Embed Size (px)
Citation preview
Programme & Abstracts
SANUM 2013
The 37th AnnualSouth African Symposium on
Numerical and Applied Mathematics
3, 4 and 5 April
Hosted by theDepartment of Mathematical Sciences
Stellenbosch UniversitySouth Africa
Welcome to SANUM!
Venue The conference is held on the third floor of Stellenbosch University's Engineering Knowledge Center (see map below). Facing the general engineering building, the entrance is about 50m behind the cafeteria ("Plakkies").
Registration takes place on 3 April from 08:30 to 09:30.
A special session will be held on 3 April in celebration of Ben Herbst's 60th birthday.
On the cover: Panoramic view of Stellenbosch from Papegaaiberg. Photo by Francois Malan.
Internet Access, reception, lunches
Wireless internet access is available in the two conference venues:
SSID: SANUM2013Password: s2.71828
As part of the session celebrating Ben Herbst's 60th birthday, please join us for a reception on Wednesday afternoon, directly following the talks, on the porch of the Paul Sauer building (see map).
Lunch is served daily at Café Go in the Neelsie student center. Please remember your lunch voucher, allocated during registration.
Conference Dinner
The conference dinner is held on Thursday evening at 19:00 at Long Table restaurant. If you require transport, please gather in front of the engineering building at 18:40. Please remember your dinner voucher, allocated during registration.
Directions to GPS coordinates:S34°0' 0.58" E18°51' 22.12"
• Take the R44 towards Somerset West.• Turn left onto the Annandale road at
the Mooiberge strawberry farm.• Continue straight, passing Jacana and
Bilton Wines.• The restaurant is at Haskell Vineyards.
Programme for SANUM 2013Wednesday 3 April Thursday 4 April Friday 5 April
08:30—09:00
Registration(K302/303, General Engineering Building)
09:00—09:30
He
rbst
Plenary I (Room K302 ) Jie Shen
Some recent advances on phase-field models for multiphase complex fluids
Wei
dem
an
Plenary I (Room K302 ) Jan Nordström
Initial boundary value problems, summation-by-parts operators and weak boundary conditions with
multi-physics applications09:30—10:00 Opening remarks (Room K302)
10:00—11:00
Wei
dem
an Plenary IMark Ablowitz
Nonlinear waves: from beaches to dispersive shocks
Plenary IISusanne Brenner
C0 interior penalty methods
Plenary IIJean-Paul Berrut
The linear barycentric rational quadrature method for Volterra integral equations
11:00—11:30 Tea
11:30—12:30
Wei
dem
an Plenary I I Bengt Fornberg
Radial Basis Functions: Freedom from meshes in scientific computations
Band
a Plenary I II Alastair Spence
The identification of Hopf bifurcations in problems from computational fluid dynamics
Mas
on
Plenary I II Siegfried Rump
Computer-assisted proofs by verification methods: rigorous results using floating-point arithmetic
12:30—14:00 Lunch
K302 K303 K302 K303 K302 K303
14:00—14:30
Van
der W
alt
AhrensA new angular
discretization for the 3D transport equation:
initial numerical results
Mak
inde
KaraConservation laws of
partial difference equations
Mile
wski
FreitagFast computation of
the stability radius of a matrix
Abel
man
WafyImprovement of visual
Cryptography techniques
Harle
y
EricksonStable, high order accurate adaptive
schemes for long time highly intermittent
geophysics problems
Band
a
AbelmanMHD flow of a Sisko fluid past a porous
plate
14:30—15:00
MaritzPlacement of linear accelerometers for
angular motion estimation - a
systematic approach
MoitshekiNonclassical symmetry
analysis of a heat conduction model for
heat transfer
RagazzoThe turbulent wake of axisymmetric bodies
OliphantTrajectory based
method for nonlinear constrained optimization
AdegboyeSuper Runge-Kutta Nyström method for direct Integration of third order ordinary
differential equations
MasonHydraulic fractures:
conservation laws and exact solutions
15:00—15:30
CoetzerDynamic fusion of
human and machine decisions for efficient
cost-sensitive biometric
authentication
FareoSymmetries and
similarity solutions for a pre-existing fracture driven by a power-law fluid in permeable rock
ChunHow to solve PDEs
numerically on Riemannian geometry?
JacobsA novel approach to text binarization via a diffusion-based model
AhmadiaPyClaw: An
accessible, extensible, and scalable tool for the solution of wave
propagation problems
15:30—16:00 Tea
16:00—16:30
Mar
itz
OlivierA numerical study of the large-period limit
of the Zakharov-Shabat
eigenvalue problem
Kara
HutchinsonTurbulent flow: Lie
group analysis
Moi
tshek
i
Van RensburgExistence theory for
general second order linear hyperbolic
equations
Laur
ie
SwanepoelWriter-specific dissimilarity
normalisation for improved off-line
signature verification
See http://dip.sun.ac.za/sanum for more information.
