes equations haveproved useful for analyzing regularity and uniquenessproperties of the Navier-Stokes equations. This talk willfocus on several recent results concerning the existence,regularity, and properties of local Leray solutions.

Boundary Control of Optimal Mixing in Navier-StokesFlowsWeiwei Hu, Oklahoma State UniversityAbstract: We discuss the problem of optimal mixing ofan inhomogeneous distribution of a scalar field via an ac-tive control of the flow velocity, governed by the Navier-Stokes equations, in a two dimensional open bounded andconnected domain. We consider the velocity field steeredby a control input that acts tangentially on the boundary ofthe domain through the Navier slip boundary conditions.This is motivated by the problem of mixing within a cav-ity or vessel by moving the walls or stirring at the bound-aries. Our main objective is to design an optimal Navierslip boundary control that optimizes mixing at a given fi-nal time. Non-dissipative scalars, both passive and active,governed by the transport equation will be discussed. Inthe absence of diffusion, transport and mixing occur dueto pure advection. This essentially leads to a nonlinearcontrol problem of a semi-dissipative system. A rigor-ous proof of the existence of an optimal controller and thefirst-order necessary conditions for optimality will be pre-sented.

Existence results for Viscoelastic fluidsHua Qiu, South China Agricultural UniversityAbstract: In this talk, I will talk about some recent workabout the existence results related to the viscoelastic flu-ids.

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Stochastic analysis on PDE in fluid dynamicsKazuo Yamazaki, University of RochesterAbstract: In this talk, I will discuss some recent devel-opments of analysis on PDE in fluid dynamics with noise,about which the author has been involved. The discussionwill concern the Navier-Stokes equations, magnetohydro-dynamics system, the Hall-magnetohydrodynamics sys-tem, and possibly the Kardar-Parisi-Zhang equations. Thetopics should concern around the well-posedness, ergod-icity (Markov selection), and possibly rough path theory,regularity structures, and para-controlled distributions.

Session III Sat, Oct. 6, 05:00pm - 06:20pm

Global well-posedness of the magnetohydrodynamics(MHD) equations with fractional diffusionYichen Dai, Xiamen UniversityAbstract: In this talk, we consider the general d-dimensional magnetohydrodynamics (MHD) equationswith the fractional dissipation (−∆)αu and (−∆)βb. Weprove the local existence and uniqueness solutions of thed-D MHD equations and the global existence and unique-ness of weak solutions (u, b) ∈ L2(Rd) as a special con-sequence when α, β = 1

2 + d4 .

Solution Regularity of the Primitive EquationsNing Ju, Oklahoma State UniversityAbstract: Some recent results on regularity propertiesfor solutions of the Primitive Equations for large scaleoceanic flow of viscous fluid will be presented and dis-cussed.

Determining Form for the 2D Rayleigh-Benard prob-lemYu Cao, Indiana UniversityAbstract: We show that the long-time dynamics (theglobal attractor) for the 2D Rayleigh-Benard problemsubject to a set of mixed boundary conditions can be em-bedded in the long-time dynamics of an ordinary differen-tial equation, called a determining form. The phase spacefor this ODE is a Banach space of trajectories of a finitedimensional projection of velocity. It is shown that steadystates of the determining form can be uniquely identifiedwith the trajectories in the global attractor of the Rayleigh-Benard problem.

MS02. Numerical Methods and Stability ofNonlinear Dynamical System

Organizer: Bo Yu, yub@uwplatt.edu, University ofWisconsin-PlattevilleCo-organizers: Albert Luo, aluo@siue.edu, SouthernIllinois UniversityYu Guo, gary.rain@gmail.com, Midwestern StateUniversity

Schedule: PHSC 117 (for both sessions)

Session I Sat, Oct. 6, 10:40am - 12:00pm

Stability and bifurcation in Nonlinear dynamical sys-temsAlbert Luo, Southern Illinois University EdwardsvilleAbstract: Stability and bifurcation in Nonlinear dy-namical systems In this talk, the stability and singularityof equilibriums in nonlinear dynamical systems are dis-cussed in eigenvector space. Based on the eigenspace, co-variant and contravariant variables are defined for stabilityand bifurcation analysis of dynamical systems. The Hopfbifucation and spiral stability are studied through Fourierseries rather than circular assumption. Such investigationcan help one better understand the stability and bifurca-tion in nonlinear dynamical systems.

On the quantitative analysis of periodic motions in atime-delayed, softening, Duffing oscillatorSiyuan Xing, Southern Illinois University EdwardsvilleAbstract: In this paper, the period-1 and period-2 mo-tions of a time-delayed, softening, Duffing oscillator aredetermined by a semi-analytical method. Such a semi-analytical method established implicit mappings for pe-riodic motions through discretization of the differentialequation of the Duffing oscillator. The stability and the bi-furcation conditions of period-1 and period-2 motions insuch a Duffing oscillator are predicted through eigenvalueanalysis of the mappings of periodic nodes; the bifurca-tion trees of period-1 to period-2 motions are completed,accordingly. From analytical predictions, numerical sim-ulations of period-1 and period-2 motions in the oscillatorare carried out. From Finite Fourier series, the frequency-amplitude analysis for period-1 to period-2 motions arecompleted. Through this method, the quantitative analy-sis of periodic motions in such a delayed, nonlinear dy-namical system can be achieved.

Semi-analytical solutions of periodic motions in a vander Pol oscillatorYeyin Xu, Southern Illinois University EdwardsvilleAbstract: In this presentation, the periodic solutions ofperiodic motions of a van der Pol oscillator are achievedfrom a discrete mapping method. The discrete map-ping method produces semi-analytical solutions of peri-odic motions by discretizing the continuous nonlinear sys-tem to form the implicit mappings. Based on these spe-cific mapping structures, the semi-analytical solutions canbe obtained very accurately. Based on the discrete map-ping method, the stable periodic motion shrinking phe-nomenon is discovered. The shrinking pattern from stableperiodic motion to limit cycle of the van der Pol oscil-lator is released. The route from independent and sym-metric periodic motion adding sequence to chaos in a van

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der Pol oscillator are achieved Stability analysis is carriedout through eigenvalue analysis. The frequency amplitudecharacteristics of periodic motions are discussed. Numer-ical results are calculated in comparison of the analyticalresults. The phase trajectories of periodic motions are pre-sented and the evolution from unstable periodic motion tostable periodic motion can be clearly observed. The har-monic spectra are shown for a better understanding of thenonlinear properties of the periodic motions of the van derPol oscillator. The results achieved in this paper have var-ious applications in physics and electronics.

Period-1 and period-2 motions in a periodically forced,damped double pendulumChuan Guo, Southern Illinois University EdwardsvilleAbstract: In this paper, period-1 and period-2 motionsin a periodically forced, damped double pendulum is an-alytically predicted through a discrete implicit mappingmethod. The implicit mapping is established via the dis-cretized differential equation. The corresponding stabilityand bifurcation conditions of the period-1 and period-2motions are predicted through eigenvalue analysis. Nu-merical simulation of the period-1 and period-2 motionsin the double pendulum is completed from analytical pre-dictions.Session II Sat, Oct. 6, 03:10pm - 04:50pm

Analytical Bifurcations Trees of a Parametrically Ex-cited PendulumYu Guo, Midwestern State UniversityAbstract: In this paper, the complete bifurcation treesof a parametrically excited pendulum is investigated us-ing discrete implicit maps obtained through a mid-timescheme. Based on these discrete maps, mapping struc-tures are developed to describe periodic motions in thesystem. Analytical bifurcation trees of various periodicmotions to chaos are obtained through the nonlinear alge-braic equations of such implicit maps. Eigenvalue anal-ysis is carried out for stability and bifurcation analysis.Finally, numerical simulation results of various periodicmotions are illustrated in verification.

Period Motions and Stability of a Nonlinear SpringPendulumYaoguang Yuan, Southern Illinois University Ed-wardsvilleAbstract: In this paper, bifurcation trees of periodicmotions in a periodically forced, nonlinear spring pen-dulum system are predicted analytically through a semi-analytic method. The stability and bifurcation of the peri-odic motions are determined through eigenvalue analysis.The frequency- amplitude characteristics of the bifurca-tion trees of periodic motions are presented. Finally, fromthe analytical prediction, numerical simulations of peri-odic motions are completed. The complex periodic mo-

tions in the nonlinear spring pendulum are illustrated fora better understanding of motions complexity.

On periodic motions of a two degree of freedom chem-ical oscillatorYutong Liu, Southern Illinois University EdwardsvilleAbstract: In this paper, a model of oscillating chemi-cal reaction system is given by a 2-D nonlinear dynamicalsystem. Analytical solutions of periodic motions are ob-tained by analytical method accurately. The correspond-ing stability and bifurcations of periodic motions are cal-culated through the eigenvalue analysis of equilibriumpoints. Numerical solutions of periodic motions are cal-culated by transforming the 2-D nonlinear dynamical sys-tem into a nonlinear dynamical system of coefficients inthe Fourier series. From this study, some insight of oscil-latory behavior in a specific chemical reaction system canbe gained.

Investigation of vertical galloping and periodic mo-tions using the method of implicit mappingBo Yu, University of Wisconsin PlattevilleAbstract: Current methods for analyzing periodic mo-tion of a structure with a noncircular cross section usea one-degree-of-freedom galloping model. This modelrequires approximating the vertical force coefficient as apolynomial using a power series expansion. This approxi-mation has limitations, including limited accuracy and re-striction to period-I motion. The current approximationderives the first two coefficients of the power series ana-lytically and uses curve-fit experimental data to determinethe rest of the coefficients. In this presentation, a tech-nique is proposed to use implicit discrete mapping to ana-lytically predict the periodic motions of the system. Thesemaps are obtained by transferring the differential equa-tion used to describe the one-degree-of-freedom gallopingmodel into discrete counterparts. Using these mappingstructures, bifurcation trees of periodic motions are pre-dicted and corresponding stability and bifurcation analy-sis are carried out utilizing eigenvalue analysis. Numeri-cal results of periodic motions are generated to verify thetheoretical analytical prediction. Employing this implicitdiscrete mapping technique will improve on the accuracyof current methods used to analyze periodic motions.

A Brief Introduction to “Mem-Models” in Engineer-ing Mechanics ApplicationsJin-Song Pei, University of OklahomaAbstract: A significant event happened for electricalengineering in 2008, when researchers at HP Labs an-nounced that they had found “the missing memristor”,a fourth basic circuit element that was postulated nearlyfour decades earlier by Dr. Leon Chua who also devel-oped the theory of memristive systems. One consequenceof this announcement has been revitalized research in all

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areas of brain-imitating computer technologies, primar-ily because memristors mimic synapses. Moreover, thetheory involving memristor devices and memristive sys-tems was extended to include memcapacitors and me-minductors, thereby introducing an entire class of “mem-models”. This model class is the foundation of the presentstudy. By applying well-known mechanical-electrical sys-tem analogies, the mathematics of mem-models may betransferred to the setting of engineering mechanics, result-ing in mechanical counterparts of memristors, memcapac-itors, etc. We identify some recent examples of “mem-dashpots” and “mem-springs”. In addition to a “zero-crossing” condition, we highlight the role played by dis-continuities in the model and/or the excitation, the combi-nation of which enables mem-models to produce numer-ous hysteresis patterns. We also consider some new prop-erties and modeling techniques that call for further im-provement so that these new types of mechanical modelscan become more usable for analyzing real-world data.

MS03. Recent Advances in Modeling, Nu-merics, and Analysis of Electrodiffusion Phe-nomena

Organizer: Mingji Zhang, mingji.zhang@nmt.edu,New Mexico Institute of Mining and TechnologyCo-organizer: Duan Chen, Duan.Chen@uncc.edu,Uiversity of North Carolina Charlotte

Schedule: PHSC 317 (for both sessions)

Session I Sat, Oct. 6. 10:40am - 12:00pm

Effects of permanent charges and ion sizes on ionicfluxesHamidreza Mofidi, University of KansasAbstract: In this talk, we will report our ongoingproject concerning combined effects of (small) permanentcharges and ionic sizes on ionic fluxes via the Poisson-Nernst-Planck models with a hard-sphere component forion channel problems. For the three-constant-piece per-manent charge with one nonzero region, we are able toderive the expansion of the fluxes in the small nonzeropermanent charge and ionic diameter, up to quadratic or-der terms. Based on the leading order terms, we deter-mine several critical potentials in terms of the other keyphysical parameters that identify boundaries of differenteffects on fluxes rom the interaction between the perma-nent charge and ionic sizes. Other effects on fluxes due tohigh order terms will be reported too.

Non-Isothermal Electrokinetics: Energetic Varia-tional ApproachPei Liu, Penn State University

Abstract: Fluid dynamics accompanies with the entropyproduction thus increases the local temperature, whichplays an important role in charged systems such as the ionchannel in biological environment and electrodiffusion incapacitors/batteries. In this article, we propose a generalframework to derive the transport equations with heat flowthrough the Energetic Variational Approach. According tothe first law of thermodynamics, the total energy is con-served and we can use the Least Action Principle to derivethe conservative forces. From the second law of thermo-dynamics, the entropy increases and the dissipative forcescan be computed through the Maximum Dissipation Prin-ciple. Combining these two laws, we then conclude withthe force balance equations and a temperature equation.To emphasis, our method provide a self consistent proce-dure to obtain the dynamical equations satisfying properenergy laws and it not only works for the charge systemsbut also for general systems.

Permanent charge effects on ionic flowsWeishi Liu, University of KansasAbstract: In this talk, we will report our analytical andnumerical studies of permanent charge effects on ionicfluxes via Poisson-Nernst-Planck type models. A flux ra-tio associated to each ion species was identified throughanalysis that quantifies a key effect of permanent chargesin the sense that > 1 (resp. < 1) corresponds to an en-hancing (resp. a reducing) effect of the permanent chargeon the flux.

Boundary layer effects on ionic flows via classicalPoisson-Nernst-Planck modelsZhenshu Wen, Huaqiao UniversityAbstract: We study a quasi-one-dimensional steady-state Poisson-Nernst-Planck model for ionic flows of mul-tiple charged ion species through a membrane channel.Of particular interest is to study boundary layer effects onionic flows in terms of individual fluxes. More precisely,we study the flow properties of interest without assum-ing electro-neutrality boundary. Two cases are carefullyanalyzed, case one involves two ion species, one nega-tively charged and one positively charged, while case twoinvolves two positively charged ion species and one nega-tively charged ion species. In both cases, new phenomenaare observed and much more rich dynamics of ionic flowsare obtained.

Session II Sat, Oct. 6, 03:10pm - 04:30pm

Model of Visco-elastic Incompressible Fluid Flow andits ApplicationZhiliang Xu, University of Notre DameAbstract: Energetic Variational Approach is used toderive a novel thermodynamically consistent three-phasemodel of a mixture of Newtonian and visco-elastic flu-

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ids. The model which automatically satisfies the energydissipation law and is Galilean invariant, consists of cou-pled Navier-Stokes and Cahn-Hilliard equations. Modi-fied General Navier Boundary Condition with fluid elas-ticity taken into account is also introduced for using themodel to study moving contact line problems. The modelcan be applied for studying various biological or biophys-ical problems such as blood clotting.

Boundary layer effects on ionic flows via Poisson-Nernst-Planck models with local hard-sphere poten-tialsJianing Chen, New Mexico TechAbstract: We study a quasi-one-dimensional steady-state Poisson-Nernst-Planck model with a local hard-sphere potential for ionic flows of two oppositely chargedion species through a membrane channel. Of particularinterest is to study qualitative properties of ionic flowsin terms of individual fluxes without assuming electro-neutrality boundary conditions (indicating the existenceof boundary layers). Compared with our previous re-sults with electro-neutrality boundary conditions, newphenomena are observed, which further indicates the richand complicated dynamics of the flow properties of inter-est in the study of ion channel problem.

Fractional Poisson-Nernst-Planck Model for IonChannelsDuan Chen, University of North Carolina at CharlotteAbstract: In this talk, we introduce a fractional Poisson-Nernst-Planck model to describe ion permeation in ionchannels. Due to the intrinsic conformational changes,crowdedness in narrow channel pores, binding and trap-ping introduced by functioning units of channel proteins,ionic transport in the channel exhibits a power-law-likeanomalous diffusion dynamics. We start from continu-ous time random walk model for a single ion, and usea long-tailed density distribution function for the parti-cle jump waiting time, to derive the fractional Fokker-Planck equation. Then it is generalized to the macro-scopic fractional Poisson-Nernst-Planck model for ionicconcentrations. Necessary computational algorithms aredesigned to implement numerical simulations for the pro-posed model and the dynamics of transmembrane currentis investigated. Numerical simulations show that the frac-tional PNP model provides a more qualitatively reason-able match to the profile of gating currents from exper-imental observations. Meanwhile, the proposed modelmotivates new challenges in terms of mathematical mod-eling and computations.

