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GAMM-Workshop onComputational Optimization with PDEs
9/10/2014 – 9/12/2014
Dortmund, Germany
Conference Information & Book ofAbstracts
Participants:
• Prof. Thomas Apel, Universitat der Bundeswehr Munchen• Thomas Betz, TU Dortmund• Katharina Bieker, Universitat Paderborn• Dr. Tobias Breiten, Karl-Franzens-Universitat Graz• Regina Dickmann, Universitat Paderborn• Hendrik Feldhordt, Universitat Duisburg-Essen• Dr. Fernando Gaspoz, Universitat Stuttgart• Silke Glas, Universitat Ulm• Dr. Andreas Gunnel, TU Chemnitz• Prof. Roland Herzog, TU Chemnitz• Prof. Christian Meyer, TU Dortmund• Prof. Michael Hinze, Universitat Hamburg• Dr. Andreas Rademacher, TU Dortmund• Dr. Jens Saak, MPI Magdeburg• Ailyn Schafer, TU Chemnitz• Maria Schutte, Universitat Paderborn• Dr. Martin Siebenborn, Universitat Trier• Prof. Thomas Slawig, Christian-Albrechts-Universitat Kiel• Simeon Steinig, TU Dortmund• Dr. Martin Stoll, MPI Magedburg• Livia Susu, TU Dortmund• Oliver Thoma, TU Dortmund• Prof. Stefan Turek, TU Dortmund• Prof. Daniel Wachsmuth, Universitat Wurzburg• Jun-Prof. Winnifried Wollner, Universitat Hamburg
Organizers:
• Prof. Christian Meyer• Prof. Stefan Turek
Important Information:
Map of Conference Location:
• All talks will take place in room E 28 on the ground floor of themath tower• You will find the registration desk in front of E 28• The GAMM-FA meeting will take place in M 614 on the sixth
floor of the math tower
Map of City Center:
Travel Information
Conference location travel information: The university is connected to Dort-mund central station by the S-line S1:
• Dortmund Hbf - Dortmund Universitat: departure times 7.53, 8.04,8.13, 8.33 ... every 20 minutes; travel time 6min• Dortmund Universitat - Dortmund Hbf : departure times 16.19, 16.39,
16.59 ... every 20 minutes; travel time 7min
Stadium tour travel information: The stadium is connected to Dortmund Hbfby U-Bahn:
• Westfallenhallen: U45; departure times 17.27, 17.37 ... every 10 minutes;travel time 10min• Theodor-Fliedner-Heim: U42; departue times 17.26, 17.36 ... every 10
minutes; change at Stadtgarten for U45 to get to Dortmund Hbf; traveltime 10min
Stadium tour participants will be transferred to the stadium from theconference location by car! The departure time will be announced atthe end of the Thursday morning session
General Information:
• Every conference participant will receive a folder containing– public transport tickets for travelling back and forth from the con-
ference location– lunch vouchers for three days– city map– conference fee reciept and confirmation of participation letter– flash drive containing general information and book of abstracts and
tourism guide for Dortmund– notepad and biro
• Internet connection: eduroam is accessible in the conference location
Book of Abstracts
DISCRETIZATION OF DIRICHLET BOUNDARY CONTROLPROBLEMS
T. ApelUniversitat der Bundeswehr Munchen, [email protected]
Several approaches are discussed how to understand the solution of the Dirichletproblem for the Poisson equation when the Dirichlet data are in L2 only. For themethod of transposition (sometimes called very weak formulation) the appropriatespace for the test functions is identified, and a regularity result is given. An ap-proach of Berggren is recovered as the method of transposition. A further idea isthe regularization of the boundary data combined with the weak solution of theregularized problem. The effect of the regularization error is studied.The regularization approach is the simplest to discretize. The discretization erroris estimated for a sequence of quasi-uniform meshes. Since this approach turns outto be equivalent to Berggren’s discretization his error estimates are rendered moreprecisely. Numerical tests show that the error estimates are sharp, in particularthat the order becomes arbitrarily small when the maximal interior angle of thedomain tends to 2π.In the case of non-convex domains one obtains a reduced convergence order. As aremedy, a dual variant of the singular complement method is proposed. The errororder of the convex case is retained. Numerical experiments confirm the theoreticalresults.In the second part of the talk we discuss the optimal control problem. We elaboratethe regularity of the optimal solution in dependence of the largest angle of the do-main. Finally, we review results and expectations of the order of the discretizationerror, again in dependence of the largest angle of the domain.