16:30—17:00Van der WaltSparse spherical
harmonics kernels for representing diffusion signals on a sphere
MolatiGroup analysis of a
generalized KdV equation
BassonConvergence of the
finite element approximation for the vibration of a beam with a damping tip
body
MitchleyOrthogonal subspace projection methods for
endmember identification in
hyperspectral data
17:00—17:30Weideman
A computational exploration of the second Painlevé
equation
MakindeComputational
modelling and optimal control of pest invasion
on maize farms
HarleyA hybrid finite sine transform and the
variational iteration method for solving a
time-fractional diffusion equation
StoltzPulse reformation,
solving the leakage problem in the Discrete
Pulse Transform domain
17:30—18:00Laurie
The interval euclidean algorithm
MatebeseImproving path quality
of RRT-based algorithms
MilewskiNonlinear-optics-like behaviour in water
waves
SathinarainNumerical investigation of the parabolic mixed
derivative diffusion equation via ADI
methods
Welcoming Reception at 18:00 Conference Dinner at 19:00
5
Wednesday 3 April
PLENARY (K302) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Nonlinear waves: from beaches to dispersive shocks
Mark Ablowitz, University of Colorado at Boulder . . . . . . . . . . . . . . . . . . . . . 6Radial Basis Functions: freedom from meshes in scientific computations
Bengt Fornberg, University of Colorado at Boulder . . . . . . . . . . . . . . . . . . . . . 6WEDNESDAY AFTERNOON I (ROOM K302) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
A new angular discretization for the 3D transport equation: initial numerical resultsCory Ahrens, Colorado School of Mines . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Placement of linear accelerometers for angular motion estimation - a systematic approachMilton Maritz, Stellenbosch University . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
Dynamic fusion of human and machine decisions for efficient cost-sensitive biometric authen-ticationHanno Coetzer, Stellenbosch University . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
A numerical study of the large-period limit of the Zakharov-Shabat eigenvalue problem withperiodic potentialsCarel Olivier, University of Cape Town . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Sparse spherical harmonics kernels for representing diffusion signals on a sphereStefan van der Walt, Stellenbosch University . . . . . . . . . . . . . . . . . . . . . . . . 7
A computational exploration of the second Painleve equationJAC Weideman, Stellenbosch University . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
The interval euclidean algorithmDirk Laurie, Stellenbosch University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
WEDNESDAY AFTERNOON II (K303) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Conservation laws of partial difference equations
Abdul Hamid Kara, Wits University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8Nonclassical symmetry analysis of a heat conduction model for heat transfer in a longitudinal
rectangular fin of a functionally graded materialJoel Moitsheki, University of the Witwatersrand . . . . . . . . . . . . . . . . . . . . . . 8
Symmetries and similarity solutions for a pre-existing fracture driven by a power-law fluid inpermeable rockAdewunmi Gideon Fareo, University of the Witwatersrand . . . . . . . . . . . . . . . . . 8
Turbulent flow: Lie group analysisAshleigh Hutchinson, Wits University . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
Group analysis of a generalized KdV equationMotlatsi Molati, National University of Lesotho/North-West University . . . . . . . . . . 8
Computational modelling and optimal control of pest invasion on maize farmsOluwole Daniel Makinde, Cape Peninsula University of Technology . . . . . . . . . . . . 9
Improving path quality of RRT-based algorithmsBelinda Matebese, Stellenbosch University . . . . . . . . . . . . . . . . . . . . . . . . . . 9
6 Wednesday 3 April
NONLINEAR WAVES: FROM BEACHES TODISPERSIVE SHOCKS
Mark Ablowitz (University of Colorado at Boulder)
The study of localized waves has a long history datingback to the discoveries in the 1800s describing soli-tary water waves in shallow water. In the 1960s re-searchers found that certain equations, including theKorteweg-deVries (KdV) and nonlinear Schrodinger(NLS) equations arise widely. Both equations admitlocalized solitary wave- or soliton solutions. Via anovel nonlocal formulation of water waves asymp-totic reductions can be readily obtained such as the1d equations mentioned above and the 2d Benney-Luke (BL) and Kadomstsev-Petviashvili (KP) equa-tions. Some solutions will be discussed as well asrecent observations. In fluids and optics ‘dispersive’shock waves can occur. Applications, properties andinteractions of will be discussed.
RADIAL BASIS FUNCTIONS: FREEDOMFROM MESHES IN SCIENTIFIC COMPU-TATIONS
Bengt Fornberg (University of Colorado at Boulder)
Most numerical computations in multiple dimensionsuse either structured grids (e.g. finite differences andspectral methods) or unstructured ones (e.g. finiteelements). Radial Basis Functions (RBFs) and, mostrecently, RBF-generated finite differences (RBF-FD)offer the opportunity of completely meshfree calcula-tions in a large number of application areas, such asgeoscience, fluid mechanics, mathematical biology,etc. The approach combines a number of strengths,such as:
• High orders of accuracy,
• High computational efficiency,
• Geometric flexibility (irregular domain shapes,local node refinements, solutions over curvedsurfaces, etc.),
• Short and simple codes,
• Excellent opportunities for distributed comput-ing.
RBFs were introduced in the 1970s, first as a toolfor interpolating scattered 2-D data. Since then, bothour knowledge about RBFs and their range of appli-cations have grown tremendously, with the numeri-cal solution of PDE systems arising in the geosciencesnow becoming a particularly important one.
A NEW ANGULAR DISCRETIZATION FORTHE 3D TRANSPORT EQUATION: INITIALNUMERICAL RESULTS
Cory Ahrens (Colorado School of Mines)
The discrete ordinate (Sn) equations have been theworkhorse of deterministic radiation transport calcu-lations for many years. Here we derive a new angulardiscretization for the 3D transport equation. The newdiscretization, derived using Lagrange interpolationon the sphere and collocation, retains the classicalSn structure, with the main difference being how thescattering source is calculated. Because of the formalsimilarity with the classical Sn equations, it should bepossible to modify existing computer codes to takeadvantage of the new formulation. Moreover, thenew Sn-like equations correctly capture delta func-tion scattering, an important property when simulat-ing charged particle transport. Preliminary numeri-cal results will be presented.
PLACEMENT OF LINEAR ACCELEROME-TERS FOR ANGULAR MOTION ESTIMA-TION - A SYSTEMATIC APPROACH
Milton Maritz (Stellenbosch University)
Linear accelerometers are often mounted in satellites(or vehicles) in order to estimate the angular veloc-ity and angular acceleration of the body. How manyof these accelerometers must be used, where in thebody must one mount them, and in which directionsmust they point? Is there an optimal placement?Previous researchers have proposed various place-ment configurations together with algorithms for ex-tracting the required angular velocity and accelera-tion.In this talk we approach the problem in a math-ematically systematic way and show the power ofthis mathematical approach by illustrating that goodplacement configurations come forth naturally fromthe analysis. We demonstrate that the previously sug-gested configurations are special cases of the general‘good’ configuration.