Channel geometry and permanent charge effects onionic flows via Poisson-Nernst-Planck modelsMingji Zhang, New Mexico TechAbstract: We study effects of permanent charges

on ionic flows through ion channels via a quasi-one-dimensional classical Poisson-Nernst-Planck (PNP)model. The geometry of the three-dimensional channelis presented in this model to a certain extent, which iscrucial for the study in this paper. Two ion species, onepositively charged and one negatively charged, are con-sidered with a simple profile of permanent charges. Theclassical PNP model can be viewed as a boundary valueproblem (BVP) of a singularly perturbed system. The sin-gular orbit of the BVP depends on Q0 in a regular way.Assuming |Q0| is small, a regular perturbation analysisis carried out for the singular orbit. Our analysis indi-cates that effects of permanent charges depend on a richinterplay between boundary conditions and the channelgeometry. Furthermore, interesting common features arerevealed: for Q0 = 0, only an average quantity of thechannel geometry plays a role; however, for Q0 6= 0, de-tails of the channel geometry matter, in particular, to op-timize effects of a permanent charge, the channel shouldhave a short and narrow neck within which the permanentcharge is confined. The latter is consistent with structuresof typical ion channels.

MS04. Recent Developments of High-orderDiscontinuous Galerkin Methods

Organizer: Yang Yang, yyang7@mtu.edu, MichiganTechnological University

Schedule: PHSC 317

Session III Sat, Oct. 6, 05:00pm - 06:40pm

High-order bound-preserving discontinuous Galerkinmethods for compressible miscible displacements inporous media on triangularYang Yang, Michigan Technological UniversityAbstract: In this talk, we develop high-order bound-preserving (BP) discontinuous Galerkin (DG) methods forthe coupled system of compressible miscible displace-ments on triangular meshes. We consider the problemwith multi-component fluid mixture and the (volumetric)concentration of the j-th component, cj , should be be-tween 0 and 1. There are three main difficulties. Firstly, cjdoes not satisfy a maximum-principle. The main idea is toapply the positivity-preserving techniques to all cj’s andenforce Σcj = 1 simultaneously. By doing so, we have totreat the time derivative of the pressure dp/dt as a sourcein the concentration equation and choose suitable fluxesin the pressure and concentration equations. Secondly,the construction of high-order BP schemes on triangularmeshes. We use interior penalty DG methods for the con-centration equations, and the first-order numerical fluxesare not easy to construct. Therefore, we will constructsecond-order BP schemes and then combine the second-

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and high-order fluxes to obtain a new one. Finally, cj’sare not the conservative variables, as a result, the classicalslope limiter cannot be applied. Moreover, for fluid mix-ture with more than two components, we cannot simplyset the upper bound of each cj to be 1. Therefore, a suit-able limiter for multi-component fluid will be introduced.Numerical experiments will be given to demonstrate thehigh-order accuracy and good performance of the numer-ical technique.

Fourier analysis of local discontinuous Galerkin meth-ods on overlapping meshesNattaporn Chuenjarern, Michigan Technological Univer-sityAbstract: The local discontinuous Galerkin (LDG)method is a popular one for solving parabolic equationssuch as the heat equation by introducing the auxiliary vari-able q to represent the derivative of the primary variableu, and solve them on the same mesh. A new LDG methodwas introduced to solve u and q on different meshes toimprove the computational cost. In this talk, we will usethe new method and Fourier analysis to analyze the errorestimate of one dimension parabolic equations on over-lapping meshes. We also demonstrate the relationship ofthe ratio of the overlapping mesh and the super conver-gence points to obtain the super convergence. Numericalexperiments will be given to demonstrate the analysis.

An implicit sparse grid discontinuous Galerkinmethod for high dimensional reaction-diffusion equa-tionsYuan Liu, Mississippi State UniversityAbstract: In this talk, we will introduce a class ofimplicit sparse grid discontinuous Galerkin(DG) methodwith the aim of breaking the curse of dimensionality. Thekey ingredient of the spatial discretization is a set of mul-tiwavelet basis functions on nested grids. Krylov implicitintegration factor method is used for temporal discretiza-tion in order to relax the CFL constraints. Numerical ex-amples in 2D and 3D cases are provided to show the per-formance of our proposed method.

An alternative formulation of discontinous Galerkinschemes for solving Hamilton-Jacobi equationsWei Guo, Texas Tech UniversityAbstract: In this talk, we will introduce an alterna-tive formulation of discontinuous Galerkin (DG) schemesfor approximating the viscosity solutions to nonlinearHamilton-Jacobi (HJ) equations. The main difficulty indesigning DG schemes lies in the inherent non-divergenceform of HJ equations. One effective approach is to ex-plore the elegant relationship between HJ equations andhyperbolic conservation laws: the standard DG schemeis applied to solve a conservation law system satisfiedby the derivatives of the solution of the HJ equation. In

this work, we consider an alternative approach to directlysolving the HJ equations, motivated by a class of success-ful direct DG schemes by Cheng et al. [J. Comput. Phys.,v223 (2009); J. Comput. Phys., 268(2014)].The proposedscheme is derived based on the idea from the central-upwind scheme by Kurganov et al. [SIAM J. Sci. Com-put., v23 (2001)]. In particular, we make use of preciseinformation of the local speeds of propagation at discon-tinuous element interface with the goal of adding adequatenumerical viscosity and hence naturally capturing the vis-cosity solutions. A collection of numerical experiments ispresented to demonstrate the performance of the methodfor solving general HJ equations with linear, nonlinear,smooth, non-smooth, convex, or non-convex Hamiltoni-ans.

Solving Interface Problems of the Helmholtz EquationQiao Zhuang, Virginia TechAbstract: This presentation reports our explorations ofsolving interface problems of Helmholtz equation by theimmersed finite elements with an interface independentmesh. Two immersed finite element (IFE) methods areinvestigated: the partial penalized IFE (PPIFE) and dis-continuous Galerkin IFE (DGIFE) methods. Optimal con-vergence rates are observed for these IFE methods oncethe mesh size is smaller than an optimal mesh size whichis mainly dictated by the wave number. Our explorationsalso suggest that higher degree IFE methods are more effi-cient than their lower degree counterparts because of theirlarge optimal mesh size and higher convergence rates.

MS05. Interactions among Analysis, Opti-mization and Network Science

Organizer: Pietro Poggi-Corradini, pietro@math.

ksu.edu, Kansas State UniversityCo-organizer: Nathan Albin, albin@math.ksu.edu,Kansas State University

Schedule: PHSC 116 (for all four sessions)

Session I Sat, Oct. 6, 10:40am - 12:20pm

SIS Epidemics in Temporal Networks: a MultilayerApproachCaterina Scoglio, Kansas State UniversityAbstract: To improve the accuracy of network-basedSIS models we introduce and study a multi-layer repre-sentation of a time-dependent network. In particular, weassume that individuals have their long-term (permanent)contacts that are always present, identifying in this waythe first network layer. A second network layer also exists,where the same set of nodes can be connected by occa-sional links created with a given probability. While linksof the first layer always exist, a link of the second layer is

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only activated with some probability and under the condi-tion that the two nodes, connected by this link, are simul-taneously participating to the temporary link. We developa model for the SIS epidemic on this time-dependent net-work, analyze equilibrium and stability of the correspond-ing mean-field equations, and shed some light on the roleof the temporal layer on the spreading process.

Period collapse in Ehrhart quasi-polynomialsTyrrell McAllister, University of WyomingAbstract: The problem of counting integer lattice pointsin rational polytopes arises in many contexts, includingthe geometry of toric varieties and the representation the-ory of semisimple Lie algebras. If a polytope P ⊂ Rnhas rational vertices, then a seminal result of E. Ehrhartsays that the number of lattice points in the kth dilate ofP (where k is a positive integer) is a quasi-polynomialfunction of k — that is, a ”polynomial” in which the co-efficients are themselves periodic functions of k. ”Periodcollapse” occurs when these coefficients have smaller pe-riods than naively expected. This phenomenon has con-nections to the deformation theory of the correspondingtoric varieties. We present tools for constructing poly-topes in which the periods of the coefficients take on pre-scribed values. The cyclic polytopes of McMullen’s upperbound theorem will make a surprise appearance.

How to prove a Poincare inequality?Sylvester Eriksson-Bique, University of California LosAngelesAbstract: I will discuss some of my recent work andcollaborations on conditions guaranteeing and character-izing Poincare inequalities, and some ways these can beused in particular applications. For example, I will dis-cuss the existence of a Semmes family of curves given a1-Poincare inequality, and an idea of iteration that can beused to prove and improve Poincare inequalities. Timepermitting some applications and consequences will bementioned.

Studying the shape of data: From geometric probabil-ity to data scienceDominique Zosso, Montana State UniversityAbstract: In this talk I will first provide a lay-man’srandom walk from geometric measure theory to integralgeometry of convex bodies, loosely based on Gian-CarloRota’s 1997 AMS colloquium lecture. An important re-sult is the following: the volume of a convex body Athickened (dilated) by r is a polynomial in r (Steiner for-mula). Its coefficients are called Minkowski functionals,mixed volumes, or Quermassintegrals (for different nor-malizations); they can be identified with properties suchas volume, perimeter, mean width, etc. of the initialbody, and fully characterize A from a geometric proba-bility point of view.I will then sketch how we now would

like to use well-established extensions of the above to theconvex ring (finite unions of convex bodies) as a startingpoint for further inquiry into studying the shape of data:Consider a data set in the form of a point cloud in somepotentially high-dimensional feature space, where indi-vidual points are samples from a nice underlying struc-ture that we are trying to uncover (think: samples from adonut in 3-space—gotta have this donut). We can gener-ate a finite union of convex bodies parametrized by thatdata set by associating a ball of radius R+r with eachdata point. By estimating the volume (with multiplicities)of this ball-cloud, for different thickenings r, we can de-termine the Quermassintegrals through polynomial fitting(at scale R).We anticipate that such a tool will be usefulin generic data science as well as in some concrete low-dimensional ”shape-from-samples” problems, such as theshape description of biofilms or the characterization ofpercolation.

Metric spaces as limits of graphs and the Analyst’sTraveling Salesman TheoremGuy C. David, Ball State UniversityAbstract: We describe a class of highly non-Euclideanmetric spaces, constructed by Cheeger and Kleiner, thathave interesting analytic properties and are built as inverselimits of finite simplicial graphs. We then report on recentjoint work of the speaker and Raanan Schul, which provesa version of Peter Jones’s ”Analyst’s Traveling SalesmanTheorem” in these spaces. This characterizes subsets ofrectifiable curves as those which are quantitatively flat in acertain sense, which must be correctly interpreted in thesenon-Euclidean spaces.

Session II Sat, Oct. 6, 03:10pm - 04:50pm

Spanning Tree ModulusNathan Albin, Kansas State UniversityAbstract: Spanning tree modulus on a graph is related toa number of interesting problems. For example, throughthe concept of blocking duality, spanning tree modulus isstrongly connected to the modulus of a transverse familyof objects, the feasible partitions. The solution to eitherof these problems immediately yields the solution to theother. On the other hand, when viewed through a proba-bilistic framework, spanning tree modulus is dual to twointerrelated problems involving random spanning trees:the fairest edge usage problem and the minimum expectedoverlap problem. The former asks for a probability distri-bution on spanning trees that spreads the edge usage prob-abilities as evenly as possible, while the latter asks for adistribution with the property that the expected overlap oftwo independent, identically distributed random spanningtrees is as small as possible. Finally, through a processknown as deflation of homogeneous subgraphs, spanningtree modulus is connected to two combinatorial problems:

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the densest subgraph problem and the minimum feasiblepartition problem. This talk will introduce all of theseproblems and show how all are connected through thespanning tree modulus.

Maximal Directional Derivatives and Universal Differ-entiability SetsGareth Speight, University of CincinnatiAbstract: Rademacher’s theorem asserts that every Lip-schitz function from Rn to Rm is differentiable except ona set of measure zero. Depending on the dimensions of thespaces involved, this theorem may or may not be sharp. Ifn > 1 then there exists a measure zero set in Rn contain-ing a point of differentiability for every Lipschitz functionfrom Rn to R. Such sets are called universal differentia-bility sets. We investigate universal differentiability setsin Euclidean spaces and Carnot groups. Carnot groups arespaces with much of the Euclidean structure (translations,dilations, Haar measure and a path distance) but wherethe shortest distance between points may not be along astraight line.

Oscillation estimates of eigenfunctions via the combi-natorics of noncrossing partitionsJeremy Martin, University of KansasAbstract: We study oscillations in the eigenfunctionsfor a fractional Schrodinger operator on the real line. Anargument in the spirit of Courant’s nodal domain theo-rem applies to an associated local problem in the upperhalf plane and provides a bound on the number of nodaldomains for the extensions of the eigenfunctions. Usingthe combinatorial properties of noncrossing partitions, weturn the nodal domain bound into an estimate for the num-ber of sign changes in the eigenfunctions. We discuss ap-plications in the periodic setting and the Steklov problemon planar domains. This is joint work with Vera Miky-oung Hur (University of Illinois) and Mat Johnson (Uni-versity of Kansas). Since I am the one combinatorialiston the team, my talk will focus more on the combinatorialtechniques used than the underlying PDE problems.

New methods for incorporating network cyclic struc-tures to improve community detectionMichael Higgins, Kansas State UniversityAbstract: A distinguishing property of communities innetworks is that cycles are more prevalent within com-munities than across communities. Hence, the detec-tion of these communities may be aided through theuse of measures of the local ”richness” of cyclic struc-tures. We investigate the use of two methods for quantify-ing this richness—loop modulus (LM) and renewal non-backtracking random walks (RNBRW)—to improve theperformance of existing community detection algorithms.LM solves a quadratic program to find an optimal alloca-tion of edge usage across cycles to minimize cycle over-

lap, thereby giving a rigorous way to quantify the impor-tance of edges. RNBRW, introduced for the first time inthis paper, quantifies edge importance as the likelihood ofan edge completing a cycle in a non-backtracking randomwalk. We argue that RNBRW provides an efficient andscalable alternative to LM. We give simulation results thatsuggest pre-weighting edges by the proposed methods canimprove the performance of popular community detectionalgorithms substantially. Our methods are especially effi-cient for the challenging case of detecting communities insparse graphs.

Spectral methods for peridynamic modelsBacim Alali, Kansas State UniversityAbstract: Linear peridynamic-type operators, with sym-metric integrable kernels, can be expressed in terms ofcompact self-adjoint operators, which makes them suit-able to study using spectral methods. A spectral numer-ical method is developed for solving scalar peridynamicequations with periodic boundary conditions. A FourierContinuation method can be used to extend the spectralmethod to study other boundary conditions. An investi-gation of sources of discontinuities and their evolution inperidynamic models is provided.

Session III Sat, Oct. 6, 05:00pm - 06:40pm

Android Malware Detection Using Graph-based LabelPropagation ApproachesDoina Caragea, Kansas State UniversityAbstract: Android devices and apps are becoming in-creasingly popular. At the same time, malware appsare becoming more prevalent, and current app vettingtechnologies are lagging behind the threats. Recent re-search has focused on automated malware detection ap-proaches that combine machine learning techniques withstatic and dynamic code analysis. However, for Androidmalware detection, precise ground truth is a rare commod-ity. As security knowledge evolves, what may be con-sidered ground truth at one moment in time may change,and apps once considered benign turn out to be mali-cious. The inevitable noise in data labels poses a chal-lenge to creating effective machine learning models. Ourwork is focused on approaches for learning classifiers forAndroid malware detection in a manner that is method-ologically sound with regard to the uncertain and ever-changing ground truth in the problem space. More specifi-cally, we study graph-based label propagation approaches,where the nodes represent apps, and the links betweennodes denote the similarity between apps. The similarityis determined based on metrics that make use of featuresextracted using static analysis. Given a graph that containsa small number of labeled apps and a large number of un-labeled apps, the goal is to propagate the labels from thelabeled apps to the unlabeled apps, while accounting for

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potential noise in the original labels. Several label prop-agation approaches are studied and compared to identifythose approaches that can best handle the noisy groundtruth and produce accurate classifications.

Finite Perimeter Sets in Metric Measure Spaces andMinimalityJames Gill, Saint Louis UniversityAbstract: We discuss sets of finite perimeter in arbi-trary metric measure spaces with doubling condition andPoincare inequality. We are specifically interested in tan-gent spaces in the blow-up. This is joint research withSylvester Eriksson-Bique, Panu Lahti, and NageswariShanmugalingam.