SECOND-ORDER SUFFICIENT OPTIMALITY CONDITIONS FOROPTIMAL CONTROL OF STATIC ELASTOPLASTICITY WITH
HARDENING
T. Betz and C. MeyerTU Dortmund, [email protected]
An optimal control problem governed by an elliptic variational inequality (VI) isconsidered. This VI models the static problem of infinitesimal elastoplasticity withhardening. It is well known that the control-to-state map associated to VIs is ingeneral not Gateaux-differentiable. Thus standard techniques to derive optimalityconditions cannot be employed. It can however be shown that the control-to-stateoperator associated to elastoplasticity is Bouligand differentiable. Based on thisresult, we establish second-order sufficient optimality conditions.
FEEDBACK CONTROL OF THE MONODOMAIN EQUATIONS
T. Breiten und K. KunischUniversity of Graz, [email protected]
We discuss feedback control strategies for the monodomain equations arising in car-diac electrophysiology. The governing system is described by a semilinear reaction-diffusion equation of Fitz-Hugh-Nagumo type that is coupled to a linear ODE. Ourgoal is to determine a feedback law that locally exponentially stabilizes the systemwith respect to a given stationary solution. We show that for a distributed controlacting on the PDE, it suffices to solve the operator Riccati equation arising for thelinearized decoupled system in order to ensure local exponential stability for thecomplete system.
AFEM FOR DIRICHLET CONTROL-CONSTRAINED OPTIMALCONTROL
F. GaspozUniversitat Stuttgart, [email protected]
The aim of this talk is to present, discuss and analyze the a priori and a posterioriestimation, and convergence for an adaptive finite element method for optimalcontrol problem with constrained Dirichlet control of the form
minu∈Uad
1
2‖y − yd‖2L2(Ω) +
α
2‖u‖2
H12 (∂Ω)
subject to−∆y = f in Ω
y = u on ∂Ω,
and Uad = u ∈ H 12 (∂Ω) : a ≤ u ≤ b.
REDUCED BASIS APPROXIMATION OF NON-COERCIVEVARIATIONAL INEQUALITIES
S. GlasUniversity of Ulm, [email protected]
We consider variational inequalities with different trial and test spaces and a pos-sibly non-coercive bilinear form. Well-posedness is shown under general conditionsthat are e.g. valid for the space-time formulation of parabolic variational inequali-ties. Using space-time formulations, we do not use a time-stepping scheme, but takethe time as an additional variable in the variational formulation of the problem.As an example for a parabolic variational inequality, we consider time-dependentobstacle problems or option pricing, e.g. for American Options.Fine discretizations that are needed for such problems resolve in high dimensionalproblems and thus in long computing times. To reduce the dimensionality of theseproblems, we use the Reduced Basis Method (RBM) [3]. The objective of theRBM is to efficiently reduce discretized parametrized partial differential equations.Problems are considered where not only a single solution is needed but solutionsfor a range of different parameter configurations.In the context of variational inequalities, RBMs have already been applied to theelliptic case [1]. Besides, recent work on American Options combines RBMs withparabolic variational inequalities, but does not provide error estimators, [2]. Indeed,error estimators for parabolic equations could be achieved by linking the RBMwith the space-time formulation [4]. In our work, error estimators in terms of theresidual could be obtained by combining RBM with a space-time formulation ofthe variational inequality. We provide numerical results for a heat inequality modelfocusing on rigorosity and efficiency of the error estimator.
References
[1] B. Haasdonk and J. Salomon and B. Wohlmuth,“A Reduced Basis Method for Parametrized
Variational Inequalities”, SIAM Journal on Numerical Analysis, 50, 2656–2676, (2012).
[2] B. Haasdonk and J. Salomon and B. Wohlmuth “A Reduced Basis Method for the Simulationof American Options”, ENUMATH 2011 Proceedings, (2012).