DYNAMIC FUSION OF HUMAN ANDMACHINE DECISIONS FOR EFFICIENTCOST-SENSITIVE BIOMETRIC AUTHENTI-CATION
Hanno Coetzer (Stellenbosch University)
We present a novel protocol for optimally combininghuman and machine decisions in a cost-sensitive en-vironment. This should benefit financial institutions
Wednesday Afternoon I (Room K302) 7
where off-line signatures, each associated with a spe-cific transaction value, require authentication. Whenpresented with labelled signatures produced by rep-resentative writers, the proficiency of a human work-force and a score-generating machine can be esti-mated and represented in ROC space. Using differentBoolean fusion functions, a specific employee’s de-cision is combined with each threshold-specific ma-chine decision, after which the performance of thecandidate ensembles is represented in ROC space.Only the optimal ensembles and associated decisiontrees are retained. When presented with a ques-tioned signature linked to an arbitrary writer, the sys-tem first uses the ROC-based cost gradient associatedwith the transaction value to select the candidate en-semble that minimises the expected cost, and thenuses the corresponding decision tree to make an op-timal combined decision for the employee in ques-tion. Finally, the optimal combined decisions for ev-ery employee in the workforce are fused through ma-jority voting. By considering the average expectedcost over all operating conditions, we show that theproposed hybrid classifier consistently outperformsan unaided machine and an unaided human work-force with more than five employees.
A NUMERICAL STUDY OF THE LARGE-PERIOD LIMIT OF THE ZAKHAROV-SHABAT EIGENVALUE PROBLEM WITHPERIODIC POTENTIALS
Carel Olivier (University of Cape Town)
Deconinck and Kutz (2007) developed an efficient al-gorithm for solving the Zakharov-Shabat eigenvalueproblem with periodic potentials numerically. It isnatural to use the same algorithm for solving theproblem for non-periodic potential (decaying poten-tials defined over the whole real line) using large pe-riods. In this paper we propose the use of a specificvalue of the Floquet exponent. Our numerical resultsindicate that it can produce accurate results long be-fore the period becomes large enough for the ana-lytical convergence results of Gardner (1997) to bevalid. We also illustrate the rather complicated pathto convergence of some nonlinear Schrodinger po-tentials.
SPARSE SPHERICAL HARMONICS KER-NELS FOR REPRESENTING DIFFUSIONSIGNALS ON A SPHERE
Stefan van der Walt (Stellenbosch University)
Magnetic resonance imaging (MRI) allows the obser-vation of protons in water molecules that traversethe human brain. Mapping their motion illuminates
white matter fiber pathways, and allows the explo-ration of the structure of the brain in-vivo (‘within theliving’). We describe why mathematical modelling ofMRI signals is needed, and present a sparse modelwith desirable properties that fits the data more ro-bustly than equivalent existing methods.
A COMPUTATIONAL EXPLORATION OFTHE SECOND PAINLEVE EQUATION
JAC Weideman (Stellenbosch University)
The Painleve equations are nonlinear, second-orderdifferential equations that have become increasinglyprominent in applications over the last two or threedecades. There are six members in the family, thefirst two of which are
PI:d2u
dz2= 6u2+z, PII:
d2u
dz2= 2u3+zu+α, z ∈ C
with α a constant. Solutions cannnot, in general, beexpressed in terms of classical special functions andthey define new transcendental functions for whichnumerical methods have to be developed. The solu-tions are characterized by extended pole fields, some-times accompanied by pole free sectors in the com-plex plane (called tronquee solutions). Both thesefeatures present their own challenges to numericalcomputation.In a talk presented at SANUM2011, an algorithm de-veloped in collaboration with Bengt Fornberg was de-scribed. It combines a pole-friendly Pade time step-ping algorithm with a novel path selection strategy.In that talk numerical results for PI was presented.Here we describe further results for PII. We shallshow computed solutions and pole field patterns,some of which we believe have not been observedbefore. (Joint work with Bengt Fornberg, Universityof Colorado at Boulder, USA.)
THE INTERVAL EUCLIDEAN ALGORITHM
Dirk Laurie (Stellenbosch University)
Finding the greatest common denominator of two in-tegers is one of the oldest problems of number theory.Its relevance is undimininished even in today’s worldof open-key cryptosystems based on the difficulty offactoring large integers.When working with very large numbers, say onemillion digits, the basic idea, first shown by D.H.Lehmer in 1938, is known to numerical analysts as”iterative improvement”: approximate answer vialow-precision calculation, high-precision evaluationof residual, another low-precision calculation usingthe residuals as data, high-precision combination ofthe two results to combine the answers.
8 Wednesday 3 April
This idea still lies at the heart of modern softwarefor the GCD in multiprecision arithmetic. It is com-mon practice to increase the precision exponentiallyat each step, for example, by doubling the number ofdigits.We discuss two variations. The first involves anotheridea from numerical analysis, namely interval arith-metic. This leads to a simpler and slightly faster al-gorithm. The second is to increase the precision lin-early, but with chunk sizes that are optimized for thedesired final number of digits.
CONSERVATION LAWS OF PARTIAL DIF-FERENCE EQUATIONS
Abdul Hamid Kara (Wits University)
We present some recently developed notions on con-servation laws of Difference Equations and discussthe similarities with Differential Equations (see theworks of Hydon and Mansfield & Hydon). The ideaof a relationship between symmetries and conserva-tion laws will be pursued.