Gradient and Harnack type inequalities for PageRankPaul Horn, University of DenverAbstract: PageRank, as introduced by Brin and Page,and more generally ‘personalized PageRank,’ has been offundamental importance in network search. A key pa-rameter in PageRank is the ‘jumping constant’ which al-lows (among other things) one to tailor the sensitivity ofPageRank to local cuts. In this talk, we describe new gra-dient estimates (akin to the Li-Yau inequality for solutionsto the heat equation) and Harnack type inequalities forgraphs satisfying certain curvature conditions. These al-low us to compare the ‘importance’ of different nodes, andhow PageRank regularizes as the jumping constant goes tozero. This is based on joint work with Lauren Nelsen.

Effect of random time changes on Loewner hullsJoan Lind, University of TennesseeAbstract: Loewner hulls are determined by their real-valued driving functions, via the Loewner differentialequation. We study the geometric effect on the Loewnerhulls when the driving function is composed with a ran-dom time change, such as the inverse of an α-stablesubordinator. In contrast to Schramm-Loewner evolu-tion (SLE), we show that for a large class of randomtime changes, the time-changed Brownian motion processdoes not generate a simple curve. Further we developcriteria which can be applied in many situations to de-termine whether the Loewner hull generated by a time-changed driving function is simple or non-simple. Toaid our analysis of an example with a time-changed de-terministic driving function, we prove a deterministic re-sult that a driving function that moves faster than atr forr ∈ (0, 1/2) generates a hull that leaves the real line tan-gentially.

Optimizing Extremal Eigenvalues in Multiplex Net-worksHeman Shakeri, Kansas State UniversityAbstract: We can learn much about graphs by calcu-lating the extremal eigenvalues of the Laplacian matrix.

These values are closely connected to the node and linkconnectivity, isoperimetric properties, max-cut and rate ofmixing in Markov chains. In this talk, we discuss how todesign interlayer links in multiplex networks to push theseextremal eigenvalues further and improve the overall per-formance. Therefore, we find the best interlayer weightdistributions that with a given budget; first, minimize thesmallest nonzero eigenvalue known as the algebraic con-nectivity; second, maximize the largest eigenvalue knownas the Laplacian spectral radius; and finally, minimize thespectral gap of the Laplacian.

Session IV Sun, Oct. 7, 10:30am - 12:10pm

Discrete approximation of continuous modulus usinggridsPietro Poggi-Corradini, Kansas State UniversityAbstract: p-Modulus of curve families is a commontool in complex analysis and more generally geometricfunction theory in metric spaces with controlled geometrysuch as higher-dimensional Euclidean spaces. A paralleltheory of p-Modulus has been developed on graphs, hencea natural question arises: can continuous modulus be ap-proximated by discrete modulus? We show that connect-ing modulus on topological quadrilateral in the complexplane can be approximated by discrete connecting mod-ulus when using very general Skopenkov-Werness-typequadrangular grids.The proof makes use of the theory ofFulkerson duality developed by the author, N. Albin et al.We will also describe several remaining open questions.

Quasisymmetric embeddings of Sierpinski spaces intothe planeHrant Hakobyan, Kansas State UniversityAbstract: We will provide some necessary and somesufficient conditions for the existence of quasisymmetric(QS) embeddings of planar domains and Sierpinski car-pets into the plane. For some classes of constructions thisgives a complete characterization of the Sierpinski carpetswhich QS embed into the plane. The main tool is a ver-sion of transboundary modulus of families of curves dueto Schramm.

A greedy approach to modulusJason Clemens, Wichita State UniversityAbstract: Given a family of objects on a network, themodulus problem can be formulated as a convex optimiza-tion problem. It has been shown that the dual problem canbe viewed as another modulus problem on a related fam-ily of objects. This talk will introduce a greedy algorithmdeveloped to solve these modulus problems, as long as aminimum object can be found.

p-modulus and p-resistor networksLukas Geyer, Montana State University

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Abstract: I will discuss the concepts of p-resistor net-works and effective p-resistance and its connection to p-modulus and p-extremal length. These concepts were in-troduced by Jarek Kwapisz in his investigation of the con-formal dimension of Sierpinski carpets. It turns out thateffective p-resistance is comparable to p-extremal length,but easier to handle computationally. The notion of p-resistor networks also naturally lead to notions of convexand topological duality, which are useful both computa-tionally and theoretically.

Spanning tree modulus for secure broadcast gamesKapila Kottegoda, Kansas State UniversityAbstract: The p-modulus is a general framework forquantifying the richness of a family of objects on a graph.When applied to the family of spanning trees, p-modulushas an interesting probabilistic interpretation. In particu-lar, the 2-modulus problem in this case has been shownto be equivalent to the problem of finding a probabilitydistribution on spanning trees that utilizes the edges ofthe graph as evenly as possible. For this reason, thereis a strong connection between 2-modulus of the familyof spanning trees and the edge-disjointness of this family.We use this fact to produce a game-theoretic interpretationof modulus and apply modulus to the problem of mini-mizing the number of broadcast messages intercepted byan eavesdropper listening on an unknown link.

MS06. Recent Development in NumericalPDEs and Applications

Organizer: Peimeng Yin, pemyin@iastate.edu,Iowa State UniversityCo-organizers: Songting Luo, luos@iastate.edu,Iowa State UniversityHailiang Liu, hliu@iastate.edu, Iowa State Univer-sity

Schedule: PHSC 313 (for all three sessions)

Session I Sat, Oct. 6, 10:40am - 12:00pm

On preconditioning the treecode-accelerated bound-ary integral (TABI) Poisson-Boltzmann solverWeihua Geng, Southern Methodist UniversityAbstract: We recently developed a treecode-acceleratedboundary integral (TABI) solver for solving Poisson-Boltzmann (PB) equation. The solver has combined ad-vantages in accuracy, efficiency, memory, and paralleliza-tion as it applies a well-posed boundary integral formu-lation to circumvent many numerical difficulties associ-ated with the PB equation and uses an O(N ∗ log(N))treecode to accelerate the GMRES iterative solver. How-ever, as observed in our previous work, occasionally when

the triangular mesh generator produces low quality trian-gles, the number of GMRES iterations required to solvethe discretized boundary integral equations Ax = b couldbe large. To address this issue, we design a precondi-tioning scheme using preconditioner matrix M such thatM (− 1)A has much improved condition while M (− 1)zcan be rapidly computed for any vector z. The numeri-cal results show that this new preconditioning scheme im-proves the TABI solver with significantly reduced itera-tion numbers and better accuracy, particularly for proteinsets on which TABI solver previously converges slowly.

Positivity preserving, conservative, and free en-ergy dissipating finite difference methods for multi-dimensional Poisson–NernstZhongming Wang, Florida International UniversityAbstract: We develop simple but effective finitedifference methods for solving the multi-dimensionalPoisson–Nernst–Planck (PNP) equations with multipleionic species. A novel central-differencing discretizationbased on harmonic-mean approximations is proposed forthe Nernst–Planck (NP) equations. The forward and back-ward Euler discretizations in time are employed to derivean explicit scheme and a linearized semi-implicit scheme,respectively. Numerical analysis proves that the numeri-cal schemes respect three desired properties that are pos-sessed by analytical solutions: I) ionic mass conservation,II) positivity of ionic concentrations, and III) free-energydissipation. The semi-implicit scheme is further shown topreserve positivity unconditionally, whereas a constrainton a mesh ratio is required for the explicit scheme to en-sure positivity. The positivity preservation is establishedbased on advantages brought by the harmonic-mean ap-proximations, In addition, theoretical and computationalinvestigations are performed to study condition numbersof the semi-implicit discretization of the NP equations,further revealing advantages of the scheme in computa-tional efficiency and stability. Our estimates on the up-per bound of condition numbers indicate that the devel-oped discretization based on harmonic-mean approxima-tions can effectively solve a known issue — a large con-dition number is often accompanied by the use of Slot-boom variables. Numerical tests verify that the numericalsolution respects desired properties, and is second-orderaccurate in space and first-order accurate in time. An ap-plication of the numerical scheme to an electrochemicalcharging system demonstrates its effectiveness in solvingrealistic problems. This is a joint work with D. Jie and S.Zhou.

A shape optimization approach for Stefan problemsJohannes Tausch, Southern Methodist UniversityAbstract: The Stefan problem is reformulated as a shapeoptimization problem for the position of the phase transi-tion as a function of time. The functional to be minimized

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is the mismatch of the Dirichlet to Neumann map and theStefan condition. The only stationary point of the func-tional is the minimum which is also the solution of theStefan problem. A gradient based optimization method isderived from shape calculus. The state and adjoint equa-tion of the heat equation are solved using integral equationtechniques which avoids a discretization in the domain. ANystrom quadrature method is analyzed and numerical re-sults are presented.

Session II Sat, Oct. 6, 03:10pm - 04:30pm

Computation of a shrinking interface problemShuwang Li, Illinois Institute of TechnologyAbstract: In this talk, we present an adaptive rescal-ing method for computing a shrinking interface in a Hele-Shaw cell with a time increasing gap width b(t). We fo-cus our study on a one-phase interior Hele-Shaw prob-lem where a blob of fluid, surrounded by air, dynami-cally responds to the changing gap width. To explorethe full nonlinear interface dynamics, we develop a spec-trally accurate boundary integral method in which a newtime and space rescaling is implemented. In the rescaledframe, the motion of the interface is slowed down, whilethe area/volume enclosed by the interface remains un-changed. This method, for the first time, enables us toadaptively remove the severe numerical stiffness imposedby the rapidly shrinking interface (especially at late times)and accurately compute the dynamics to far longer timesthan could previously be accomplished. Numerical testsdemonstrate that the method is stable, efficient and accu-rate.

A high order well-balanced particle-in-cell method forshallow water equationsWei Wang, Florida International UniversityAbstract: A hybrid Eulerian–Lagrangian parti-cle–in–cell (PIC) type numerical method is developedfor the solution of advection dominated flow problems.For smooth flows the method presented is of formalhigh–order accuracy in space. The method is applied tosolve the non–linear shallow water equations resulting in anew, and novel, shock capturing shallow water solver. Theapproach is able to simulate complex shallow water flowswhich can contain an arbitrary number of discontinuities.Both trivial and non–trivial bottom topography is consid-ered and it is shown that the new scheme is inherentlywell–balanced exactly satisfying the C–property. Thescheme is verified against several 1D benchmark shallowwater problems.

A residual-based a posteriori error estimation for im-mersed finite element methodXu Zhang, Mississippi State UniversityAbstract: Interface problems arise in many applications

in science and engineering. Partial differential equations(PDEs) are often used to model interface problems. So-lutions to these PDE interface problems often involveskinks, singularities, discontinuities, and other non-smoothbehaviors. The immersed finite element method (IFEM)is a class of numerical methods for solving PDE interfaceproblems with unfitted meshes. In this talk, we introduce aresidual-based a posteriori error estimation for IFEM. Weshow that the error estimator is globally reliable and lo-cally efficient, with the reliability and efficiency constantsindependent of the interface location. This error estima-tor can provide accurate assessment of the numerical ap-proximation and can also be used as a guidance for adap-tive mesh refinement for IFEM. Numerical results are pro-vided to demonstrate the features of the adaptive IFEM.

A fast algorithm for radiative transportYimin Zhong, University of California IrvineAbstract: We propose in this work a fast numerical algo-rithm for solving the equation of radiative transfer (ERT)in isotropic media. The algorithm has two steps. In thefirst step, we derive an integral equation for the angu-larly averaged ERT solution by taking advantage of theisotropy of the scattering kernel, and solve the integralequation with a fast multipole method (FMM). In the sec-ond step, we solve a scattering-free transport equation torecover the original ERT solution. Numerical simulationsare presented to demonstrate the performance of the algo-rithm for both homogeneous and inhomogeneous media.

Session III Sat, Oct. 6, 05:00pm - 06:20pm

An augmented matched interface and boundary(MIB) method for solving elliptic interface problemHongsong Feng, University of AlabamaAbstract: A second order accurate augmented matchedinterface and boundary (MIB) is introduced for solv-ing two-dimensional (2D) elliptic interface problems withpiecewise constant coefficients. The augmented MIBseamlessly combines several key ingredients of the stan-dard MIB, augmented immersed interface method (IIM),and explicit jump IIM, to produce a new fast interface al-gorithm. Fictitious values generated by MIB method pro-vide a simple procedure to reconstruct Cartesian deriva-tive jumps as auxiliary variables and couple them withthe jump-corrected Taylor series expansions, which allowus to restore the order of the central difference across theinterface to two. Moreover, by using the Schur comple-ment to disassociate the algebraic computation of auxil-iary variables and function values, the discrete Laplaciancan be efficiently inverted by using the fast Fourier trans-form (FFT). Our numerical experiments shows that theiteration number in solving the auxiliary system weaklydepends on the mesh size. As a consequence, the totalcomputational cost of the augmented MIB is about O(n2)

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for a Cartesian grid with dimension n × n in 2D. There-fore, the augmented MIB outperforms the classical MIBin all cases by significantly reducing the CPU time, whilekeeping the same second order of accuracy in dealing withcomplicated interfaces. Moreover, our experiments indi-cate that the AMIB scheme can produce a second order ofaccuracy in approximating the solution gradients.

A fourth order accurate bound-preserving compact fi-nite difference scheme for scalar convection diffusionequationsHao Li, Purdue UniversityAbstract: We demonstrate that the classical fourth or-der accurate compact finite difference scheme satisfiesa natural weak monotonicity property for solving scalarconvection-diffusion equations. Based on such a prop-erty, we design a simple limiter to enforce the bound-preserving or positivity-preserving property of numericalsolutions without losing conservation or high order ac-curacy. Higher order schemes including 6th and 8th or-der schemes satisfying the weak monotonicity can also beconstructed. We show that the bound-preserving propertyis still valid when a total-variation-bounded (TVB) lim-iter is used to reduce oscillations. All these results can beeasily extended to higher dimensions and passive convec-tion such as incompressible flows. More general bound-ary conditions such as the inflow-outflow boundary con-dition will also be discussed. This is a joint work withProf. Shusen Xie at Ocean University of China and Prof.Xiangxiong Zhang at Purdue University.

The Application of Immersed Finite Element Methodson Elasticity Interface ProblemsRuchi Guo, Virginia TechAbstract: In this talk, we present a group of immersedfinite element (IFE) spaces for solving planar elasticityequation involving interface. The shape functions in theseIFE spaces are constructed through some specific approx-imate jump conditions such that the unisolvence of thebilinear and rotated Q1 IFE shape functions are alwaysguaranteed regardless of the Lame parameters and the in-terface location. A group of nice properties for these IFEshape functions are established. The optimal approxima-tion capabilities of these IFE spaces are analyzed thor-ough the multiple point Taylor expansion technique. TheIFE spaces can be used in the partially penalized IFE(PPIFE) method to solve the elasticity interface problemswith the optimal convergence rate.

A mixed discontinuous Galerkin method without in-terior penalty for time-dependent fourth order prob-lemsPeimeng Yin, Iowa State UniversityAbstract: A novel discontinuous Galerkin (DG) methodis developed to solve time-dependent bi-harmonic type

equations involving fourth derivatives in one and multi-ple space dimensions. We present the spatial DG dis-cretization based on a mixed formulation and central in-terface numerical fluxes so that the resulting semi-discreteschemes are L2 stable even without interior penalty. Fortime discretization, we use Crank-Nicolson so that the re-sulting scheme is unconditionally stable and second or-der in time. We present the optimal L2 error estimate ofO(hk+1) for polynomials of degree k for semi-discreteDG schemes, and the L2 error of O(hk+1 + (∆t)2) forfully discrete DG schemes. Extensions to more generalfourth order partial differential equations and cases withnon-homogeneous boundary conditions are provided. Nu-merical results are presented to verify the stability and ac-curacy of the schemes. Finally, an application to the one-dimensional Swift-Hohenberg equation endowed with adecay free energy is presented.

MS07. Applied and Computational TopologyOrganizer: Henry Adams, henry.adams@

colostate.edu, Colorado State UniversityCo-organizers: Mehmet Aktas, maktas@uco.edu,University of Central OklahomaWenwen Li, wli11@ou.edu, University of OklahomaMurad Ozaydin , mozaydin@math.ou.edu, Universityof Oklahoma

Schedule: PHSC 222 (for all four sessions)

Session I Sat, Oct. 6, 10:40am - 12:00pm

Persistence-based summaries of metric graphsEllen Gasparovic, Union CollegeAbstract: This talk will focus on topological summaryinformation that one can capture from metric wedge sums,gluings, and graphs. We will give a complete charac-terization of the persistence diagrams in dimension 1 formetric graphs under a particular intrinsic setting. We willlook at two persistence-based distances that one may de-fine for metric graphs and discuss progress toward estab-lishing their discriminative capacities. We will show thatthe Vietoris-Rips (resp., Cech) complex of a wedge sum,equipped with a natural metric, is homotopy equivalent tothe wedge sum of the Vietoris-Rips (resp., Cech) com-plexes. We also provide generalizations for when twometric spaces are glued together along a common iso-metric subset. As a result, we can describe the persistenthomology, in all homological dimensions, of the Vietoris-Rips complexes of a wide class of metric graphs. This talkcovers joint work with Michal Adamazsek, Henry Adams,Maria Gommel, Emilie Purvine, Radmila Sazdanovic, BeiWang, Yusu Wang, and Lori Ziegelmeier.