[3] A. T. Patera, and G. Rozza, “Reduced Basis Approximation and A Posteriori Error Estimation
for Parametrized Partial Differential Equations”, Version 1.0, Copyright MIT, to appear in(tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering (2006).
[4] K. Urban and A. T. Patera,“An Improved Error Bound for Reduced Basis Approximation of
Linear Parabolic Problems”, Mathematics of Computation, S 0025-5718(2013)02782-2, (2013).
OPTIMAL CONTROL OF LARGE DEFORMATION ELASTICITY
A. Gunnel and R. Herzog
TU Chemnitz, [email protected]
Nonlinear models describing large elastic deformations of a solid body Ω usuallyinclude external volume and boundary loads, which are considered dead loads inmost cases. In our presentation we address an alternative model in which internalstresses are caused by a turgor pressure or a tension along a bundle of fibres insidethe body. These differ from classical loads by being internal stresses which enter thebalance of forces in a different way. Still, both can still be modelled as conservativeloads.We present an optimal control problem where the displacement U : Ω→ R3 is thestate and an internal or external load is the control C. The objective functionalI(U,C) = Q(U)+ ||C||L2
consists of a quality term Q for the state U and a penaltyterm for the control C. We show that a standard tracking-type quality term mightbe unsatisfactory for large deformations and give an alternative that is based onthe penalization of certain regions which should be avoided by the deformed body.In order to solve the optimal control problem, we derive formal first-order opti-mality conditions and present two algorithms: a Lagrange-Newton method for anall-at-once approach and a Quasi-Newton method with an inverse BFGS-updateapplied to the reduced system. Both algorithms use a multigrid hierarchy to builda preconditioner for iterative Krylov subspace solvers. Numerical results will begiven.
PRECONDITIONING OF TRUST-REGION SQP METHODS INPDE-CONSTRAINED OPTIMIZATION
R. HerzogTU Chemnitz, [email protected]
Trust-region methods are widely used for the numerical solution of nonlinear con-strained and unconstrained optimization problems. Similar as line-search methods,they take turns in minimizing local models of the objective. In contrast to line-search methods, however, trust-region approaches do not require nor promote thepositive definiteness of the local Hessians or their approximations.The presented sequential quadratic programming method employs a composite-steptrust-region approach [2] to handle the nonlinear (PDE) constraints. Fast iterativesolvers which exploit the saddle point structure of the arising linear systems in theresulting quasi-normal, tangential and multiplier step subproblems are essential [1,3] to cope with the numerical complexity of such problems. We will present analysisand numerical results for challenging applications based on a joint Matlab/FEniCSimplementation.
References
[1] M. Heinkenschloss and D. Ridzal. Integration of sequential quadratic programming and domain
decomposition methods for nonlinear optimal control problems. submitted, 2006.[2] E. Omojokun. Trust region algorithms for optimization with nonlinear equality and inequlity
Constraints. PhD thesis, University of Colorado, Boulder, 1989.[3] D. Ridzal. Trust-Region SQP Methods with Inexact Linear System Solvers for Large-Scale
Optimization. PhD thesis, Rice University, 2006.
RECONSTRUCTION OF MATRIX PARAMETERS FROM NOISYMEASUREMENTS
M. HinzeUniversitat Hamburg, [email protected]
We consider identification of the diffusion matrix in elliptic PDEs from measure-ments. We prove existence of solutions using the concept of H-convergence. Wediscretize the problem using variational discretization and prove Hd-convergenceof the discrete solutions by adapting the concept of Hd-convergence introduced byEymard and Gallouet for finite-volume discretizations to finite element approxi-mations. Furthermore, we prove strong convergence of the discrete coefficients inL2, and of the associated discrete states in the norm of the oberservation space.Finally, assuming a projected source condition we prove error estimates for thediscrete coefficients.