NONCLASSICAL SYMMETRY ANALYSISOF A HEAT CONDUCTION MODEL FORHEAT TRANSFER IN A LONGITUDINALRECTANGULAR FIN OF A FUNCTIONALLYGRADED MATERIAL
Joel Moitsheki (University of the Witwatersrand)
We consider a heat conduction model arising in tran-sient heat transfer through longitudinal fins of a het-erogeneous (functionally graded) material. In thiscase, the thermal conductivity depends on the spa-tial variable. The heat transfer coefficient dependson the temperature and is given by the power lawfunction. The resulting nonlinear partial differentialequation is analyzed using both classical and non-classical symmetry techniques. Both the transientstate and the steady state result in a number of exoticsymmetries being admitted by the governing equa-tion. Furthermore, nonclassical symmetries are alsoadmitted. Both classical and nonclassical symmetryanalysis results in some useful reductions and someremarkable exact solutions are constructed.
SYMMETRIES AND SIMILARITY SOLU-TIONS FOR A PRE-EXISTING FRACTUREDRIVEN BY A POWER-LAW FLUID IN PER-MEABLE ROCK
Adewunmi Gideon Fareo (University of the Witwater-srand)
In this talk we derive group invariant, exact andnumerical solution for the evolution of a two-dimensional permeable fracture with non-zero ini-tial length and driven by a non-Newtonian fluid ofpower-law rheology. The two- dimensional fractureis driven by the injection of an incompressible power-law fluid and the fluid flow in the fracture in consid-ered laminar. With the aid of lubrication theory andthe PKN approximation a nonlinear diffusion equa-tion for the fracture half-width is derived. Since thefluid-rock interface is permeable the nonlinear dif-fusion equation has a leak-off velocity sink term. Acondition, in the form of a first order partial differ-ential equation for the leak-off velocity, is obtainedfor the nonlinear diffusion equation to have Lie pointsymmetries. The general form of the leak- off veloc-ity is derived. Using the Lie point symmetries theproblem is reduced to a boundary value problem fora second order ordinary differential equation. Theleak- off velocity is further specified by assuming thatit is proportional to the fracture half-width. The ef-fect of power-law rheology on hydraulic fracturing isinvestigated and the evolution of a two-dimensionalfracture with non-zero initial length and driven by apower-law fluid is analysed. In this talk, hydraulicfracturing with shear thinning, Newtonian and shearthickening fluids are compared.
TURBULENT FLOW: LIE GROUP ANALY-SIS
Ashleigh Hutchinson (Wits University)
Kolmogorov’s theory of turbulence describes turbu-lent flow as consisting of eddies of varying sizes. En-ergy is transferred from larger to smaller eddies un-til sufficiently small scales are reached. The viscos-ity dominates and energy is effectively dissipated atthese small scales. A brief review of Kolmogorov’stheory of turbulence is given and it is shown how Liesymmetry methods can be applied. A simple time de-pendent flow problem is considered and laminar andturbulent solutions are compared.
GROUP ANALYSIS OF A GENERALIZEDKDV EQUATION
Motlatsi Molati (National University of Lesotho/North-West University)
In this work the Korteweg-de Vries equation whichcontains an arbitrary function in the nonlinear termis considered and it is referred to as a generalizedKdV. The Lie group analysis approach is employed toobtain the possible forms of the arbitrary parameter.
Wednesday Afternoon II (K303) 9
COMPUTATIONAL MODELLING AND OP-TIMAL CONTROL OF PEST INVASION ONMAIZE FARMS
Oluwole Daniel Makinde (Cape Peninsula University ofTechnology)
In Africa, maize is one of the most commonly culti-vated crops in all major agro-ecological zones. It isone of the main food sources for the people in thecontinent. Maize also finds its way in various formsinto animal feed rations and is industrially processedfor the production of starches, syrups and alcohol. Inspite of its importance, maize yields are generally lowdue to pest invasion. Among the insect pests associ-ated with maize, Lepidopteran stem borers are themost dangerous threatening for the production. Inthis paper, a compartmental deterministic model forpest invasion on maize farm is proposed. The non-linear problem is tackled using stability theory of dif-ferential equations and an optimal control strategy isdetermined computationally. Numerical simulationis performed and the pertinent results are presentedgraphically and discussed quantitatively.
IMPROVING PATH QUALITY OF RRT-BASED ALGORITHMS
Belinda Matebese (Stellenbosch University)
Sampling-based methods such as Rapidly-exploringRandom Tree (RRT) have been successfully usedin solving motion planning problems in high-dimensional and complex environments. RRT-basedalgorithms are the most popular since they havethe ability to find a feasible and collision-free pathquickly. At the same time, the algorithms are not ableto find optimal solutions and moreover produce non-smooth solutions. To improve the quality of these so-lutions, optimal control based methods are being in-vestigated and will be integrated with the algorithm.This approach will be discussed and preliminary re-sults presented.