Automated Learning of Topological Features: Predict-ing Protein Stability

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Francis Motta, Florida Atlantic UniversityAbstract: Considerable effort has been spent develop-ing tools that are designed to strengthen the applicabilityof computational statistics and machine learning to persis-tent homology. For example, several methods that trans-form persistence diagrams into vectors, which aim to mapthe algebraic content of a persistence diagram to featurevectors to be subsequently used with machine learningparadigms, have been proposed. Often these transforma-tions require choices, which may be regarded as modelhyperparameters to be tuned to a given problem or deter-mined by prior subject-matter expertise. We show howto use the Cover-tree Differencing via Entropy Reduc-tion algorithm (CDER)—a general tool for performingsupervised learning on labeled point clouds—to automat-ically learn discriminating and parsimonious sets of coor-dinates to vectorize persistence diagrams. We apply thistechnique to the problem of predicting synthetic proteinstability and discuss its potential to reveal new biologi-cal insights via interpretation of the learned coordinates.This material is based upon work supported by the De-fense Advanced Research Projects Agency (DARPA) andthe Air Force Research Laboratory under Contract No.FA8750-17-C-0054 (and related contracts by SD2 Pub-lication Consortium Members).

Multirank: a combinatorial multiparameter persis-tent homology invariantAshleigh Thomas, Duke UniversityAbstract: This talk introduces a novel multiparameterpersistence invariant – the multitrank function – that cap-tures geometric information about how homology classesmove through a filtration, including how classes persist,merge, and branch. Multiranks of persistence modulesare ranks of homomorphisms between finite direct sumsof homology groups. In the case where the source andtarget direct sums have one summand each, multirank isthe usual rank function. Multirank extends the behaviorof the rank function to multiparameter persistence in thatit 1) directly measures how many homology classes per-sist across subposets that are “wider” than intervals and 2)determines the degree and multiplicity at which all birthsand deaths occur. A computationally friendly version ofmultirank, called m-rank, restricts the number of sourceand target direct summands to at most m; it determines allbirths and deaths when m is sufficiently large.

Persistent homology on dynamically changing func-tional brain networkMoo K. Chung, University of Wisconsin-MadisonAbstract: Advances in functional magnetic resonanceimaging (fMRI) techniques enabled us to measure spon-taneous fluctuations of neural signals in the brain. Manyprevious studies on resting-state fMRI have mainly fo-cused on the topological characterization of static graph

theory features that will not fluctuate over time. In thistalk, we present a simple but very effective data-driven ap-proach to assess the dynamic pattern of resting state func-tional connectivity using persistent homology.Persistenthomology has been successfully applied to various staticbrain networks by building graph filtrations by sequen-tially thresholding edge weights. For dynamic networks,we propose to build graph filtrations over time and edgeweights at the same time. Stable connectivity patternthat behaves like the backbone of networks through fluc-tuating functional connectivity are also identified. Therecently proposed exact combinatorial inference proce-dure for static network was adapted for statistically quan-tifying dynamic brain networks. This talk is based onarXiv:1509.04771.

Session II Sat, Oct. 6, 03:10pm - 04:30pm

Topological modeling of biomolecular dataKelin Xia, Nanyang Technological UniversityAbstract: The understanding of biomolecular structure,flexibility, function, and dynamics is one of the most chal-lenging tasks in biological sciences. In this talk, I will in-troduce molecular topological fingerprints (MTFs), whichare derived from the persistent homology analysis andprovide a unique representation of the biomolecular struc-ture. Further, I will talk about a special multidimensionalpersistent homology and its application in protein foldinganalysis. The simulation of the unfolding process is doneby using steered molecular dynamics. An excellent con-sistence between my persistent homology prediction andfolding energy is found. More importantly, topologicalquantities from this unique topological representation canbe used as features in machine learning models and havebeen proved to be highly efficient, particularly in drug de-sign. Finally, I will discuss a weighted persistent homol-ogy model which can incorporate the weight informationof simplicial complexes into the filtration process.

A Topological Approach to Spatio-temporal PatternFormationMelissa McGuirl, Brown UniversityAbstract: Spatio-temporal pattern formation appears ina range of natural applications, from Rayleigh–Benardconvection to vegetation patches. Many dynamical sys-tem models have been created and analyzed to better un-derstand the evolution of these patterns. In this talk weapply methods from topological data analysis to analyzean agent-based model for pattern formation on zebrafishand the Rossler system for the study of spiral wave pat-terns. In particular, we will use topological propertiesof the spatio-temporal patterns for both classification andstability analyses.

Classifying fingerprints with persistent homology

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Noah Giansiracusa, Swarthmore CollegeAbstract: Classifying fingerprints (dividing them intoloops, whorls, arches, etc.) is a natural problem to applytopological methods to since these classes are indepen-dent of choice of coordinates. I’ll discuss joint work withChul Moon where we apply persistent homology and per-form machine learning feature selection in several differ-ent ways and compare the results.

Learning non-positively curved cubical complexesDan Guralnik, University of PennsylvaniaAbstract: I will show how Sageev-Roller duality offinite CAT(0) cubical complexes (AKA ”cubings”) maybe exploited for learning and maintaining a dynamicallyevolving, modality-agnostic internal representation of anagent’s interactions with its environment. The emphasisof this representation method is on computational effi-ciency, to be achieved by distributing the tasks of repre-sentation and control among multiple binary agents, eachdeciding whether to act or not to act, at any given time,according to a private prediction extracted from an inter-nal representation which takes the form of the dual poc setof a cubing. I will present some results on learning suchrepresentations, relate them to the homotopy type of thesystem’s configuration/state space, together with simula-tions of such agents ”at work”. This is a joint work withDaniel Koditschek.

Session III Sat, Oct. 6, 05:00pm - 06:20pm

On Kernels and Cokernels in Persistent HomologyJan Segert, University of MissouriAbstract: In a recent paper (arXiv:1701.02055[math.AT]) we introduced an alternative categoricalframework for persistent homology, and studied barcodedecompositions in the alternate framework. In this talkI will consider kernels and cokernels of a morphism inthe alternate categorical framework, explained with sim-ple concrete examples.

Learning Simplicial Complexes from Persistence Dia-gramsRobin Belton, Montana State UniversityAbstract: Topological Data Analysis (TDA) studies the”shape” of data. A common topological descriptor is thepersistence diagram, which encodes topological featuresin a topological space at different scales. Turner, Mukher-jee, and Boyer showed that one can reconstruct a simpli-cial complex embedded in R3 using persistence diagramsgenerated from all height filtrations (an uncountably in-finite number of directions). In this talk, we present analgorithm to reconstruct plane graphs, K = (V,E) inR2, i.e., a planar graph with vertices in general positionand a straight line embedding, from a quadratic numberof height filtrations and their respective persistence dia-

grams. If time allows, we will also discuss current workon how one can use collections of persistence diagramsgenerated from height filtrations for shape alignment.

The importance of forgetting: Limiting memory im-proves recovery of topological characteristics fromneural dataSamir Chowdhury, The Ohio State UniversityAbstract: Place cells in the hippocampus are known fortheir importance in spatial learning. Previously, Curtoand Itskov showed that knowledge of which groups ofplace cells cofire in a rodent hippocampus is sufficientto recover the topology of the rodent’s physical environ-ment. The topological method here comprises the fol-lowing steps: represent the place cells as vertices of anabstract simplicial complex, insert higher simplices cor-responding to cells that fire together, and then computethe homology of the resulting complex. This follows theclassical Hebbian “fire together, wire together” learningmodel. Prior studies have used models where only synap-tic potentiation (“wire together”) events are considered.However, depotentiation (“unwiring” or “forgetting”) is akey complementary event that has recently been proposedas an important component of spatial learning. We ex-plore this possibility by first simulating a model of rodentplace cell activity that incorporates depotentiation accord-ing to a memory parameter, and then computing the ac-curacy with which the place cell representations can tellapart different environments as a function of memory. Weobtain the surprising conclusion that some amount of for-getting is beneficial for the learning process—likely byerasing “errors” that occur when learning place cell repre-sentations.

Vietoris-Rips thickenings of the circle and centrally-symmetric orbitopesJohnathan Bush, Colorado State UniversityAbstract: Given a metric space X and scale parameterr > 0, the Vietoris-Rips simplicial complex V R(X; r)has as its simplices all finite subsets of X of diameterless than r. A result of Jean-Claude Hausmann statesthat the homotopy type ofX often may be recovered fromV R(X; r) for sufficiently small r. However, much less isknown about the topological behavior of these complexesfor large values of r, despite the fact that they arise nat-urally in applications of persistent homology. One resultalong these lines, due to Adams and Adamaszek, providesthe homotopy type of the Vietoris-Rips complex of thecircle for all r. On the other hand, a simplicial complexV R(X; r) does not preserve metric information about X ,and this deficiency motivates the consideration of a sepa-rate but related construction, called the Vietoris-Rips met-ric thickening. We will discuss the homotopy type of theVietoris-Rips thickening of the circle for a range of scaleparameters. Our primary tools will be an embedding of

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the Vietoris-Rips thickening into Euclidean space via asymmetric moment curve, together with the facial struc-ture of the Barvinok-Novik orbitope.

Session IV Sun, Oct. 7, 10:30am - 11:50am

On homology reconstruction of geodesic subspacesRafal Komendarczyk, Tulane UniversityAbstract: In this work in progress, we develop a persis-tence based algorithm for the homology/homotopy groupsreconstruction of the unknown underlying geodesic sub-space of Rn from a point cloud. In the case of a metricgraph, we can output a subspace which is an arbitrarilygood approximation of the underlying graph.

Enhancing witness complexes modeling chaotic dy-namical systemsNicole Sanderson, University of Colorado BoulderAbstract: Topological data analysts often model pointcloud data with a simplicial complex and compute the per-sistent homology as a topological signature. When pointcloud data comes from a dynamical system, the objectwe seek to model topologically is inherently not stagnant.Properties of the dynamics can inform us about the un-derlying topology of the attractor of the dynamical sys-tem that may be obscured by the geometry of the pointcloud. Encoding the local dynamics as unit tangent vec-tor approximations at each data point and penalizing thedistance between points in the reconstructed state spacetraveling in dissimilar directions, we can both cheapenthe construction of filtrations of witness complexes andenhance the topological signature of the attractor as re-ported by the persistent homology. We demonstrate thisthrough a set of new topological statistics on the persistenthomology of delay reconstructions of chaotic dynamicalsystems.

Restricted Configuration Spaces of a Metric GraphJames Dover, Cameron UniversityAbstract: We consider the space of configurations ofn ordered points on a finite, connected metric graph witha vector-valued restraint parameter r dictating the mini-mum distance permitted between each pair of points. Thisrestricted configuration space arises naturally in topologi-cal robotics when the n points represent robots of varyingsizes. In this talk, we discuss the homeomorphism and ho-motopy types of this space, which are indexed by the facesof a hyperplane arrangement in the space of parameters r.

Thin position motivated weight regularization for deepneural networksDoug Heisterkamp, Oklahoma State UniversityAbstract: This talk presents an initial investigation ofa topologically motivated weight regularization for deepneural networks. The topological approach uses the al-gorithms of topologically intrinsic lexicographic ordering

(TILO), pinch ratio clustering (PRC), and thin tree posi-tion (TTP) which are inspired by thin position for knotsand 3-dimensional manifolds. Artificial neural networksconduct computations distributed over the nodes of thenetwork and this computation is typically opaque (hardto interpret). Current weight regularization is typicallyapplied uniformly over the weights of the network. Theapproach of this talk is to use topological information tocreate a relative importance scores for weights and nodesthat are then used for non-uniform regularization. Thelong term goal of this approach is to bias the distributedcomputation into functional submodules which are inter-pretable and reusable.

MS08. Recent Advances of Numerical Meth-ods in Fluid Mechanics with Applications

Organizer: Jia Zhao, jia.zhao@usu.edu, Utah StateUniversityCo-organizer: Daozhi Han, handaoz@mst.edu, Mis-souri University of Science and Technology

Schedule: PHSC 323 (for all four sessions)

Session I Sat, Oct. 6, 10:40am - 12:00pm

Partitioning algorithms for the system of a fluid and aporoelastic structureHyesuk Lee, Clemson UniversityAbstract: Computational algorithms for the generalizedStokes-Biot coupled system are proposed for the inter-action of a free fluid with a poroelastic structure. Thetalk is focused on decoupling schemes that allow the non-Newtonian fluid and the poroelastic structure computedindependently using a common stress force along the in-terface. This approach is based on a nonlinear operatorequation, where the operator measures violation of someinterface conditions. Numerical algorithms based on aNewton-type updating technique are discussed and nu-merical results are provided to validate the accuracy andefficiency of the proposed algorithms.

Well-Balanced Positivity Preserving Central-UpwindScheme with a Novel Wet/Dry Reconstruction on Tri-angular Grids for the Saint-Yekaterina Epshteyn, The University of UtahAbstract: In this talk we will introduce and discusssecond-order central-upwind scheme for the Saint-Venantsystem of shallow water equations on triangular grids.The Saint-Venant system is widely used in many scien-tific and engineering applications related to modeling ofwater flows in rivers, lakes and coastal areas. The de-velopment of robust and accurate numerical methods forthe simulation of shallow water equations is importantand challenging problem. In our talk we will show that

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the designed central-upwind scheme preserves ”lake atrest” steady states, guarantees the positivity of the com-puted fluid depth and delivers accurate reconstructionof “dry”/“almost dry states” (waves arriving or leavingthe shore). Moreover, it can be applied to models withdiscontinuous bottom topography and irregular channelwidths. We will demonstrate these features of the pro-posed scheme, as well as its high resolution and robust-ness in a number of numerical examples.

Modeling and computation of an elastic tumor-host in-terfaceShuwang Li, Illinois Institute of TechnologyAbstract: We consider the nonlinear dynamics of anavascular tumor at the tissue scale using a two-fluid flowStokes model, where the viscosity of the tumor and hostmicro-environment may be different. In an effort to under-stand the role the interface stiffness plays on tumor evolu-tion, here we model the tumor-host interface as an elasticmembrane governed by the Helfrich bending energy. Us-ing an energy variation approach, we derive a modifiedLaplace-Young condition for the stress jump across theinterface in the Stokes equation. We then perform a lin-ear morphological stability analysis of the tumors, and in-vestigate the role of nonlinearity using boundary-integralsimulations in two dimensions. Simulation results areconsistent with linear theory for nearly circular tumors.As perturbations develop and grow, preliminary nonlinearresults show that the tumors tend to develop invasive fin-gers and a branched- like structure, if the tumor is moreviscous than its environment.

A Second Order Fully-discrete Linear Energy StableNumerical Scheme of a Binary Compressible ViscousFluid ModelXueping Zhao,University of South CarolinaAbstract: We derive a thermodynamically consistenthydrodynamic phase field model of binary compressiblefluid flow mixtures using the generalized Onsager Princi-ple, warranting not only the variational structure, but alsothe mass, linear momentum conservation, and the energydissipation law in an isothermal case. To solve this math-ematical model, we present a linear, second order fullydiscrete numerical scheme on a staggered grid. We provethe unique solvability of the linear scheme rigorously. Fi-nally, we present several numerical examples, includingphase separation due to the spinodal decomposition of twopolymeric fluids and the gas-liquid mixture, to show theconvergence property, stability, and efficiency of the newscheme.

Session II Sat, Oct. 6, 03:10pm - 04:30pm

Optimal rate convergence analysis for Cahn-Hilliardequation coupled with fluid flow

Cheng Wang, University of Massachusetts DartmouthAbstract: A few fully discrete numerical schemes forthe Cahn-Hilliard-Hele-Shaw and Cahnn-Hilliard-stokesequations, a modified Cahn-Hilliard model coupled withwith the the Darcy or Stokes flow law, are analyzed indetails. The unique solvability comes from the varia-tional calculation, while and unconditional energy stabil-ity comes from the energy estimate. In turn, a uniformin time H1 bound for the numerical solution becomesavailable. Moreover, with the help of discrete Gagliardo-Nirenberg type inequality, derived in terms of discreteFourier analysis, we are able to obtain the L2(0, T ;H3)stability of the numerical solution. Subsequently, weperform a discrete L∞(0, T ;H1) and L2(0, T ;H3) errorestimate, instead of the L∞(0, T ;L2) and L2(0, T ;H2)one, which represents the typical analysis. Such an ap-proach allows us to treat the nonlinear convection term ina positive way, since one essential nonlinear inner productturns out to be non-negative. Some numerical simulationresults are also presented in the talk.