STRONG STATIONARITY FOR OPTIMAL CONTROL OF AVARIATIONAL INEQUALITY OF SECOND KIND
J.C. De Los Reyes and C. MeyerTU Dortmund, [email protected]
The talk is concerned with an optimal control problem governed by a variationalinequality (VI) of the 2nd kind involving the subdifferential of the L1-norm. Thefirst part of the talk deals with the directional derivative of the solution map associ-ated with the VI. Based on a Lipschitz continuity result in L∞, we derive the weakconvergence of the difference quotients to a solution of a VI of 1st kind, providedthat the solution satisfies certain regularity assumptions and the active set fulfillsa rather restrictive structural assumption. In the second part of the talk the direc-tional derivative is used to establish first-order necessary optimality conditions. Wepresent two optimality system, which turn out to be comparable to Bouligand andstrong stationarity conditions known from finite dimensional MPECs. Finally, thedirectional derivative is used to design a trust region algorithm for the numericalsolution of the optimal control problem. Preliminary numerical results show theefficiency of the approach.
ADAPTIVE OPTIMAL CONTROL OF THE OBSTACLE PROBLEM
C. Meyer, A. Rademacher and W. Wollner
TU Dortmund, [email protected]
In this talk, we present an adaptive method for the optimal control of the obsta-cle problem. Due to the inherent indifferentiability of the obstacle problem, theneed to regularize the problem arises in order to apply fast adjoint based solutionalgorithms. By the regularization, an additional error beside the usual finite ele-ment discretization and numerical error is introduced. The aim is to construct anadaptive algorithm balancing the different error contributions based on a poste-riori error estimators, which are derived using the dual weighted residual (DWR)method. In contrast to the estimation of the discretization and the numerical error,which is quite standard, c.f. for instance [1], the estimation of the regularizationerror is more involved. Using the DWR ansatz, a representation of the regular-ization error up to a remainder term of higher order arising from the applicationof the trapezoidal rule is derived, which cannot directly be evaluated. Hence, anumericlal approximation based on extrapolation methods using the derivative ofthe path w.r.t. to regularization parameter is constructed. One advantage of thisapproach is that the numerical effort correspond to the one of a single Newtonstep. The error estimators are finally utilized in an adaptive algorithm balancingthe error contributions. The talk concludes with the application of the presentedalgorithms on a challenging numerical example given in [2] substantiating the goodperformance of the proposed algorithm.
References
[1] R. Rannacher, J. Vihharev: Adaptive fnite element analysis of nonlinear problems: balancing
of discretization and iteration errors, J. Numer. Math., 21, 23–62, 2013[2] C. Meyer, O. Thoma: Finite element error analysis for optimal control of the obstacle problem,
SIAM J. Numer. Analysis, 51(1), 605–628, 2013
ON AN INEXACT NEWTON-ADI SOLVER FOR ALGEBRAICRICCATI EQUATIONS RELATED TO THE LQR PROBLEM FOR
LINEARIZED NAVIER-STOKES EQUATIONS
P. Benner, M. Heinkenschloss, J. Saak and H. WeicheltMPI Magdeburg, [email protected]
We consider the numerical solution of the generalized algebraic Riccati equation(GARE)
0 = CTC +ATXM+MXA−MXBBTXM =: R(X).(1)
Here
M := ΠMΠT , A := ΠAΠT , B := ΠB and C := CΠT(2)
result from the linear quadratic regulator problem minimizing
J (z(t),u(t)) :=1
2
∫ ∞0
(z(t)TCTCz(t)
)+ u(t)Tu(t) dt
with respect to the linearized Navier Stokes system
Md
dtz(t) = Az(t) +Gp(t) +Bu(t),(3a)
0 = GT z(t),(3b)
y(t) = Cz(t).(3c)
Furthermore ΠT := I −M−TG(GTM−1G)−1GT represents the discretized Lerayprojection under which we expect the solution to be invariant, i.e., X = ΠTXΠ.In our context ΠT is acting as the projection onto the hidden manifold of thedifferential algebraic system (3).Our aim is an iterative solver for (1) based on the Newton-ADI method. Featuring atriple nested loop of Newton-, low-rank ADI- and iterative solver processes, in orderto achieve short runtimes, the overall solver requires an accuracy control for theinner loops. Here, we present an inexact Newton method exploiting the structureof the matrices in (2) and low-rank representations for the Lyapunov and Riccatiresiduals to allow efficient solution and to control the accuracy of the ADI iteration.The open problem of controlling the accuracy of the solver for the shifted linearsystems in saddle point form appearing in each ADI step will also be addressedshortly.