10 Wednesday 3 April
Thursday 4 April
PLENARY (K302) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Some recent advances on phase-field models for multiphase complex fluids
Jie Shen, Purdue University . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11C0 interior point methods
Susanne Brenner, Louisiana State University . . . . . . . . . . . . . . . . . . . . . . . . 11The identification of Hopf bifurcations in problems from computational fluid dynamics
Alastair Spence, University of Bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11THURSDAY AFTERNOON I (K302) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Fast computation of the stability radius of a matrixMelina Freitag, University of Bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
The turbulent wake of axisymmetric bodiesClodoaldo Ragazzo, Universidade de Sao Paulo . . . . . . . . . . . . . . . . . . . . . . . 11
How to solve PDEs numerically on Riemannian geometry?Sehun Chun, African Institute for Mathematical Sciences . . . . . . . . . . . . . . . . . . 12
Existence theory for general second order linear hyperbolic equationsNic Janse van Rensburg, University of Pretoria . . . . . . . . . . . . . . . . . . . . . . . 12
Convergence of the finite element approximation for the vibration of a beam with a dampingtip bodyMadelein Basson, University of Pretoria . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
A hybrid finite sine transform and the variational iteration method for solving a time-fractionaldiffusion equation on a bounded domainCharis Harley, University of the Witwatersrand . . . . . . . . . . . . . . . . . . . . . . . 12
Nonlinear-optics-like behaviour in water wavesPaul Milewski, University of Bath . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
THURSDAY AFTERNOON II (K303) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Improvement of visual cryptography techniques
Maged Wafy, Aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Trajectory based method for nonlinear constrained optimization
Terry-Leigh Oliphant, University of the Witwatersrand . . . . . . . . . . . . . . . . . . . 13A novel approach to text binarization via a diffusion-based model
Byron Jacobs, University of the Witwatersrand . . . . . . . . . . . . . . . . . . . . . . . 13Writer-specific dissimilarity normalisation for improved writer-independent off-line signature
verificationJacques Swanepoel, Stellenbosch University . . . . . . . . . . . . . . . . . . . . . . . . . 13
Orthogonal subspace projection methods for endmember identification in hyperspectral dataMichael Mitchley, University of the Witwatersrand . . . . . . . . . . . . . . . . . . . . . 13
Pulse Reformation, solving the leakage problem in the Discrete Pulse Transform domainGene Stoltz, TUKS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Numerical investigation of the parabolic mixed derivative diffusion equation via ADI methodsMelisha Sathinarain, University of the Witwatersrand . . . . . . . . . . . . . . . . . . . 13
Thursday Afternoon I (K302) 11
SOME RECENT ADVANCES ON PHASE-FIELD MODELS FOR MULTIPHASE COM-PLEX FLUIDS
Jie Shen (Purdue University)
I shall present some recent work on phase-fieldmodel for multiphase complex fluids. Particular at-tention will be paid to develop models which arevalid for problems with large density ratios and obeyan energy dissipation law.I shall present efficient and accurate numericalschemes for solving the coupled nonlinear system forthe multiphase complex fluid, in many case provethat they are energy stable, and show ample numeri-cal results which not only demonstrate the effective-ness of the numerical schemes, but also validate theflexibility and robustness of the phase-field model.
C0 INTERIOR POINT METHODS
Susanne Brenner (Louisiana State University)
C0 interior penalty methods are discontinuousGalerkin methods for fourth order problems that usestandard finite element spaces for second order prob-lems. These methods have some advantages overclassical approaches for fourth order problems. Theyare much simpler than C1 finite element methods.The lowest order C0 interior penalty methods areas simple as classical nonconforming finite elementmethods. But unlike classical nonconforming finiteelement methods that only use low order polynomi-als, C0 interior penalty methods come in a naturalhierarchy and higher order C0 interior penalty meth-ods can capture smooth solutions efficiently. Com-pared with mixed finite element methods, the sta-bility of C0 interior penalty methods can be estab-lished in a straightforward manner and the symmet-ric positive definiteness of the continuous problemsis preserved by C0 interior penalty methods. More-over, since the underlying finite element spaces arestandard spaces for second order problems, multigridsolves for the Laplace operator can be used as naturalpreconditioners for C0 interior penalty methods, andproblems on smooth domains can be easily handledby isoparametric C0 interior penalty methods.In this talk we will discuss the formulation, erroranalysis, fast solution techniques for these methodsand present applications to boundary value prob-lems, obstacle problems and optimal control prob-lems.
THE IDENTIFICATION OF HOPF BIFUR-CATIONS IN PROBLEMS FROM COMPU-TATIONAL FLUID DYNAMICS
Alastair Spence (University of Bath)
The detection of a Hopf bifurcation in a large-scaledynamical system that depends on a physical pa-rameter often consists of computing the right-mosteigenvalues of a sequence of large sparse eigenvalueproblems. This is not only an expensive operationbut many of the common numerical methods for thisproblem may be unreliable for large sparse matrices.This talk will summarise some of the methods com-monly used in applications and discuss their advan-tages and disadvantages. Next, we shall describe arecent approach that reformulates the problem usingKronecker products of matrices (see Meerbergen &Spence, SIMAX, Vol 31, 2010, and Elman, Meerber-gen, Spence & Wu, SISC, Vol 34, 2012). This ap-proach is based on inverse iteration but requires thesolution of Lyapunov equations with low-rank right-hand sides at each step of the iteration. Numericalresults will be presented for some large- scale prob-lems arising from fluid dynamics and aeroelasticity.
FAST COMPUTATION OF THE STABILITYRADIUS OF A MATRIX
Melina Freitag (University of Bath)
A new fast algorithm for the computation of the dis-tance of a stable matrix to the unstable matrices isprovided. The method is based on finding a two-dimensional Jordan block corresponding to a pureimaginary eigenvalue in a certain two-parameterHamiltonian eigenvalue problem introduced by Byers(SIAM J. Sci. Statist. Comput., 9 (1988), pp. 875-881). The solution is achieved by an extension ofthe Implicit Determinant Method. Numerical resultsshow the performance of this algorithm for severalexamples and comparison is made with other meth-ods for the same problem.
THE TURBULENT WAKE OF AXISYMMET-RIC BODIES
Clodoaldo Ragazzo (Universidade de Sao Paulo)
In this talk experimental and theoretical results onaxisymmetric selfsimilar turbulent wakes are pre-sented and discussed with emphasis on the wakewidth growth rate. Some applications of the existingtheory to the problem of hydrodynamic interactionbetween ships is made.
HOW TO SOLVE PDES NUMERICALLY ONRIEMANNIAN GEOMETRY?