Stabilized-IEQ and Stabilized SAV method for gradi-ent flow models with strong anisotropyXiaofeng Yang, University of South CarolinaAbstract: We consider numerical approximations forgradient flow models with strong anisotropy by takingthe anisotropic Cahn-Hilliard/Allen-Cahn equations withtheir applications to the faceted pyramids on nano scalecrystal surfaces and the dendritic crystal growth problems,as special examples. The main challenge of constructingnumerical schemes with unconditional energy stabilitiesfor these type of models is how to design proper tempo-ral discretizations for the nonlinear terms with the stronganisotropy. We combine the recently developed IEQ/SAVapproach with the linear stabilization approach, wheresome linear stabilization terms are added. These termsare shown to be crucial to remove the oscillations causedby the anisotropic coefficients, numerically. The noveltyof the proposed schemes is that all nonlinear terms can betreated semi-explicitly, and one only needs to solve somecoupled/decoupled, but linear equations at each time step.We further prove the unconditional energy stabilities rig-orously, and present various 2D and 3D numerical simu-lations to demonstrate the stability and accuracy.

A dual-porosity-Stokes model and finite elementmethod for coupling dual-porosity flow and free flowXiaoming He, Missouri University of Science and Tech-nologyAbstract: We propose and numerically solve a newmodel considering confined flow in dual-porosity me-dia coupled with free flow in embedded macro-fracturesand conduits. Such situation arises, for example, forfluid flows in hydraulic fractured tight/shale oil/gas reser-voirs. The flow in dual-porosity media, which consists of

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both matrix and micro-fractures, is described by a dual-porosity model. And the flow in the macro-fractures andconduits is governed by the Stokes equation. Then thetwo models are coupled through four physically valid in-terface conditions on the interface between dual-porositymedia and macro-fractures/conduits, which play a keyrole in a physically faithful simulation with high accu-racy. All the four interface conditions are constructedbased on fundamental properties of the traditional dual-porosity model and the well-known Stokes-Darcy model.The weak formulation is derived for the proposed modeland the well-posedness of the model is analyzed. A finiteelement semi-discretization in space is presented basedon the weak formulation and four different schemes arethen utilized for the full discretization. The convergenceof the full discretization with backward Euler scheme isanalyzed. Four numerical experiments are presented tovalidate the proposed model and demonstrate the featuresof both the model and numerical method, such as the opti-mal convergence rate of the numerical solution, the detailflow characteristics around macro-fractures and conduits,and the applicability to the real world problems.

Session III Sat, Oct. 6, 05:00pm - 06:20pm

Efficient algorithms for simulating ensembles of pa-rameterized flow problemsZhu Wang, University of South CarolinaAbstract: Many computational fluid dynamics applica-tions require multiple simulations of a flow under differ-ent input conditions. In this talk, we consider such set-tings for which one needs to perform a sequence of simu-lations based on the Navier-Stokes equations, each havingdifferent initial condition data, boundary condition data,forcing functions, and/or coefficients such as the viscos-ity. For such settings, we propose ensemble methods toaccelerate the solutions. The main idea is to manipulatethe time-stepping scheme so that all the problems couldshare a common coefficient matrix, then, instead of solv-ing a sequence of linear systems with one right-hand-sidevector, the method needs to solve one linear system withmultiple right-hand-sides. The computational efficiencyis then improved by using block iterative algorithms. Rig-orous analyses are given proving the conditional stabil-ity and establishing error estimates for the proposed algo-rithms. Numerical experiments are presented to illustratethe analyses.

Eddy viscosity models for turbulence not at statisticalequilibriumNan Jiang, Missouri University of Science and Technol-ogyAbstract: Standard eddy viscosity models, while ro-bust, cannot represent backscatter and have severe diffi-culties with complex turbulence not at statistical equilib-

rium. In this talk, we give a new derivation of eddy vis-cosity models from an equation for the evolution of vari-ance in a turbulent flow. The new derivation also showshow to correct eddy viscosity models. We prove that thecorrected models preserve important features of the trueReynolds stresses and give algorithms for their discretiza-tion including a minimally invasive modular step to adaptan eddy viscosity code to the extended models.

Conservative Explicit Local Time-Stepping Schemesfor the Shallow Water EquationsThi-Thao-Phuong Hoang, Auburn UniversityAbstract: In this talk, we present explicit local time-stepping schemes of second and third order accuracy forthe shallow water equations. The system is discretizedin space by a C-grid staggering method, namely theTRiSK scheme adopted in MPAS-Ocean, a global oceanmodel with the capability of resolving multiple resolu-tions within a single simulation. The time integration isdesigned based on the strong stability preserving Runge-Kutta methods but with different time step sizes in dif-ferent regions of the domain restricted by respective lo-cal CFL conditions. The proposed local time-steppingschemes preserve all important properties in the discretesense, such as exact conservation of the mass and potentialvorticity and conservation of the total energy within time-truncation errors. Extensive numerical tests are presentedto illustrate the performance of the proposed algorithms.

Incremental POD mode algorithm for fluids with acomputable error boundJohn Singler, Missouri University of Science and Tech-nologyAbstract: We discuss an incremental algorithm forproper orthogonal decomposition (POD) computations.Specifically, we develop an incremental matrix SVD al-gorithm with respect to a weighted inner product for PODcomputations of data arising from Galerkin-type simula-tion methods for time dependent PDEs. The algorithm ini-tializes and efficiently updates the POD eigenvalues andmodes during the time stepping in a PDE solver withoutstoring the simulation data. Also, the algorithm returnsan easily computed error bound. We demonstrate the ef-fectiveness of the algorithm using finite element compu-tations for fluid flows.

Session IV Sun, Oct. 7, 10:30am - 12:10pm

Fully Discrete Second-order Linear Schemes for Hy-drodynamic Phase Field Models of Binary ViscousFluid Flows with Variable DensityJia Zhao, Utah State UniversityAbstract: In this talk, we present spatial-temporallysecond-order, energy stable numerical schemes for twoclasses of hydrodynamic phase field models of binary vis-

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cous fluid mixtures of different densities and viscosities.One is quasi-incompressible while the other is incom-pressible. We introduce a novel invariant energy qudra-tization (IEQ) technique to arrive at fully discrete linearschemes, where in each time step only a linear systemneeds to be solved. These schemes are then proved to beunconditionally energy stable rigorously so that a largetime step is plausible. Both spatial and temporal mesh re-finements are conducted to illustrate the second- order ac-curacy of the schemes. Several Numerical examples andconducted, and predictions by the two fluid-mixture mod-els are compared and discussed. As a conclusion, we be-lieve the quasi-incompressible model is more reliable thanthe incompressible one.

An HDG Method for Tangential Boundary Control ofStokes EquationsYangwen Zhang, University of DelawareAbstract: We propose a hybridizable discontinuousGalerkin(HDG) method to approximate the solution of atangential Dirichlet boundary control problem governedby Stokes equation. Dirichlet boundary control problemsare naturally studied in the L2-setting, and therefore well-known to be challenging due to its non variational formand low regularity, even for simpler elliptic PDEs. Al-though there are many works in the literature on Dirichletboundary control problems for fluids flow, the authors arenot aware of any existing theoretical or numerical analysisfor Dirichlet boundary control of Stokes or Navier-Stokesequation in such setting. To avoid the extreme low regu-larity of solutions for Stokes Dirichlet boundary control,this work gives a first attempt to the analysis of tangen-tial boundary control of fluid flows with controls in L2.The contribution of this paper is twofold. First, we ob-tain well-posedness and regularity results for the tangen-tial Dirichlet control problem. Second, under certain as-sumptions on the domain and the target state, we obtainoptimal a priori error estimates for the control in 2D forthe HDG method. We present numerical experiments todemonstrate the performance of the HDG method.

MS09. Modeling and Analysis in Ecology andEpidemiology

Organizer: Matthew Beauregard, beauregama@

sfasu.edu, Stephen F. Austin State University

Schedule: PHSC 115 (for all three sessions)

Session II Sat, Oct. 6, 03:10pm - 04:30pm

Analysis of Modified Trojan Y-Chromosome Model forEradicating Nonnative FishSarah Boon, Stephen F Austin State UniversityHarley Conaway, Stephen F Austin State University

Thomas Griffin, Stephen F Austin State UniversityAbstract: The Trojan Y-Chromosome (TYC) strategy isa proposed method to eradicate an invasive species. Thestrategy consists of introducing sex-reversed males con-taining two Y chromosomes into a habitat to skew the sexratio of subsequent generations toward an increasing num-ber of males. A new mathematical model that incorpo-rates both a strong Allee effect and intraspecies competi-tion between supermales and wild-type males for femalesmates is provided. The efficacy of the strategy is exam-ined through the mathematical model. The influence ofthe frequency and amplitude of introduction on the strat-egy effectiveness will also be discussed. This is joint talkwith Sarah Boon, Harley Conaway, and Thomas Griffin

A comparison of the TYC Strategy to pure harvestingJingjing Lyu, Clarkson UniversityAbstract: The Trojan Y Chromosome Strategy (TYC)is a promising method for biological control of invasivespecies that works on manipulating the sex ratio of aninvasive population towards all males. In the currentmanuscript we compare the classical TYC strategy to apure harvesting strategy, as well as explore hybrid meth-ods that combine the pure strategy with harvesting meth-ods.The dynamical analysis leads to results on stability,global boundedness and bifurcations of the model. Next,via optimal control methods we draw several conclusionsabout the different methods. Our results have key implica-tions for preserving native species under invasion by non-natives. In particular they we affirm that a pure harvestingstrategy or a hybrid method works better than the classicalTYC method.

The effect of additional food on introduced predatorsfor biological controlRana Parshad, Iowa State UniversityAbstract: Biological control, the use of predators andpathogens to control target pests, is a promising alterna-tive to chemical control. It is hypothesized that the preda-tors efficacy can be boosted by providing it with an ad-ditional food source. In the current literature it is provedthat if the additional food is of sufficient constant quantityand quality then pest eradication is possible in finite time.We show to the contrary that pest eradication is not pos-sible in finite time, and will occur only in infinite time towhich end we derive decay rates. However for a densitydependent quantity of additional food, we show pest erad-ication in finite time is indeed possible. We also considerthe case of both predator and prey evolution taking place,and its effect on the dynamics of the system. Our resultshave many consequences for designing effective biologi-cal control strategies.

The ”destabilizing” effect of cannibalism in a spa-tially explicit three-species age structured predator-

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prey modelAladeen Al Basheer, The University of GeorgiaAbstract: Cannibalism, which is the act of killing andconsumption of conspecifics, has been considered primar-ily in the predator, despite strong ecological evidence thatit exists among prey. In the current manuscript, we in-vestigate both the ODE and spatially explicit forms of aHolling–Tanner model, with ratio dependent functionalresponse, and show that cannibalism in the predator pro-vides a stabilizing influence as expected. However, whencannibalism in the prey is considered, we show that it can-not stabilize the unstable interior equilibrium in the ODEcase, in certain parameter regime, but can destabilize thestable interior equilibrium, leading to a stable limit cycleor ”life boat” mechanism, for prey. We also show that preycannibalism can lead to pattern forming Turing dynamics,which is an impossibility without it.

Session III Sat, Oct. 6, 05:00pm - 06:20pm

Landscape heterogeneityPatrick Shipman, Colorado State UniversityAbstract: Landscape heterogeneity and long-distancetransportation connections can be factors in the spreadof an infectious disease or an invasive species. We ana-lyze mathematical models including these factors for thespread of the invasive cheatgrass and for the spread ofwhite-nose syndrome in bat populations.

Flapping Flight of Ecologically Important Tiny In-sectsArvind Santhanakrishnan, Oklahoma State UniversityAbstract: Miniature flying insects with body lengthsless than 1 mm, such as thrips and some parasitoid wasps,serve important ecological roles that include: transmittingpollen during feeding, invasive pests of agriculturally im-portant plants, and biological vectors of microbial plantpathogens. However, the aerodynamics of flapping flightof these insects has been relatively unexplored. Under-standing single-organism level aerodynamics is neededprior to the development of mathematical models of theircollective dispersal. The flapping wing kinematics inthrips is characterized by wing-wing interaction (clap andfling), such that the wings clap at the end of the upstrokeand fling apart at the beginning of the downstroke. Theseinsects also show unique adaptations in their wing design,consisting of a thin solid membrane with long bristles atthe fringes. Given the challenges of acquiring real-timerecordings of freely flying insects at length scales of un-der a millimeter while resolving fast wing beats on the or-der of 100 Hz, we use experiments on dynamically scaledphysical models and 2D numerical simulations for inves-tigation. In this talk, I will discuss the importance of bris-tled wing morphology and wing-wing interaction in theaerodynamics of tiny insect flight.

The Spatio-Temporal Spread of a Vector Host Diseaseover a Large DomainWilliam Fitzgibbon, University of HoustonAbstract: We are concerned with a class of modelsdescribing the spatio-temporal spread of vector host dis-ease over a large domain. Previous work has consideredthe case where the host and vector population inhabit thesame domain or the case of vectors inhabiting subdomainsof a larger domain inhabited by the host. Here we shallconsider the case of the host occupying subdomains of alarger domain. The work contained herein is further dif-ferentiated from previous work by virtue of the fact wedefine the vector habitat as all of R2. Of particular inter-est is the spread of the disease over a large geographic dis-ease free region by virtue of the introduction of infectedhosts into a relatively small subregion. Our work is mo-tivated by Blue Tongue Disease. Blue Tongue disease isa non-contagious viral disease transmitted via bites of themidges of the genus culicoides to a variety of both do-mestic and wild ruminants including cattle, deer, goats,dromedaries, and antelopes.

Optimal Control Strategies for Dengue Transmissionin PakistanFolashade Agusto, University of KansasAbstract: This paper presents a deterministic model fordengue virus transmission. The model was parameter-ized using data from the 2017 dengue outbreak in Pak-istan. We estimated the basic reproduction number (R0)without any interventions for the 2017 dengue outbreakin Peshawar district of Pakistan as R0 ≈ 2.64, the dis-tribution of the reproduction number lies in the rangeR0 ∈ [1.21, 5.24] (with a mean R0 ≈ 2.64). Optimalcontrol theory was then applied to investigate the optimalstrategy for curtailing the spread of the disease using twotime-dependent control variables determined from sensi-tivity analysis. These control variables are insecticide useand vaccination. The results show that the two controlsavert the same number of infections in the district regard-less of the weights on the costs this is due to the reciprocalrelationship between the cost of insecticide use and vac-cination. A strong reciprocal relationship exists betweenthe use of insecticide and vaccination; as the cost of insec-ticide increases the use of vaccination increases. The useof insecticide on the other hand slightly increases whenvaccination level decreases due to increase in cost.

Session IV Sun, Oct. 7, 10:30am - 11:50am

Tree Harvesting In Age-structured Forests Subject ToBeetle InfestationsBenito Chen-Charpentier, University of Texas at Arling-tonAbstract: In this presentation we investigate a math-ematical model for age-structured forest-beetle interac-

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tions. In the first part of this study, we consider differ-ent scenarios of the forest infestation by the beetle andobserve that the quantitative age profile of the forest issignificantly dependent on whether the beetle populationis in an endemic or epidemic state. In the second partwe include harvesting of the forest trees and analyze twodifferent harvesting strategies: clear cutting trees olderthan a certain age, and using a cut rate proportional tothe number of trees older than a certain age. Numeri-cal simulations are implemented to determine the optimalcutting age for both harvesting strategies. The numericalsimulations reveal that, independent of the beetle popu-lation’s steady state (that is, no beetles, endemic or epi-demic state) clear cutting all trees older than a given ageprovides a higher harvesting benefit. Our numerical sim-ulations further indicate that in order to obtain a fixed har-vesting yield, a forest under an beetle epidemic state haveto be cut at a younger age than if the forest were at anendemic beetle state or a no beetle state.

Numerical Realizations of Nonlinear Population Mod-els with Self and Cross DiffusionMatthew Beauregard, Stephen F. Austin State UniversityAbstract: Self- and cross-diffusion are important and of-ten neglected nonlinear spatial derivative terms that are in-cluded into population models in ecology. Self-diffusionmodels overcrowding effects, while cross-diffusion incor-porates the response of one species in light of the concen-tration of another. The underlying diffusion matrix is nolonger positive definite which complicates the theoreticalanalysis and development of numerical approximations.In this talk, we explore a convergent nonlinear operatorsplitting routine and then compare this to a newly devel-oped positivity-preserving numerical approximation.