OPTIMAL CONTROL PROBLEMS IN THERMOPLASTICITY
R. Herzog, C. Meyer and A. SchaferTU Chemnitz, [email protected]
Elastoplastic deformations play a tremendous role in industrial forming. Many ofthese processes happen at non-isothermal conditions. Therefore, the optimizationof such problems is of interest not only mathematically but also for applications.In this talk we will motivate and develop a model of thermoviscoelastoplasticityand present first steps in its analysis. We will point out the difficulties arising fromthe nonlinear coupling of the heat equation with the mechanical part of the model.An outlook on future work with regard to optimization will be given.The talk is based on joint work with Roland Herzog and Christian Meyer.
ON THE IDENTIFICATION OF HIDDEN GEOMETRIC OBJECTSUSING SHAPE DERIVATIVES
S. Schmidt, M. Schutte and A. Walther
Universitat Paderborn, [email protected]
In many industrial fields such as biology and aircraft, the identification of hiddengeometric objects plays an important role. In a few cases, one has a comprehensiveknowledge of the form and the structure of the object. There it is possible to usee.g. an indicator function to identify the position of an object.But usually this entire information is not known by the user. Even if the userhas a concrete idea of the form of the object, exterior influcences might have de-formed the object slightly. In order to perform an accurate identification, a moresophisticated approach has to be applied. This problem can be formulated as anshape optimization problem. The target function is to minimize the L2-norm of thesimulated and real measured data. Starting with an initial guess for the geometricobject, the forward simulation is performed.Afterwards the shape gradient, where also the adjoint equation takes part in, willbe computed. Similar to the method of steepest descent, the boundary of the cur-rent object will be modified until a convergence criterion is reached. In this talk, atheoretical analysis of the Maxwell’s equations will be shown as well as numericalresults, computed by the software tool FEniCS.
STRUCTURED INVERSE MODELING IN DIFFUSIVEPROCESSES
M. Siebenborn and V. SchulzUniversitat Trier, [email protected]
Recent studies have shown advantages of the application of second order derivativesand Hessian approximations in PDE constrained shape optimization. This talkintroduces a limited memory BFGS approach for shape optimization in diffusiveflow processes. The standard procedure is to fit a parabolic model to observeddata by optimizing with respect to a distributed diffusion coefficient. Here it isdemonstrated how this fitting can be improved by also treating the shape of theparameter distribution as a variable. It is shown that superlinear convergence canbe achieved which is a significant speedup against optimizations based only onshape gradients. These techniques are utilized in order to fit a model of the humanskin to data measurements and not only estimate the permeability coefficients butalso the shape of the cells.
PARAMETER ESTIMATION IN MARINE ECOSYSTEM MODELS
T. Slawig
Christian-Albrechts-Universitat Kiel, [email protected]
We discuss parameter estimation problems in 3-D marine ecosystem models. Thesemodels are used to enclose the carbon cycle in global climate simulations. Forthese models, parameters are usually estimated or optimized for a stable annuallyperiodic solution before they are used in transient, prognostic runs. Simulation ofsuch a stable annual cycle in the coupled system of ocean circulation and marineecosystem is an iterative procedure that is computationally very costly. Naturally,these high computational costs are growing when simulation-based parameter opti-mization or model calibration runs are necessary. Each optimization run may needseveral hundreds of function evaluations. As a consequence, methods to reducethe computational effort in both simulation and optimization are highly desirable.After a mathematical discussion of a typical ecosystem model w.r.t. existence anduniqueness of solutions, we present two optimization methods and their numericalresults. The first one is based on an Lagrange multiplier approach and simultane-ously iterates the state and optimizes the parameters. The second method is basedon surrogate models. In this method, the original and computationally expensivefine model is replaced by a so-called surrogate, which is created from a less accuratebut computationally cheaper coarse model with a additional correction approach.
RELIABLE A POSTERIORI ERROR ESTIMATION FORSTATE-CONSTRAINED OPTIMAL CONTROL PROBLEMS
S. Steinig
TU Dortmund, [email protected]
In this talk we will present reliable a posteriori error estimator for a finite elementdiscretisation of an elliptic state-constrained optimal control problem. The esti-mator us guaranteed to converge under certain modest assumptions. A modifiedmaximum strategy will be introduced guiding the adaptive algorithm. Besides,some numerical examples will be shown demonstrating the merit of our approach.