Sehun Chun (African Institute for Mathematical Sci-ences)
Modern scientific era faces unprecedented demandsof computational modeling on complex geometry, es-
12 Thursday 4 April
pecially in the area of biological fluid dynamics, im-age processing, and computational biology. In mostnumerical schemes, geometry has been regarded asthe synonym of ’mesh’ and computational techniqueshave been developed almost independent of the ge-ometrical properties of the domain or its materialproperty. But, these approaches soon revealed its lim-itations on the order of its convergence on curvedsurfaces as the simplest Riemannian geometry. Inthis talk, we briefly review the previous numericalschemes to solve PDEs on curved surfaces and howthe incorporation of the concepts of differential ge-ometry can be used to develop or improve the numer-ical solutions on Riemannian geometry that we facein the actual world. The recently-proposed methodof moving frames will be introduced as an example.
EXISTENCE THEORY FOR GENERALSECOND ORDER LINEAR HYPERBOLICEQUATIONS
Nic Janse van Rensburg (University of Pretoria)
The existence of solutions for problems involving par-tial differential equations is not a black or white is-sue; there exist a number of possibilities. For a start,there is a distinction between so called weak solu-tions and classical solutions. But there are different”levels of weakness” and definitions also differ. Us-ing the one- dimensional wave equation, it is easy toidentify four possibilities. The properties of a solu-tion are not merely of academic interest, since therate of convergence of numerical approximations de-pends on the ”smoothness” of a solution. General-ized, second order linear hyperbolic partial differen-tial equations and vibration of systems of elastic bod-ies are of the same type which we refer to as generalsecond order linear hyperbolic equations. In this pre-sentation we consider existence results for this typeof problem. Different approaches to the theory aresketched and the usefulness of the results is con-sidered. We argue that there should be at least anawareness of the possibility that a solution may notbe smooth enough or not exist at all when comput-ing a numerical approximation. Elementary exam-ples are presented to illustrate some of the possibili-ties.
CONVERGENCE OF THE FINITE ELE-MENT APPROXIMATION FOR THE VIBRA-TION OF A BEAM WITH A DAMPING TIPBODY
Madelein Basson (University of Pretoria)
A model for the vibration of a beam with a dampingtip body that appeared in a recent article is consid-
ered. A variational form for the motion of the beamis derived and it is used to prove that the model prob-lem has a unique solution. Using a so called energymethod, we prove that the Galerkin approximationconverges to the solution and derive error estimates.
A HYBRID FINITE SINE TRANSFORM ANDTHE VARIATIONAL ITERATION METHODFOR SOLVING A TIME-FRACTIONAL DIF-FUSION EQUATION ON A BOUNDED DO-MAIN
Charis Harley (University of the Witwatersrand)
The aim of this research is to introduce a methodbased on the hybridization of the Finite Sine Trans-form (FST) and the Variational Iteration Method(VIM). The nature of the boundary conditions in-forms the choice of transform method: in this caseDirichlet boundary conditions are required by theFST for instance. The VIM considers only the ini-tial condition as an iteration seed. It is this compati-bility that suggests this hybridization to be effective.As a source of comparison we implement both theFST-VIM hybrid method and the FST-FFD (FractionalFinite Difference) method and report on the results.
NONLINEAR-OPTICS-LIKE BEHAVIOURIN WATER WAVES
Paul Milewski (University of Bath)
A sufficiently high intensity beam of light in amedium whose refractive index is intensity depen-dent (such as air or water) will exhibit self focussinguntil higher order effects, noise, or plasma genera-tion come into play. The cross-sectional profile ofthe focussed beam depends on the initial profile. Itturns out that a very similar phenomenon occurs ina patch of water waves until nonlinearity arrests thefocussing and the patch breaks up into a complex setof localised structures. Whilst water may be too vis-cous for the phenomena to be observed, computa-tions suggest that the behaviour should occur in mer-cury.
IMPROVEMENT OF VISUAL CRYPTOGRA-PHY TECHNIQUES
Maged Wafy (Aims)
The Visual cryptography scheme (VCS) is a securemethod that uses mathematical techniques to encodesecret image. The idea is to convert the written ma-terial into an image, then breaking this image into Nshares. A distinctive property of VCS is that one canvisually decode the secret image by superimposingshares without computation.
Thursday Afternoon II (K303) 13
Recently, research problems in visual cryptographyare related to several extension of visual cryptogra-phy research that includes VCS for general accessstructure, contrast optimization and the concept ofrandomness, Multi secret messages, and meaningfulshares. I shall introduce some new ideas to improvecontrast, and meaningful shares.
TRAJECTORY BASED METHOD FOR NON-LINEAR CONSTRAINED OPTIMIZATION
Terry-Leigh Oliphant (University of the Witwater-srand)
The trajectory-based method for solving nonlinearconstrained programming problems is proposed andthe convergence properties of this method are inves-tigated. The augmented Lagrangian problem refor-mulation is used to covert the constrained probleminto an equivalent unconstrained problem and a newupdating scheme for the penalty parameter is dis-cussed
A NOVEL APPROACH TO TEXT BINARIZA-TION VIA A DIFFUSION-BASED MODEL
Byron Jacobs (University of the Witwatersrand)
We present here a new approach to document im-age binarization. The method is based on the dy-namic process of diffusion, coupled with a nonlinearFitzhugh-Nagumo type source term that exhibits bi-narizing properties. These desirable properties leadto a method that is robust to noise and is able tosuccessfully binarize an input document image. Wemeasure the efficacy of our proposed method againstindustry standards by two methods, a pixel by pixelcomparison with the ground truth image and a stan-dard optical character recognition test. Throughthese measures we illustrate a progressive methodthat performs at the highest level in the field.