MS10. Complex PatternsOrganizer: Patrick Shipman, shipman@math.

colostate.edu, Colorado State University

Schedule: PHSC 115

Session I Sat, Oct. 6, 10:40am - 12:00pm

Pattern Structures in Vapor-to-Particle ReactionsBahaudin Hashmi, Texas State UniversityAbstract: When ammonia and hydrogen chloride reactin a tube, an oscillatory reaction front of the vapor ammo-nium chloride is formed that nucleates to the formationof solid ammonium chloride. After corroborating exper-imental data with our threshold function, additional sim-ulations were conducted with parametric variations thatdemonstrated pattern structures observed in other similarreactions. We will also demonstrate how the topologicalpositioning of the reactants affects the resulting pattern ofthe solid ammonium chloride.

Effects of anisotropy in pattern forming systemsDerek Handwerk, Colorado State UniversityAbstract: The isotropic complex Ginzburg-Landau(CGL) equation has been derived as an amplitude equa-tion for Poiseuille flows and reaction-diffusion equations.The Kuramoto-Sivashinky (KS) equation can be extractedvia a multiple-scales expansion from the CGL as a phaseequation. Both of these equations have been well stud-ied and exhibit wide ranging dynamics including spatio-temporal chaos. The anisotropic versions of these equa-tions, ACGL and AKS, have arisen in the study of elec-troconvection of nematic liquid crystals, surface nanopat-terning by ion-beam erosion, chemical waves in cat-alytic surface reactions, epitaxial growth, and solidifica-tion from a melt. The effects of the added anisotropy willbe discussed.

Spatiotemporal Complex Patterns in Anisotropic Sys-temsIuliana Oprea, Colorado State UniversityAbstract: Unlike low dimensional systems, where dif-ferent scenarios for the transition to chaos have beenestablished, in spatially extended systems the transitionfrom regular stationary patterns to complex spatiotempo-ral regimes is only partially understood. In this presen-tation we identify and characterize the instability mech-anisms generating the spatiotemporal complex patterns,such as spatiotemporal intermittency and chaos, in the nu-merical simulations of a system of four globally coupledcomplex Ginzburg Landau equations. Applications to theelectroconvection of nematic liquid crystals are also pre-sented.

MS11. Partial Differential Equations: Anal-ysis, Modeling, and Applications

Organizer: John Singler, singlerj@mst.edu, Mis-souri University of Science and TechnologyCo-organizer: Weiwei Hu, weiwei.hu@okstate.edu,Oklahoma State University

Schedule: PHSC 316 (for all three sessions)

Session II Sat, Oct. 6, 03:10pm - 04:30pm

Degenerate FCH Functional and Defects in Am-phiphilic StructuresShibin Dai, The University of AlabamaAbstract: We introduce a modified functionalized Cahn-Hilliard (FCH) functional to model the free energy of am-phiphilic mixtures. We prove that the modified FCH func-tional admits geometrically localized minimizers, whichcorrespond to lipid bilayers, filamentous pores, or mi-celles. In addition, we identify the leading order profileof the bilayers under the assumption that the geometri-

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cally localized minimizers have bounded variations alongthe tangential directions. We will also describe how tohandle defects caused by the presence of nanoparticles ormolecules.

Stability of solitary waves for the 1d NLS with a deltapotentialJason Murphy, Missouri University of Science and Tech-nologyAbstract: In this talk we will discuss recent joint workwith S. Masaki (Osaka University) and J. Segata (To-hoku University) on the problem of asymptotic stabilityof small solitary wave solutions to the one-dimensionalnonlinear Schr odinger equation with a delta potential.

Recent Performance Optimizations of Front TrackingBased Simulation PackageTulin Kaman, University of ArkansasAbstract: Performance studies are conducted to iden-tify the optimization potential in the computational fluiddynamics (CFD) code used for efficient numerical simula-tions of compressible turbulent mixing problems on highperformance computing systems. For these studies, one ofthe most powerful supercomputers, Blue Waters, is used.A case study on the use of CrayPat Profiler Toolkit for per-formance analysis and the recent enhancements to fronttracking based CFD code are presented.

Session III Sat, Oct. 6, 05:00pm - 06:20pm

On the long–time behavior of a perturbed conservativesystem with degeneracyWenqing Hu, Missouri University of Science and Tech-nologyAbstract: We consider in this work a model conservativesystem subject to dissipation and Gaussian–type stochas-tic perturbations. The original conservative system pos-sesses a continuous set of steady states, and is thus degen-erate. We characterize the long–time limit of our modelsystem as the perturbation parameter tends to zero. Thedegeneracy in our model system carries features found insome partial differential equations related, for example, toturbulence problems.

Partially dissipative 2D Boussinesq equations withNavier type boundary conditionsBei Xiao, Oklahoma State UniversityAbstract: This talk will focus on the system of the 2DBoussinesq equations with only kinematic dissipation inbounded domains with the Navier type boundary condi-tions. Firstly, because the Navier boundary conditionsgenerate boundary terms and require compatibility con-ditions compared to the whole space and the periodic do-mains, we will provide a direct and transparent approachthat explicitly reveals the impacts of the Navier boundaryconditions. Then, we will present a proof of the global

existence and uniqueness of the aforementioned systemunder minimal regularity assumptions on the initial data.The proof resorts to the existence and regularity results onthe associated Stokes problem with Navier boundary con-ditions and Yudovich techniques with the introduction ofa lower regularity counterpart of the temperature.

Boundary layers for primitive equations in a cubeDaozhi Han, Missouri University of Science and Technol-ogyAbstract: It is well-known that the inviscid primitiveequations (IPE) are ill-posed for any local boundary con-ditions. We propose a set of nonlocal boundary conditionsfor the IPE in a cube by investigating the boundary layersassociated with the viscous primitive equations. We es-tablish the convergence in energy norms.

An Analysis of Parameter Sensitivity in ContinuousData Assimilation of Turbulent FlowElizabeth Carlson, University of Nebraska-LincolnAbstract: One of the challenges of the accurate simula-tion of turbulent flows is that initial data is often incom-plete. Data assimilation circumvents this issue by contin-ually incorporating the observed data into the model. Re-cently a new approach to data assimilation known as theAzouani-Olson-Titi (AOT) algorithm introduced a feed-back control term to the 2D incompressible Navier-Stokesequations in order to incorporate sparse measurements. Itwas proven that the solution to the AOT algorithm for con-tinuous data assimilation converges exponentially to thetrue solution of 2D incompressible Navier-Stokes equa-tions with respect to the given initial data. In this talkwe examine the parameter sensitivity of viscosity. Specif-ically, we analyze how perturbations of viscosity affectthe convergence of the AOT algorithm to the 2D incom-pressible Navier-Stokes equations. Analytical and com-putational results will be presented.

Session IV Sun, Oct. 7, 10:30am - 11:50am

On the velocity-vorticity-Voigt formulation of the 3DNavier-Stokes equationsYuan Pei, Western Washington UniversityAbstract: In this talk, we propose a new regularizationof the 3D Navier-Stokes equations, which we call the 3Dvelocity-vorticity-Voigt (VVV) model, with a Voigt regu-larization term added to momentum equation in velocity-vorticity form, but with no regularizing term in the vor-ticity equation. We prove global well-posedness and reg-ularity of this model along with an energy identity. Wealso show convergence of the model’s velocity and vor-ticity to their counterparts in the 3D Navier-Stokes equa-tions as the Voigt modeling parameter tends to zero. Fur-ther, we provide a criterion for finite-time blow-up of the3D Navier-Stokes equations based on this inviscid regu-

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larization. This is joint work with Adam Larios and LeoRebholz.

A surface moving mesh method based on the equidis-tribution and alignment

Avary Kolasinski, University of Kansas

Abstract: We will propose an approach for mesh adap-tation on surfaces that, to our best knowledge, is the onlysurface moving mesh method that can be directly appliedto a general surface. We will formulate a surface func-tional based on equidistribution and alignment conditionsand employ an MMPDE to find its minimizer. We willthen discuss various theoretical properties, which enableus to prove mesh nonsingularity. Finally, we will presentnumerical results in two and three-dimensions.

The self-similar very singular solution for nonlineardiffusion equations with gradient absorption

Hailong Ye, Shenzhen University

Abstract: In this talk, I will present the uniqueness ofself-similar very singular solutions with compact supportfor the following diffusion equation with gradient absorp-

tion∂u

∂t= div

(|∇um|p−2∇um

)− |Ou|q , x ∈ RN , t >

0, where m > 0, p > 1, m(p− 1) > 1 and q > 1.

Effects of Permanent Charge and Boundary Conditionon Ionic Flow via a Quasi-1D Poisson-Nernst-PlanckModel

Yufei Yu, University of Kansas

Abstract: Ionic channels – large proteins on cell mem-brane – are a major way for ions to transport through cellmembrane that carries electric signals for cells to commu-nicate with each other. The permanent charge of an ionchannel is the crucial structure for ionic flow propertiesof the channel. The effects of permanent charges inter-acting with boundary conditions have been studied ana-lytically via the Quasi-1D Poisson-Nernst-Planck (PNP)model for small permanent charge and for large perma-nent charge. In this talk, we will present results of nu-merical investigation to bridge between the two extrema.As expected, our numerical results verify the analyticalpredictions for small and large permanent charges. Onthe other hand, non-trivial behavior emerges as one variesthe permanent charge from small to large, in particular,bifurcations are revealed, showing the rich phenomenaof permanent charge effects by the power of combiningthe analytical and numerical studies. An adaptive mov-ing mesh finite element method has been applied whichis critical due to the presence of Debye layers at the inter-face between the permanent charge regions and unchargedregions of ion channels.

MS12. Novel Mathematical Methods withApplications to Continuum Mechanics

Organizer: Anna Zemlyanova,, azem@ksu.edu,Kansas State UniversityCo-organizer: Thomas DeLillo, delillo@math.

wichita.edu, Wichita State University

Schedule: PHSC 114 (for both sessions)

Session III Sat, Oct. 6, 05:00pm - 06:20pm

Potential flow in a multiply connected circle domainusing series methodsJustin Mears, Wichita State UniversityAbstract: We study series solutions to potential flowover an assembly of disks in the complex plane. Bound-ary value problems are solved for both streaming flow uni-form at infinity and flow with circulation using Laurent se-ries. The circles are the images of multi-element airfoilsunder conformal mapping and the circulations around thedisks can be found so that the combined flow satisfiesthe Kutta condition at the preimages of the trailing edges.Some comparisons with other methods for computing theflow, such as reflection methods, will be given.

Discrete-to-Continuum Modeling of Weakly Interact-ing Incommensurate ChainsMalena Espanol, The University of AkronAbstract: In this talk, we present a formal discrete-to-continuum procedure to derive a continuum variationalmodel for two chains of atoms with slightly incommen-surate lattices. The chains represent a cross-section of athree-dimensional system consisting of a graphene sheetsuspended over a substrate. The continuum model re-covers both qualitatively and quantitatively the behaviorobserved in the corresponding discrete model. We showthat numerical solutions for both models demonstrate thepresence of large commensurate regions separated by lo-calized incommensurate domain walls. Joint work withDmitry Golovaty and Patrick Wilber.

Dispersion of waves in periodic mediaYuri Godin, University of North Carolina at CharlotteAbstract: We consider propagation of acoustic wavesin a three-dimensional medium containing a periodic lat-tice of spherical inclusions. Propagation of waves isdescribed by the Helmholtz equation with transmissionboundary conditions on inclusions’ interfaces. We assumethat the dimensionless wave frequency is small that al-lows us to view the governing equation as a perturbationof the Laplace equation. The approach is based on the em-ployment of periodic harmonic functions. Solution of theproblem is sought in the form of a power series in termsof the magnitude of the wave vector of the Bloch wave.We provide a uniform explicit asymptotic expansion of

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the dispersion relation, analyze it, and rigorously estimatethe remainder.

Session IV Sun, Oct. 7, 10:30am - 11:50am

Surface elasticity in Steigmann-Ogden form in contactproblemsAnna Zemlyanova, Kansas State UniversityAbstract: A problem of a frictionless or adhesive con-tact of a rigid stamp with an elastic semi-plane is treatedanalytically with the help of complex analysis techniques.The surface of the semi-plane is subjected to surface elas-ticity in the form proposed by Steigmann and Ogden. Theboundary conditions on the banks of the semi-plane con-nect the stresses and the derivatives of the displacements.The mechanical problem is reduced to systems of singularintegro-differential equations which are further reduced tothe systems of equations with logarithmic singularities.The existence and uniqueness of the solution for almost allthe values of the parameters is proved. Additionally, it isshown that introduction of the surface mechanics into thecontact problems leads to the size-dependent equations.A numerical scheme of the solution of the systems of sin-gular integro-differential equations is suggested, and thenumerical results are presented for different values of themechanical and the geometric parameters.

Rapidly Rotating StarsYilun (Allen) Wu, University of OklahomaAbstract: A rotating star may be modeled as a fluid un-der self gravity and with a given total mass and prescribedangular velocity. Mathematically this leads to the Euler-Poisson system. In this talk, we present an existence the-orem for such stars that are rapidly rotating, dependingcontinuously on the speed of rotation. No previous resultsusing continuation methods allowed rapid rotation. Thekey tool for the result is global continuation theory viatopological degree, combined with a delicate limiting pro-cess. The solutions form a connected set K in an appro-priate function space. As the speed of rotation increases,we prove that either the supports of the stars in K becomeunbounded or the density somewhere within the stars be-comes unbounded. We permit any equation of state of theform p = ργ ; 6/5 < γ < 2, so long as γ 6= 4/3. Thisresult is joint work with Walter Strauss.

Modeling and data inversion applied to a girderLynn Greenleaf, Stephen F. Austin State UniversityAbstract: Dynamic models of elastic structures are usedto capture the behavior of a system under study. In thiswork, a concrete girder is loaded with different forces andthe resulting displacements are measured. Given a prioriengineering requirements, an underlying model with spa-tially dependent elastic properties is derived based on en-ergy considerations. Least squares estimation techniques

to calibrate a family of models with experimental data areconsidered. A probabilistic inversion approach focusingon the ability of models to capture the information con-tained in the data is discussed.

MS13. No scheduled talk

MS14. Recent Advances in Numerical PDEs

Organizer: Xiaoming He, hex@mst.edu, MissouriUniversity of Science and TechnologyCo-organizer: Erik Van Vleck, erikvv@ku.edu, Uni-versity of Kansas

Schedule: PHSC 314 (for all four sessions)

Session I Sat, Oct. 6, 10:40am - 12:20pm

Modeling moire-driven mechanical relaxation in in-commensurate multilayer structuresPaul Cazeaux, University of KansasAbstract: We discuss novel mathematical models forthe analysis and computational prediction of mechanicalrelaxation of two-dimensional layered atomic crystals inthe presence of large-scale moire patterns. The conceptof configuration space or hull, previously introduced forthe study of transport properties in aperiodic materials byBellissard et al., is shown to allow for a unified descrip-tion of continuum as well as atomistic models of elasticrelaxation for a wide range of materials in the truly in-commensurate (aperiodic) regime.In the case of twistedbilayers with identical materials, we will present somepreliminary analysis and numerical results in the asymp-totic regime of small twist angle (inducing a large-scalemoire pattern) and small interlayer Van der Waals forces,in particular the well-known case of graphene/graphenebut also MoS2/MoS2.

Weak Galerkin finite element method for Poisson’sequation on polytopal meshes with small edges or facesQingguang Guan, Missouri University of Science andTechnologyAbstract: The weak Galerkin finite element methodfor second order elliptic problems employing polygonalor polyhedral meshes with small edges or faces was an-alyzed. With new shape regular assumptions, optimalconvergence rate for H1 and L2 error estimates were ob-tained. Also element based and edge based error estimateswere proved.

BDDC domain decomposition for Stokes equationXuemin Tu, University of KansasAbstract: A Balancing domain decomposition by con-straints (BDDC) algorithm is studied for solutions oflarge sparse linear algebraic systems arising from the dis-cretization of incompressible Stokes. The condition num-

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ber for the preconditioned system is estimated and numer-ical results are provided to confirm the results.

A linearized energy preserving finite element methodfor the dynamical incompressible magnetohydrody-namics equationsHuadong Gao, University of South CarolinaAbstract: We present and analyze a linearized finite el-ement method (FEM) for the dynamical incompressiblemagnetohydrodynamics (MHD) equations. The finite el-ement approximation is based on mixed conforming ele-ments, where Taylor–Hood type elements are used for theNavier–Stokes equations and Nedelec edge elements areused for the magnetic equation. The divergence free con-ditions are weakly satisfied at the discrete level. Due to theuse of Nedelec edge element, the proposed method is par-ticularly suitable for problems defined on non-smooth andmulti-connected domains. For the temporal discretization,we use a linearized scheme which only needs to solve alinear system at each time step. Moreover, the linearizedmixed FEM is energy preserving. We establish an optimalerror estimate under a very low assumption on the exactsolutions and domain geometries.Numerical results whichincludes a benchmark lid-driven cavity problem are pro-vided to show its effectiveness and verify the theoreticalanalysis.