A LOW-RANK IN TIME APPROACH TO PDE-CONSTRAINEDOPTIMIZATION
M. StollMPI Magedburg, [email protected]
A possible drawback of all-at-once discretizations of time-dependent PDE-constrainedoptimization problems is the high storage demand for the vectors representing thediscrete space-time cylinder. We here introduce a low-rank in time technique thatexploits the low-rank nature of the solution. The theoretical foundations for this ap-proach originate in the numerical treatment of matrix equations and can be carriedover to PDE-constrained optimization.We illustrate how three different problemscan be rewritten and used within a low-rank Krylov subspace solver with appropri-ate preconditioning.
OPTIMAL CONTROL OF PARTICLE ACCELERATORS
C. Meyer, S. Schnepp and O. ThomaTU Dortmund, [email protected]
The talk deals with the optimal control of particle accelerators by means of ex-terior magnetic fields. The forward problem is modeled by a nonlinearly coupledsystem consisting of the instationary Maxwell’s equations, an ODE for the relativis-tic particle dynamics, and an additional elliptic equation for the scalar magneticpotential. The control enters the problem via the Dirichlet data in the elliptic equa-tion. First-order necessary optimality conditions and preliminary numerical resultsare presented.
A NUMERICAL STUDY OF HIERARCHICAL SOLUTIONCONCEPTS FOR FLOW CONTROL PROBLEMS
S. Turek
TU Dortmund, [email protected]
NONSTATIONARY OPTIMIZATION PROBLEMS WITHGRADIENT CONSTRAINTS
F. Ludovici and W. WollnerUniversitat Hamburg, [email protected]
In this talk, we will consider the discretization on parabolic optimization problemsubject to pointwise inequality constraints on the gradient of the solution to theparabolic PDE.The work presented is focused on the consideration of pointwise in time but averagedin space constraints on the gradient of the solution as it is motivated by constraintson average stresses. Within this setting we show convergence rates in terms oftemporal and spatial mesh size.
CONFERENCE PROGRAM
Wed 9/10 Thu 9/11 Fri 9/12
9.00-10.00Registration (in front of E 28)
(commences 9.15)
T. ApelDiscretization of Dirichlet Boundary
Control Problems
M. HinzeReconstruction of Matrix Parameters
from Noisy Measurements
10.00-10.30 Opening Coffee Break
10.30-11.00W. Wollner
Nonstationary Optimization Problemswith Gradient Constraints
F. GaspozAFEM for Dirichlet
Control-Constrained Optimal Control
A. GunnelOptimal Control of Large
Deformation Elasticity
11.00-11.30O. Thoma
Optimal Control of a Particle
Accelerator
A. RademacherAdaptive Optimal Control of the
Obstacle Problem
A. SchaferOptimal Control Problems in
Thermoplasticity
11.30-12.00S. Glas
Reduced Basis Approximation of
Non-Coercive Variational Inequalities
S. SteinigReliable A Posteriori Error
Estimation for State-Constrained
Optimal Control Problems
T. BetzSecond-Order Sufficient OptimalityConditions for Optimal Control of
Static Elastoplasticity with Hardening
12.00-12.30C. Meyer
Strong Stationarity for Optimal
Control of a VI of 2nd Kind
S. TurekA Numerical Study of Hierarchical
Solution Concepts for Flow ControlProblems
T. BreitenFeedback Control of the Monodomain
Equations
12.30-13.30 Lunch
13.30-14.30
R. HerzogPreconditioning of Trust-Region SQP
Methods in PDE-ConstrainedOptimization
T. SlawigParameter Estimation in Marin
Ecosystem Models
14.30-15.00
GAMM-FA-Meeting
(Room M 614)
M. StollA Low-Rank in Time Approach to
PDE-Constrained Optimization
M. SchutteOn the Identification of Hidden
Geometric Objects Using ShapeDerivatives
15.00-15.30
J. SaakOn an Inexact Newton-ADI Solver for
Algebraic Riccati Equations Relatedto the LQR Problem for Linearized
Navier-Stokes Equations
M. SiebenbornStructured Inverse Modeling in
Diffusive Processes
15.30-16.00
Stadium Tour
Closing
16.00-18.00