WRITER-SPECIFIC DISSIMILARITY NOR-MALISATION FOR IMPROVED WRITER-INDEPENDENT OFF-LINE SIGNATUREVERIFICATION
Jacques Swanepoel (Stellenbosch University)
We present a novel writer-independent off-line signa-ture verification system. This system utilises the dis-crete Radon transform and a dynamic time warpingalgorithm for writer-independent signature represen-tation in dissimilarity space. The system also con-siders writer-specific statistics for dissimilarity nor-malisation. A discriminant function, either linear orquadratic, is utilised for signature modelling and ver-ification.
We show that the proposed feature extraction anddissimilarity representation framework provides asuccessful platform for signature modelling and veri-fication. We also show that the inclusion of writer-specific statistics during dissimilarity normalisationimproves the proficiency of the proposed writer-independent verification system.When evaluated on Dolfing’s data set, a signaturedatabase containing 1530 genuine signatures and3000 amateur skilled forgeries, we show that the pro-posed system outperforms existing systems also eval-uated on this data set.
ORTHOGONAL SUBSPACE PROJECTIONMETHODS FOR ENDMEMBER IDENTIFI-CATION IN HYPERSPECTRAL DATA
Michael Mitchley (University of the Witwatersrand)
Endmembers of a linear mixing model in hyperspec-tral data span the signal subspace. Using a pseu-dorank estimation procedure, the sonar- based MU-SIC method is used to test endmember candidacythrough projection of signal vectors onto the orthog-onal noise subspace.
PULSE REFORMATION, SOLVING THELEAKAGE PROBLEM IN THE DISCRETEPULSE TRANSFORM DOMAIN
Gene Stoltz (TUKS)
The Discrete Pulse Transform is derived from the con-nected operators, called the LULU operators. Con-nected operators presents a framework for the ex-traction of meaningful structures from an image.Two or more of these structures sometimes getsjoined to present a nugatory structure; this phe-nomenon is known as the leakage problem. PulseReformation is a proposed solution to undo the unionbetween meaningful structures.
NUMERICAL INVESTIGATION OF THEPARABOLIC MIXED DERIVATIVE DIFFU-SION EQUATION VIA ADI METHODS
Melisha Sathinarain (University of the Witwatersrand)
The two-dimensional analog of a two-parametermixed derivative equation is considered via two nu-merical methods: the ADI and compact ADI method.These methods are compared and the influence of thekinematic viscosity and viscoelasticity is discussed inrelation to the solutions obtained. The two schemesare proven to be conditionally stable and the meth-ods compared via an error analysis.
14 Thursday 4 April
Friday 5 April
FRIDAY PLENARY (K302) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Initial boundary value problems, summation-by-parts operators and weak boundary condi-
tions with multi-physics applicationsJan Nordstrom, Linkoping University . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
The linear barycentric rational quadrature method for Volterra integral equationsJean-Paul Berrut, Fribourg University . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Computer-assisted proofs by verification methods: rigorous results using floating-point arith-meticSiegfried Rump, Hamburg University of Technology . . . . . . . . . . . . . . . . . . . . 15
FRIDAY AFTERNOON I (K302) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Stable, high order accurate adaptive schemes for long time highly intermittent geophysics
problemsBrittany Erickson, San Diego State University . . . . . . . . . . . . . . . . . . . . . . . . 15
Super Runge-Kutta Nystrom method for direct integration of third order ordinary differentialequationsZamurat Ayobami Adegboye, Kaduna Polytechnic, Kaduna . . . . . . . . . . . . . . . . . 15
FRIDAY AFTERNOON II (K303) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15MHD flow of a Sisko fluid past a porous plate
Shirley Abelman, University of the Witwatersrand . . . . . . . . . . . . . . . . . . . . . 16Hydraulic fractures: conservation laws and exact solutions
David Mason, University of the Witwatersrand . . . . . . . . . . . . . . . . . . . . . . . 16PyClaw: An accessible, extensible, and scalable tool for the solution of wave propagation
problemsAron Ahmadia, Continuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Friday Afternoon II (K303) 15
INITIAL BOUNDARY VALUE PROBLEMS,SUMMATION-BY-PARTS OPERATORS ANDWEAK BOUNDARY CONDITIONS WITHMULTI-PHYSICS APPLICATIONS
Jan Nordstrom (Linkoping University)
During the last decade, stable high order finite differ-ence methods as well as finite volume methods ap-plied to initial-boundary-value-problems have beendeveloped. The stability is due to the use of so-called summation-by-parts operators, penalty tech-niques for implementing boundary and interface con-ditions, and the energy method for proving stability.In this talk we present the theory by using simplemodel problems, and exemplify with complex multi-physics applications. By reusing the main ideas be-hind the development mentioned above, new time-integration procedures and increased accuracy aswell as new types of boundary conditions have beenderived.
THE LINEAR BARYCENTRIC RATIONALQUADRATURE METHOD FOR VOLTERRAINTEGRAL EQUATIONS
Jean-Paul Berrut (Fribourg University)
We shall first introduce linear barycentric rationalinterpolation to the unaware audience: it can beviewed as a small modification of the classical in-terpolating polynomial. Then we present two directquadrature methods based on linear rational interpo-lation for solving general Volterra integral equationsof the second kind. The first, deduced by a directapplication of linear barycentric rational quadraturegiven in former work, is shown to converge at thesame rate, but is costly on long integration intervals.The second, based on a composite version of the ra-tional quadrature rule, looses one order of conver-gence, but is much cheaper. Both require only a sam-ple of the involved functions at equispaced nodes andyield a stable, infinitely smooth solution of most clas-sical examples with machine precision.
COMPUTER-ASSISTED PROOFS BY VER-IFICATION METHODS: RIGOROUS RE-SULTS USING FLOATING-POINT ARITH-METIC
Siegfried Rump (Hamburg University of Technology)
The classical mathematical proof is performed bypencil and paper. However, there are many waysin which computers may be used in a mathemati-cal proof, where, however, “proofs by computers” or
even the use of computers in the course of a proofare not so readily accepted.In this talk we discuss how so-called verificationmethods may assist in achieving a mathematicallyrigorous result. In particular we emphasize howfloating-point arithmetic is used.