Bernardi-Raugel Elements Renovated for Linear Elas-ticityGraham Harper, Colorado State UniversityAbstract: The Bernardi-Raugel finite elements wereoriginally developed for Stokes problems. Now they canbe renovated for solving linear elasticity problems. Thiscan be done on simplicial, quadrilateral, and hexahedralmeshes. This yields locking-free solvers for linear elas-ticity. We will show error estimates for the elementsand present numerical results for several benchmark prob-lems.This is a joint work with Drs. James Liu, SimonTavener (Colorado State), Ruishu Wang, Ran Zhang (JilinUniversity, China).

Session II Sat, Oct. 6, 03:10pm - 04:30pm

Flux based finite element methodsJaEun Ku, Oklahoma State UniversityAbstract: In this talk, we present a new finite elementmethod based on flux variables. In many applications,the flux variables are often the quantity of interest. Toapproximate the flux variable accurately and efficiently,one transforms the second-order equations into a systemof first-order and approximates both the primary and fluxvariables simultaneously. While this indeed produces ac-curate approximations for the flux variables, the result-ing algebraic system is large and expensive to solve. Wepresent a new method approximating the flux variables

only without approximations of the primary variable. Ifnecessary, the primary variable can be recovered from theflux approximation with the same order of accuracy.

Machine Learning and Transport Simulators forGroundwater Anomaly DetectionJiangguo (James) Liu, Colorado State UniversityAbstract: This talk presents some preliminary work ondevelopment of models, algorithms, and code modules forgroundwater anomaly detection. Specifically, dissolvedoxygen along with four other surrogates are used for iden-tifying anomaly, support vector machine is utilized formodel training, and real data from Colorado Water Watchis used for testing the models. Numerical simulators forDarcy and transport equations are also used for testing thecode. This talk is based on a joint work with Jianli Gu(DuPont Pioneer R&D), Huishu Li (ColoState), and KenCarlson (ColoState).

Asymptotic Methods for Time-DependentSchrodinger EquationSongting Luo, Iowa State UniversityAbstract: We will present an asymptotic method for nu-merically solving the time-dependent Schrodinger equa-tion with perfectly matched layers. The method is basedon a short-time propagator of the wavefunction. Theshort-time propagator is given in a form of integral thatcombines Huygen’s principle and WKB approximation.In the integral, the phase and amplitude can be obtainedthrough eikonal and transport equations, respectively. Theintegral can be evaluated efficiently by fast Fourier trans-form. Numerical experiments will be presented.

Positive and free energy satisfying schemes for diffu-sion with interaction potentialsWumaier Maimaitiyiming, Iowa State UniversityAbstract: We design and analyze a second order accu-rate free energy satisfying finite volume method for solv-ing diffusion equations with interaction potentials. Bothsemi-discrete and fully discrete schemes are shown toconserve total mass, preserve non-negativity for the solu-tion, and satisfy the free energy dissipation law at the dis-crete level. These properties guarantee that the numericalsolution is a probability density and the schemes obtainedare energy stable. In particular, the fully-discrete schemeis shown to satisfy the three properties without a strict re-striction on time steps, in both one and two-dimensionalcases with nonuniform meshes. In addition, the schemeis easy to implement, and efficient in computing numer-ical solutions over long time. This is a joint work withprofessor Hailiang Liu.

Session III Sat, Oct. 6, 05:00pm - 06:40pm

A posteriori error estimates for hp-adaptive approxi-mations of non-selfadjoint PDE eigenvalue problems

27

Agnieszka Miedlar, University of KansasAbstract: In this talk we present new residual aposteriori eigenvalue/eigenvector error estimates basedon Kato’s square root theorem, applied to diagonaliz-able non-selfadjoint differential operators of convection-diffusion-reaction type with real spectrum. Under cer-tain condition, the existence of the square root operatorprovides a Bauer-Fike type estimate. We present an hp-adaptive finite element algorithm with residual a posteri-ori hp-finite element error estimates which are of higherorder in the residual norm and, together with the Bauer-Fike theorem, guarantee the first order convergence of theeigenvalues. We illustrate our statements with several nu-merical examples. This is a joint work with S. Giani, L.Grubii and J. S. Ovall.

Physically-Constrained Data-Driven Reduced OrderModeling of Fluid FlowsMuhammad Mohebujjaman, Virginia TechAbstract: In this talk, we present two approaches forenforcing better conservation properties for reduced or-der models (ROMs) of fluid flows. In the first approach,to construct the centering trajectory, we use the Stokesextension instead of the standard snapshot average. Weshow that the Stokes extension yields significantly moreaccurate results. In the second approach, we enforce phys-ical constraints in the data-driven modeling of the ROMclosure term. The constrained data-driven ROM is signif-icantly more accurate than its unconstrained counterpart.

An efficient ensemble algorithm for numerical approx-imation of stochastic Stokes-Darcy equationsChangxin Qiu, Missouri University of Science and Tech-nologyAbstract: Many engineering and geological applicationsrequire effective simulations of the coupling of ground-water flows (in porous media) and surface flows. Accu-rate simulations are usually not feasible due to the factit is physically impossible to know the exact parametervalues, e.g., the hydraulic conductivity tensor, at everypoint in the domain as the realistic domains are of largescale and natural randomness occur at small scales. Thispresentation discusses an efficient ensemble algorithm forfast computation of multiple realizations of the stochasticStokes-Darcy model with a random hydraulic conductiv-ity tensor. The algorithm results in a common coefficientmatrix for all realizations at each time step making solvingthe linear systems much less expensive while maintainingcomparable accuracy to traditional methods that computeeach realization separately. Moreover, it decouples theStokes-Darcy system into two smaller sub-physics prob-lems, which reduces the size of the linear systems andallows parallel computation of the two sub-physics prob-lems. Numerical examples are presented to support thetheoretical results and illustrate the application of the al-

gorithm.

Locally-implicit schemes with realizability limiters formoment-closure approximations of kinetic equationsJames Rossmanith, Iowa State UniversityAbstract: In many applications, the dynamics of gas andplasma can be accurately modeled using kinetic Boltz-mann equations. These equations are integro-differentialsystems posed in a high-dimensional phase space, whichis typically comprised of the spatial coordinates and thevelocity coordinates. If the system is sufficiently colli-sional the kinetic equations may be replaced by a fluidapproximation that is posed in physical space (i.e., alower dimensional space than the full phase space). Theprecise form of the fluid approximation depends on thechoice of the moment-closure. In general, finding asuitable robust moment-closure is still an open scientificproblem.In this work we consider a specific moment-closure called the Quadrature-based Method of Moments(QMOM). The distribution function is approximated byDirac deltas with variable weights and abscissas. The re-sulting fluid approximations have differing properties de-pending on the detailed construction of the Dirac deltas.We then develop a high-order, locally-implicit, discontin-uous Galerkin scheme to numerically solve resulting fluidequations. We also develop limiters that guarantee that theinversion problem between moments of the distributionfunction and the weights and abscissas of the Dirac deltasis well-posed. This work is joint with Christine Wiersma(Iowa State) and Erica Johnson (Bexar County).

PIFE-PIC: A 3-D Parallel Immersed-Finite-ElementParticle-in-Cell Method for Plasma SimulationsDaoru Han, Missouri University of Science and Technol-ogyAbstract: We present a recently developed simulationframework, IFE-PIC, a 3-D Parallel Immersed Finite Ele-ment (IFE) Particle-in-Cell (PIC) code for particle simu-lations of plasmas. This code is based on a recently devel-oped non-homogeneous electrostatic IFE-PIC algorithmfor particle simulations of plasma-material interactions,which is designed to handle complex interface bound-ary conditions associated with irregular geometries whilemaintaining the computational speed of the Cartesian-mesh-based PIC. 3-D domain decomposition is used inboth field-solve and particle-push to distribute the com-putation among multiple processors. A verification caseof orbital-motion limited (OML) sheath of a dielectricsphere immersed in a stationary plasma is carried out todemonstrate the capability of the code. Applications ofPIFE-PIC to lunar surface plasma interactions are also go-ing to be presented.

Session IV Sun, Oct. 7, 10:30am - 11:50am

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A Two-field Finite Element Solver for Linear Poroelas-ticity on Quadrilateral MeshesZhuoran Wang, Colorado State UniversityAbstract: This talk presents a finite element solver forlinear poroelasticity on quadrilateral meshes based on thetwo-field model (solid displacement and fluid pressure).This solver combines the Bernardi-Raugel elements forthe displacement in elasticity and the weak Galerkin el-ements for the pressure in Darcy flow through the im-plicit Euler discretization. The solver is penalty-free andhas fewer unknowns compared to other existing methods.Numerical experiments on two widely tested benchmarkswill be presented to show the solver is free of nonphysicalpressure oscillations. This is a joint work with GrahamHarper, Dr. James Liu, and Dr. Simon Tavener.

Weak Galerkin Finite Element Methods for the Time-dependent Biharmonic EquationNolisa Malluwawadu, Colorado State UniversityAbstract: This talk presents a novel weak Galerkinfinite elements method for solving the time-dependentbiharmonic equation. The biharmonic operator is dis-cretized on general convex polygonal meshes. Degree-kpolynomial basis functions are used to approximate thescalar unknown in element interiors and on inter-elementboundaries and also its normal derivative on inter-elementboundaries. The discrete weak Laplacians of these basisfunctions are established as degree-k polynomials as well,which are used to approximate the classical Laplacian inthe variational form. For temporal discretization, we uti-lize Backward Euler method or Crank-Nicolson method.Numerical experiments are presented to corroborate thetheoretical analysis.

A Defect-Deferred Correction Method for Fluid-FluidInteractionDilek Erkmen, Michigan Tech UniversityAbstract: A method is proposed to improve two aspectsof numerical simulations for a model of two fluids cou-pled across a at interface. This problem is motivated byatmosphere-ocean interaction. A deferred correction ap-proach lifts the numerical order of accuracy formally fromfirst order (very common in applications) to second order,in terms of the time interval of communication betweenthe fluid code components. This is accomplished in atwo-step predictor-corrector type method. In the secondstep, a further defect correction is included as well. The”defect” represents artificial diffusion used in the fluidsolvers, which is often included to control numerical noiseor to model subscale mixing processes. The addition ofthe defect correction adds only marginally to the expense,but in exchange may provide a significant reduction ofoverdiffusive effects.The defect and deferred correctionapproaches are combined into a so-called defect-deferredcorrection (DDC) method. A full DDC algorithm is stud-

ied using finite elements in space, including an analysisof the stability and convergence. The method is uncondi-tionally stable, optimally convergent and also enforces aformal reduction in artificial diffusion effects. A computa-tional example using a known (manufactured) solution il-lustrates the theoretical predictions. We observe a compu-tational benefit in this example even for coarse time stepsand over a wide range of artificial viscosity values. Somediscussion is provided regarding the possibility to gener-alize the approach for appli-cation codes. Briefly, legacyatmosphere and ocean codes may be used as-is over acoupling time interval for a predictor computation. Thecorrector step would then potentially be implemented asa straight-forward modification of the predictor step thatleverages the existing code structure.

Optimal control of stochastic flow using the adaptivedynamically low-dimensional approximation methodZhipeng Yang, Beijing Computational Science ResearchCenterAbstract: We explore the dynamically bi-orthogonalmethod (DyBO) for solving the stochastic optimal controlproblem for incompressible Newtonian channel flow pasta circular cylinder. We assume the inlet flow and the ro-tation speed of the cylinder have stochastic perturbations.To achieve objectives of control, we adjust the rotationspeed of the cylinder. Based on the effective gradient-based optimization algorithm to solve the optimality sys-tems, the computational cost of the classical Monte Carlomethod (MC) is unaffordable. The polynomial chaos ex-pansions method (PCE) is a wise choice to reduce thecost, but it still requires high computational cost as thenumber of expansion terms increase quickly if we wantmore accurate solutions. The DyBO method could get ahigh-accuracy with lower computational cost based on theKarhunen-Loeve expansion (KLE) that gives the sparsestrepresentation of the stochastic solutions. To perform theDyBO method for the incompressible Navier Stokes equa-tions (NSE) with open boundary conditions, we combineDyBO method with the pressure Poisson equation for-mulation. Numerical tests are performed to validate ourmethodology.

MS15. Inverse Problems in Tomography andWave Propagation

Organizer: Yang Yang, yangy5@msu.edu, MichiganState University

Schedule: PHSC 114 (for both sessions)

Session I Sat, Oct. 6, 10:40am - 12:00pm

Stability of Stationary Inverse Transport Equation inDiffusion Scaling

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Ke Chen, University of Wisconsin-MadisonAbstract: We consider the inverse problem of recon-structing the optical parameters for stationary radiativetransfer equation (RTE) from velocity-averaged measure-ment. The RTE often contains multiple scales character-ized by the magnitude of a dimensionless parameter—theKnudsen number (Kn). In the diffusive scaling (Kn ¡¡1), the stationary RTE is well approximated by an ellip-tic equation in the forward setting. However, the inverseproblem for the elliptic equation is acknowledged to beseverely ill-posed as compared to the well-posedness ofinverse transport equation, which raises the question ofhow uniqueness being lost as Kn goes to 0. We tacklethis problem by examining the stability of inverse prob-lem with varying Kn. We show that, the discrepancy intwo measurements is amplified in the reconstructed pa-rameters at the order of Knˆp (p=1 or 2), and as a resultlead to ill-posedness in the zero limit of Kn. Our resultsapply to both continuous and discrete settings. Some nu-merical tests are performed in the end to validate thesetheoretical findings.

Breaking the curse of dimensionality: sparse quadra-ture for high-dimensional Bayesian inverse problemsPeng Chen, The University of Texas at AustinAbstract: It is one of the central topics to evaluate statis-tics, such as mean and variance of a given quantity of in-terest, with respect to the posterior distribution of a (high-dimensional) parameter in the context of Bayesian inverseproblems. In this talk, we present a sparse quadraturemethod for this problem and demonstrate that under cer-tain assumptions on the prior distribution and the forwardmap, the convergence rate of the sparse quadrature erroris independent of the parameter dimensions and can bemuch higher than that of Monte Carlo average approxi-mation error.

Some advances in photoacoustic tomography - attenu-ation and media propertiesSebastian Acosta, Texas Children’s HospitalAbstract: Concerning mathematical analysis of photoa-coustic tomography, we will talk about some advances inaccounting for attenuation effects and the simultaneousrecovery of some acoustic properties of the media.

Session II Sat, Oct. 6, 03:10pm - 04:30pm

Ultrasound-switchable fluorescence imaging in deeptissuesBaohong Yuan, University of Texas at ArlingtonAbstract: Tissue imaging via fluorescence has beenwidely used for many years. It stands out due to highsensitivity, high specificity, low cost, use of non-ionizingradiation, capability of multiplex imaging and provid-ing tissue structural, functional and molecular informa-

tion. However, it suffers from limitations: poor spa-tial resolution (i.e., fluorescence diffuse optical tomog-raphy) or small penetration depth (i.e., fluorescence mi-croscopy) due to the strong tissue light scattering. Thus,high-resolution fluorescence imaging in deep tissue hasbeen sought after for many years. Recently, we devel-oped a new imaging technique—ultrasound-switchablefluorescence (USF) which has been demonstrated to over-come the limitations. We successfully imaged centimeter-large mammary tumors in live mice with much higherspatial resolution than conventional 2D planar fluores-cence imaging or 3D fluorescence diffuse optical tomog-raphy and demonstrated the capability of USF for high-resolution in vivo imaging. We also compared USF im-ages with micro-CT to further validate the technology. Inthis talk, I will introduce USF and demonstrate recentlyacquired images from in vitro and in vivo deep tissues.

Fluorescence Ultrasound-Modulated Optical Tomog-raphy in the Diffusive Regime

Yang Yang, Michigan State University

Abstract: Fluorescence optical tomography (FOT) is animaging technology that localizes fluorescent targets intissues. FOT is unstable and of poor resolution in highlyscattering media, where the propagation of multiply-scattered light is governed by the smoothing diffusionequation. We study a hybrid imaging modality called flu-orescent ultrasound-modulated optical tomography (fU-MOT), which combines FOT with acoustic modulationto produce high-resolution images of optical propertiesin the diffusive regime. The principle of fUMOT is toperform multiple measurements of photon currents at theboundary as the optical properties undergo a series of per-turbations by acoustic radiation, in which way internalinformation of the optical field is obtained. We set upa mathematical model for fUMOT, prove well-posednessfor certain choices of parameters, and present reconstruc-tion algorithms and numerical experiments for the well-posed cases.

Inversion of the star transform

Gaik Ambartsoumian, University of Texas at Arlington

Abstract: The star transform maps a function of twovariables to a two-dimensional family of its integralsalong unions of n rays emanating from a common vertex.Such transforms have been studied in relation to imagingmodalities using single-scattered particles. The talk willpresent new exact inversion formulas for this transformand discuss the novel approach for deriving these formu-las. The efficiency of these results will be demonstratedby numerical examples.

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Contributed Talks

Schedule: PHSC 212 (for all three sessions)

Session I Sat, Oct. 6, 10:40am - 12:00pm

Effect of vertical variations in diffusivity and resistiv-ity due to convection in a porous mediumDambaru Bhatta, University of Texas Rio Grande ValleyAbstract: We consider a hydro-thermal convective flowin a horizontal porous medium. The governing systemconsists of the continuity equation for conservation ofmass, the heat equation for conservation of energy, and theconservation of momentum equation which is governedby Darcy’s law. We investigate the effect of linear varia-tions in diffusivity and resistivity in the vertical directionfor two dimensional case. Applying weakly nonlinear ap-proach, we derive the linear and first-order systems. Thenthese systems are solved numerically for quantities suchas temperature and vertical velocity. Numerical resultsfor those quantities are presented.

Numerical study of pattern formation in the fractionalGray–Scott modelHatim Khudhair, Missouri university of Science and Tech-nologyAbstract: In this talk, we numerically study pattern for-mation of the spatial-fractional Gray-Scott equation to un-derstand the nonlocal effects of the fractional Laplacian.The linear stability analysis is provided to predict the con-ditions of the Turing pattern. To investigate the nonlineareffects, we discretize the fractional Gray-Scott equationby the Fourier pseudo-spectral method in space and the4th order Runge-Kutta method in time. Then we numer-ically study the effects of the fractional exponent on thespatiotemporal patterns formation and pattern selection inGray-Scott system. The pattern formations in the standardand fractional Gray-Scott models are compared to providemore insights of the nonlocal effects.

Stability of solutions in reproducing kernel Hilbertspaces on a semi-infinite domain with applications tothe telegraph equationJabar Hassan, Missouri University of Science and Tech-nologyAbstract: In this talk, stability of solutions in certaintype of reproducing kernel Hilbert space methods will beintroduced to the non-homogeneous telegraph equation ona semi-infinite domain. These spaces are convenient fornumerical approximation because the kernels are piece-wise polynomial functions. In this presentation, numeri-cal results are also implemented to demonstrate the effi-ciency and accuracy of the method.

Session II Sat, Oct. 6, 03:10pm - 04:30pm

Rate Transient Analysis of Arbitrarily Oriented Hy-draulic FracturesUmer Farooq, University of TulsaAbstract: Objective: An analytical solution in three-dimensional Cartesian space, generated for pressure con-strained production from an arbitrarily inclined well in ananisotropic reservoir, is extended to the case of arbitrarilyoriented, fully penetrating, vertical fractures. Thus, weproduce a forward discrete fracture model for applicationto well performance in unconventional reservoirs.

Methods: A solution is generated from 3D heat equa-tion using the Newman product method. The solution isthen extended to add a fully penetrating fracture orthogo-nal to the well. The model uses convolution in Laplace do-main to capture constant bottomhole pressure constrainedproduction. Analytic Laplace inversion is performed toyield a result in time with exponential decay terms.

Conclusions: This analytical solution to the Diffusiv-ity Equation under constant pressure constraint gives anew mathematical structure for analyzing rate decline inunconventional wells.The created analytic model grants areservoir engineer the ability to analyze the case of ratetransient production from fractured horizontal well whileaccounting for the permeability anisotropy. This lays theground work for analytically modeling the most generaland widely studied case of having a number of fracturestages in a horizontal well.

Max-min Fairness Hierarchical Dissection of Net-worksZizhen Chen, Southern Methodist UniversityAbstract: Given a graph of relations between entities amax-min fairness hierarchical dissection is shown to re-veal a fundamental contraction of the entities into a rel-atively small number of components with their own re-lational network structure. Our network flow/cut dual-ity based model determines both these components andwhich component pairs remain adjacent in the contractedgraph. The structure follows from the series of max-mindetermined cuts that create the component partition.Thereduced graph, known as a “minor” of the original graphof relations, exhibits a topology of relations between thecomponents. We use novel two and three-dimensionalvisualizations to show how the graph minor, termed a”backbone”, is generated and also to illustrate the itera-tive refinement of backbones over levels. In summary,this process, determined by a simple all pairs networktraffic flow linear programming formulated model, allowsthe large domain of relations between all pairs of entitiesto be reduced to a subset of significant relations betweenhighly connected component communities of the originalgraph. We consider the impact of these results for con-gestion studies of transportation networks and communitydetection investigation of social networks.

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Unconditionally energy stable linear schemes for amulti-component two-phase diffuse interface modelwith Peng-Robinson equationChenfei Zhang, University of South CarolinaAbstract: Numerical solution of the diffuse interfacemodel with Peng-Robinson equation of state can be ap-plied to describe real states of hydrocarbon fluids inpetroleum industry. How to develop appropriate tempo-ral discretizations to overcome the strong nonlinearity ofthe source term and preserve energy dissipation law of thediscretized system is a major challenge. We develop effi-cient first and second order time stepping schemes for thismodel based on the “Invariant Energy Quadratization” ap-proach and stabilized method. Both schemes lead to lin-ear systems which are symmetric and positive definite ateach time step, and their unconditional energy stabilitiesare rigorously proved. Numerical experiments in 2D and3D are presented to demonstrate accuracy and stability ofthe proposed schemes.

Session III Sat, Oct. 6, 05:00pm - 06:40pm

An ensemble based projection method with sparsegrad div stabilization for solving the Navier-StokesequationsShuai Yuan, University of South Carolina at ColumbiaAbstract: We developed a first order projection methodfor ensemble-based simulations of the NSE in which notonly the initial data and body force function, but also theviscosity coefficient, may vary from one ensemble mem-ber to another. Numerical experiments are provided thatillustrate the accuracy and efficiency of the computing byusing this method.

Mathematics of the Significant Tornado ParameterAaron Baker, Stephen F. Austin State UniveristyAbstract: The significant tornado parameter (STP) isone of many models recently created by the NationalStorm Prediction Center (SPC) to predict the possibility ofsevere weather. This parameter looks at the winds, poten-tial energy, and rotation in the atmosphere and performscalculations based on its predicted state. As more re-search on tornadoes has become available, this relativelynew model has become more refined. The purpose of thisproject was to learn how physics and math work in meteo-rology, specifically in severe weather systems. Derivationof each component of the STP is a basic multivariable cal-culus function using basic concepts of thermodynamics.

Accurate numerical methods on the variable-orderfractional LaplacianYixuan Wu, Missouri University of Science and Technol-ogyAbstract: In the last decades, fractional partial differ-ential equations have been widely applied in sciences and

engineering. In particular, recently it has been found thatsome complex phenomena, e.g., viscoelastic mechanicsin geosciences, are governed by the fractional PDEs withvariable-order fractional Laplacian. In this talk, we willpresent an accurate finite difference methods to solve thevariable-order fractional Laplacian. Numerical accuracyin discretizing the operator and in solving the fractionalPoisson problems will be discussed. This is a joint workwith Ying Li (Missouri S&T) and Yanzhi Zhang (MissouriS&T).

Posters

Schedule: Sat, Oct. 6, 5:00pm - 6:20pm at PHSC 209

Mathematical models linking within-host andbetween-host HIV DynamicsCarlos Villanueva, University of OklahomaAbstract: Lack of HIV vaccines have made therapy es-sential to the reduction of HIV transmission and control ofepidemics. For the implementation of control strategies,it is critical to understand between-host transmission dy-namics involving proper risk of infection, which dependson the within-host HIV dynamics of source-host. In thisstudy we develop mathematical models linking within-host and between-host HIV Dynamics. In particular, weincorporate antibody response into within-host viral dy-namic models to estimate the probability of infection froman infected individual to an uninfected individual. Ourmodels predict that this probability is largely dependentupon the source-host’s disease status, including viral loadand antibody level. Using the probability of infection re-sulting from within-host models, we then develop mod-els to describe the dynamics of between-host transmis-sion, which is consistent with HIV prevalence data fromSouth Africa. With these models, we evaluate how within-host disease status of infected individuals influences thebetween-hosts spread of HIV within communities.

A Mechanistic Model of Plant-Symbiont InteractionsRebekah Wagner, University of KansasAbstract: The influence of microbial symbionts onthe phenotype of their plant host is not well understooddespite their known importance. Microbial symbionts,such as mycorrhizal fungi and rhizobia bacteria, residewithin plant roots and exchange resources with their planthost. These relationships can be classified as mutualis-tic or non-mutualistic depending on environmental con-ditions and life history characteristics of plants and sym-bionts. Mycorrhizal fungal mutualists supply their hostwith phosphorus in exchange for allotted carbon producedthrough photosynthesis by the host plant. Rhizobia bacte-ria also deliver fixed nitrogen in exchange for allotted car-bon. The resource complimentary of these relationships

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can facilitate increased host growth and can generate syn-ergisms. A mechanistic model of plant-symbiont inter-actions was developed and shows when a pool resourceis available to the host plant (i.e. soil fertility), the rela-tive abundance of each symbiont decreases. The cost ofpreferentially allocated carbon to the symbionts decreaseswhen the resource pool of phosphorus or nitrogen is avail-able. Productivity of the host plant increases as a measureof total plant biomass increases over time before reach-ing the saturating equilibrium when interacting with eithersymbiont.

Performance Studies for the Numerical Simulations ofRayleigh-Taylor InstabilityAlaina Edwards, University of ArkansasAbstract: The Rayleigh-Taylor instability is a fluid in-stability which occurs in turbulent mixing problems suchas oceanography and supernovae. Usually, it involvesthe interaction between two fluids of differing densities.As the two fluids mix, the interaction forms bubbles andspikes on the interface. We use Front Tracking and LargeEddy Simulations with subgrid-scale models for the nu-merical simulations. With the help of the Blue Waterssupercomputing system, we focus on performance studiesand improvements of the numerical simulations for theRayleigh-Taylor instability. Utilizing both the Cray andGNU programming environments, we have performedscaling studies and used performance tools to explore pro-filing and optimization techniques.

Development of a COMSOL microdialysis model, to-wards creation of microdialysis on a chip with im-proved geometries and recoveryPatrick Pysz, University of ArkansasAbstract: Microdialysis (µD) is diffusion-limited sam-pling method used in basic and clinical research. Re-search needs for collecting low, non steady-state concen-tration (pg/mL) and large (>10 kDa) signaling proteins(cytokines) are hampered by significant limitations withthe technique. Low protein diffusivity leads to low valuesfor the device calibration even under controlled bench-top conditions. Microdialysis device designs have not im-proved upon the cylindrical geometry currently availablecommercially for over 35 years. COMSOL Multiphysicsfinite element method (FEM) software was used to itera-tively model and refine microfluidic-based (µF) µD de-vice designs with the primary focus on optimizing in-novative channel geometry for improved calibration met-rics. The current µF µD design uses a simple asymmet-ric linear-looped (LL) geometry optimized to use 68% ofthe total cross-sectional area and an equal 10 mm lengthin comparison to a commercial CMA 20 µD device. Thesimulated LL µF µD achieves a 35.7% relative increase inRR vs. experimental data at a 1.0 µL/min inlet flow rateusing a model 10 kDa dextran as the analyte. LL µF µDdevices are currently being fabricated, characterized, andcompared to simulation data. This mathematical model-ing work is a critical first step towards uniquely innovativeand improved microdialysis devices.

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Index

Acosta, Sebastian, 30Adams, Henry, 15Agusto, Folashade, 22Aktas, Mehmet, 15Al Basheer, Aladeen, 22Alali, Bacim, 11Albin, Nathan, 9, 10Ambartsoumian, Gaik, 30

Baker, Aaron, 32Beauregard, Matthew, 21, 23Belton, Robin, 17Bhatta, Dambaru, 31Boon, Sarah, 21Bradshaw, Zachary, 4Bush, Johnathan, 17

Cao, Yu, 5Caragea, Doina, 11Carlson, Elizabeth, 24Cazeaux, Paul, 26Chen, Duan, 7, 8Chen, Jianing, 8Chen, Ke, 30Chen, Mingtao, 4Chen, Peng, 30Chen, Zizhen, 31Chen-Charpentier, Benito, 22Chowdhury, Samir, 17Chuenjarern, Nattaporn, 9Chung, Moo K., 16Clemens, Jason, 12Conaway, Harley, 21

Dai, Shibin, 23Dai, Yichen, 5David, Guy C., 10DeLillo, Thomas, 25Dover, James, 18

Edwards, Alaina, 33Epshteyn, Yekaterina, 18Eriksson-Bique, Sylvester, 10Erkmen, Dilek, 29Espanol, Malena, 25

Farooq, Umer, 31Feng, Hongsong, 14Fitzgibbon, William, 22

Gao, Huadong, 27Gasparovic, Ellen, 15Geng, Weihua, 13

Geyer, Lukas, 12Giansiracusa, Noah, 17Gill, James, 12Godin, Yuri, 25Greenleaf, Lynn, 26Griffin, Thomas, 21Guan, Qingguang, 26Guo, Chuan, 6Guo, Ruchi, 15Guo, Wei, 9Guo, Yu, 5, 6Guralnik, Dan, 17

Hakobyan, Hrant, 12Han, Daoru, 28Han, Daozhi, 18, 24Handwerk, Derek, 23Harper, Graham, 27Hashmi, Bahaudin, 23Hassan, Jabar, 31He, Xiaoming, 19, 26Heisterkamp, Doug, 18Higgins, Michael, 11Hoang, Luan, 4Hoang, Thi-Thao-Phuong, 20Horn, Paul, 12Hu, Weiwei, 4, 23Hu, Wenqing, 24

Jiang, Nan, 20Ju, Ning, 4, 5

Kaman, Tulin, 24Khudhair, Hatim, 31Kolasinski, Avary , 25Komendarczyk, Rafal, 18Kottegoda, Kapila, 13Ku, JaEun, 27

Lee, Hyesuk, 18Li, Hao, 15Li, Shuwang, 14, 19Li, Wenwen, 15Lind, Joan, 12Liu, Hailiang, 13Liu, Jiangguo (James), 27Liu, Pei, 7Liu, Weishi, 7Liu, Yuan, 9Liu, Yutong, 6Luo, Albert, 5Luo, Songting, 13, 27Lyu, Jingjing, 21

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Maimaitiyiming, Wumaier, 27Malluwawadu, Nolisa, 29Martin, Jeremy, 11McAllister, Tyrrell, 10McGuirl, Melissa, 16Mears, Justin, 25Miedlar, Agnieszka, 28Mofidi, Hamidreza, 7Mohebujjaman, Muhammad, 28Motta, Francis, 16Murphy, Jason, 24

Oprea, Iuliana, 23Ozaydin, Murad, 15

Parshad, Rana, 21Pei, Jin-Song, 6Pei, Yuan, 24Poggi-Corradini, Pietro, 9, 12Pysz, Patrick, 33

Qiu, Changxin, 28Qiu, Hua, 4

Rossmanith, James, 28

Sanderson, Nicole, 18Santhanakrishnan, Arvind, 22Scoglio, Caterina, 9Segert, Jan, 17Shakeri, Heman, 12Shipman, Patrick, 22, 23Singler, John, 20, 23Speight, Gareth, 11

Tausch, Johannes, 13Thomas, Ashleigh, 16Tu, Xuemin, 26

Van Vleck, Erik, 26Villanueva, Carlos, 32

Wagner, Rebekah, 32

Wang, Cheng, 19Wang, Wei, 14Wang, Zhongming, 13Wang, Zhu, 20Wang, Zhuoran, 29Wen, Zhenshu, 7Wu, Jiahong, 4Wu, Yilun (Allen), 26Wu, Yixuan, 32

Xia, Kelin, 16Xiao, Bei, 24Xing, Siyuan, 5Xu, Yeyin, 5Xu, Zhiliang, 7

Yamazaki, Kazuo, 5Yang, Wanrong, 4Yang, Xiaofeng, 19Yang, Yang (MSU), 29, 30Yang, Yang (MTU), 8Yang, Zhipeng, 29Ye, Hailong, 25Yin, Peimeng, 13, 15Yu, Bo, 5, 6Yu, Xinwei, 4Yu, Yufei, 25Yuan, Baohong, 30Yuan, Shuai, 32Yuan, Yaoguang, 6

Zemlyanova, Anna, 25, 26Zhang, Chenfei, 32Zhang, Mingji, 7, 8Zhang, Xu, 14Zhang, Yangwen, 21Zhao, Jia, 18, 20Zhao, Xueping, 19Zhong, Yimin, 14Zhuang, Qiao, 9Zosso, Dominique, 10

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