STABLE, HIGH ORDER ACCURATE ADAP-TIVE SCHEMES FOR LONG TIME HIGHLYINTERMITTENT GEOPHYSICS PROB-LEMS
Brittany Erickson (San Diego State University)
We use high-order finite difference methods throughsummation-by-parts operators and weak enforce-ment of boundary conditions to derive a provably sta-ble discretization for the elastic wave equation. Timestepping is achieved through the so-called theta-method and the second order BDF which allow usto take large time steps during the intermittent pe-riod between earthquakes. Numerical experimentsillustrate and corroborate the theoretical results.
SUPER RUNGE-KUTTA NYSTROMMETHOD FOR DIRECT INTEGRATIONOF THIRD ORDER ORDINARY DIFFEREN-TIAL EQUATIONS
Zamurat Ayobami Adegboye (Kaduna Polytechnic,Kaduna)
In this work, we extend the Runge-Kutta- Nystrom(RKN) method which is an extension of the Runge-Kutta (RK) method to Super Runge-Kutta- Nystrom(SRKN) method for direct integration of Special andGeneral third order initial valued ODEs via the ideaas those invented by Nystrom.The theory of Nystromwas adopted in the derivation of the method. Theproposed method was tested with Numerical exper-iment to illustrate its efficiency and the method canbe extended to solve higher order differential equa-tions. The scheme is simple to implement and con-verges better with the exact solution.
MHD FLOW OF A SISKO FLUID PAST APOROUS PLATE
Shirley Abelman (University of the Witwatersrand)
We consider magnetohydrodynamic (MHD) flow ofan electrically conducting Sisko fluid past a porousplate in the presence of a constant applied magneticfield. The induced magnetic field is not taken intoaccount by considering a small magnetic Reynoldsnumber. The resulting nonlinear problem is solved
16 Friday 5 April
numerically. Variations of various parameters of in-terest are examined. A comparative study is madebetween the results of viscous and Sisko fluids.
HYDRAULIC FRACTURES: CONSERVA-TION LAWS AND EXACT SOLUTIONS
David Mason (University of the Witwatersrand)
Two conservation laws for the non-linear diffusionequation describing a hydraulic fracture are derived.The Lie point symmetry associated with each conser-vation law is calculated. The exact analytical solutionfor the hydraulic fracture generated by the Lie pointsymmetry is obtained. The analytical solutions areinterpreted physically.
PYCLAW: AN ACCESSIBLE, EXTENSIBLE,AND SCALABLE TOOL FOR THE SOLU-TION OF WAVE PROPAGATION PROB-LEMS
Aron Ahmadia (Continuum)
Development of scientific software involves tradeoffsbetween ease of use, generality, and performance.We review the design of a general hyperbolic PDEsolver using Python that can be operated with theconvenience of high-level MATLAB yet achieves ef-ficiency near that of hand-coded Fortran and scalesto the largest supercomputers. Our approach usesthe petsc4py bindings to PETSc for distributed scal-ability, while simultaneously allowing for several on-node tuning strategies.
Friday 5 April 17
Participants
Surname First name(s) Affiliation E-mail
Abelman Shirley University of the Witwatersrand [email protected] Mark University of Colorado at Boulder [email protected] Zamurat Ayobami Kaduna Polytechnic, Nigeria [email protected] Aron Continuum [email protected] Cory Colorado School of Mines [email protected] Emmanuel Stellenbosch University [email protected] Mapundi Stellenbosch University [email protected] Madelein University of Pretoria [email protected] Jean-Paul Fribourg University [email protected] Susanne Louisiana State University [email protected] Sehun African Institute for Mathematical Sciences [email protected] Lize Stellenbosch University [email protected] Hanno Stellenbosch University [email protected] Jan-at Stellenbosch University [email protected] Brittany San Diego State University [email protected] Adewunmi Gideon University of the Witwatersrand [email protected] Bengt University of Colorado at Boulder [email protected] Melina University of Bath [email protected] Charis University of the Witwatersrand [email protected] Nadine Adelle Stellenbosch University [email protected] Ben Stellenbosch University [email protected] McElory Stellenbosch University [email protected] Ashleigh Wits University [email protected] Byron University of the Witwatersrand [email protected] van Rensburg Nic University of Pretoria [email protected] Abdul Hamid Wits University [email protected] Dirk Stellenbosch University [email protected] Jeanne-marie Stellenbosch University [email protected] Oluwole Daniel Cape Peninsula University of Technology [email protected] Milton Stellenbosch University [email protected] David University of the Witwatersrand [email protected] Belinda Stellenbosch University [email protected] Paul University of Bath [email protected] Michael University of the Witwatersrand [email protected] Joel University of the Witwatersrand [email protected]
18 Friday 5 April
Molati Motlatsi National University of Lesotho/North-West Uni-versity
Muller Neil Stellenbosch University [email protected] Patrick UNIVEN [email protected] Jan Linkoping University [email protected] Terry-Leigh University of the Witwatersrand [email protected] Carel University of Cape Town [email protected] Clodoaldo Universidade de Sao Paulo [email protected] Faraniaina African Institute for Mathematical Sciences [email protected] Pieter Mxit [email protected] Siegfried Hamburg University of Technology [email protected] Melisha University of the Witwatersrand [email protected] Jie Purdue University [email protected] Alastair University of Bath [email protected] Gene TUKS [email protected] Li-yeng Louisiana State University [email protected] Jacques Stellenbosch University [email protected] Mvuyisi Humphrey Stellenbosch University [email protected] Maged Aims [email protected] JAC Stellenbosch University [email protected] der Walt Stefan Stellenbosch University [email protected]
