@misc{WeiserGhosh, author = {Weiser, Martin and Ghosh, Sunayana}, title = {Theoretically optimal inexact SDC methods}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-53140}, abstract = {In several inital value problems with particularly expensive right hand side computation, there is a trade-off between accuracy and computational effort in evaluating the right hand sides. We consider inexact spectral deferred correction (SDC) methods for solving such non-stiff initial value problems. SDC methods are interpreted as fixed point iterations and, due to their corrective iterative nature, allow to exploit the accuracy-work-tradeoff for a reduction of the total computational effort. On one hand we derive an error model bounding the total error in terms of the right hand side evaluation errors. On the other hand, we define work models describing the computational effort in terms of the evaluation accuracy. Combining both, a theoretically optimal tolerance selection is worked out by minimizing the total work subject to achieving the requested tolerance.}, language = {en} } @misc{LubkollSchielaWeiser, author = {Lubkoll, Lars and Schiela, Anton and Weiser, Martin}, title = {An affine covariant composite step method for optimization with PDEs as equality constraints}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-53954}, abstract = {We propose a composite step method, designed for equality constrained optimization with partial differential equations. Focus is laid on the construction of a globalization scheme, which is based on cubic regularization of the objective and an affine covariant damped Newton method for feasibility. We show finite termination of the inner loop and fast local convergence of the algorithm. We discuss preconditioning strategies for the iterative solution of the arising linear systems with projected conjugate gradient. Numerical results are shown for optimal control problems subject to a nonlinear heat equation and subject to nonlinear elastic equations arising from an implant design problem in craniofacial surgery.}, language = {en} } @misc{WeiserScacchi, author = {Weiser, Martin and Scacchi, Simone}, title = {Spectral Deferred Correction methods for adaptive electro-mechanical coupling in cardiac simulation}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-50695}, abstract = {We investigate spectral deferred correction (SDC) methods for time stepping and their interplay with spatio-temporal adaptivity, applied to the solution of the cardiac electro-mechanical coupling model. This model consists of the Monodomain equations, a reaction-diffusion system modeling the cardiac bioelectrical activity, coupled with a quasi-static mechanical model describing the contraction and relaxation of the cardiac muscle. The numerical approximation of the cardiac electro-mechanical coupling is a challenging multiphysics problem, because it exhibits very different spatial and temporal scales. Therefore, spatio-temporal adaptivity is a promising approach to reduce the computational complexity. SDC methods are simple iterative methods for solving collocation systems. We exploit their flexibility for combining them in various ways with spatio-temporal adaptivity. The accuracy and computational complexity of the resulting methods are studied on some numerical examples.}, language = {en} } @misc{GoetschelvonTycowiczPolthieretal., author = {G{\"o}tschel, Sebastian and von Tycowicz, Christoph and Polthier, Konrad and Weiser, Martin}, title = {Reducing Memory Requirements in Scientific Computing and Optimal Control}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-42695}, abstract = {In high accuracy numerical simulations and optimal control of time-dependent processes, often both many time steps and fine spatial discretizations are needed. Adjoint gradient computation, or post-processing of simulation results, requires the storage of the solution trajectories over the whole time, if necessary together with the adaptively refined spatial grids. In this paper we discuss various techniques to reduce the memory requirements, focusing first on the storage of the solution data, which typically are double precision floating point values. We highlight advantages and disadvantages of the different approaches. Moreover, we present an algorithm for the efficient storage of adaptively refined, hierarchic grids, and the integration with the compressed storage of solution data.}, language = {en} } @misc{GoetschelNagaiahKunischetal., author = {G{\"o}tschel, Sebastian and Nagaiah, Chamakuri and Kunisch, Karl and Weiser, Martin}, title = {Lossy Compression in Optimal Control of Cardiac Defibrillation}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-18566}, abstract = {This paper presents efficient computational techniques for solving an optimization problem in cardiac defibrillation governed by the monodomain equations. Time-dependent electrical currents injected at different spatial positions act as the control. Inexact Newton-CG methods are used, with reduced gradient computation by adjoint solves. In order to reduce the computational complexity, adaptive mesh refinement for state and adjoint equations is performed. To reduce the high storage and bandwidth demand imposed by adjoint gradient and Hessian-vector evaluations, a lossy compression technique for storing trajectory data is applied. An adaptive choice of quantization tolerance based on error estimates is developed in order to ensure convergence. The efficiency of the proposed approach is demonstrated on numerical examples.}, language = {en} } @misc{GoetschelWeiser, author = {G{\"o}tschel, Sebastian and Weiser, Martin}, title = {Lossy Compression for PDE-constrained Optimization: Adaptive Error Control}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-18575}, abstract = {For the solution of optimal control problems governed by nonlinear parabolic PDEs, methods working on the reduced objective functional are often employed to avoid a full spatio-temporal discretization of the problem. The evaluation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the ad-joint equation. The state enters into the adjoint equation, requiring the storage of a full 4D data set. If Newton-CG methods are used, two additional trajectories have to be stored. To get numerical results which are accurate enough, in many case very fine discretizations in time and space are necessary, which leads to a significant amount of data to be stored and transmitted to mass storage. Lossy compression methods were developed to overcome the storage problem by reducing the accuracy of the stored trajectories. The inexact data induces errors in the reduced gradient and reduced Hessian. In this paper, we analyze the influence of such a lossy trajectory compression method on Newton-CG methods for optimal control of parabolic PDEs and design an adaptive strategy for choosing appropriate quantization tolerances.}, language = {en} } @misc{Weiser, author = {Weiser, Martin}, title = {Faster SDC convergence on non-equidistant grids by DIRK sweeps}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-18662}, abstract = {Spectral deferred correction methods for solving stiff ODEs are known to converge rapidly towards the collocation limit solution on equidistant grids, but show a much less favourable contraction on non-equidistant grids such as Radau-IIa points. We interprete SDC methods as fixed point iterations for the collocation system and propose new DIRK-type sweeps for stiff problems based on purely linear algebraic considerations. Good convergence is recovered also on non-equidistant grids. The properties of different variants are explored on a couple of numerical examples.}, language = {en} } @misc{MoualeuNgangueWeiserEhrigetal., author = {Moualeu-Ngangue, Dany Pascal and Weiser, Martin and Ehrig, Rainald and Deuflhard, Peter}, title = {Optimal control for a tuberculosis model with undetected cases in Cameroon}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-43142}, abstract = {This paper considers the optimal control of tuberculosis through education, diagnosis campaign and chemoprophylaxis of latently infected. A mathematical model which includes important components such as undiagnosed infectious, diagnosed infectious, latently infected and lost-sight infectious is formulated. The model combines a frequency dependent and a density dependent force of infection for TB transmission. Through optimal control theory and numerical simulations, a cost-effective balance of two different intervention methods is obtained. Seeking to minimize the amount of money the government spends when tuberculosis remain endemic in the Cameroonian population, Pontryagin's maximum principle is used to characterize the optimal control. The optimality system is derived and solved numerically using the forward-backward sweep method (FBSM). Results provide a framework for designing cost-effective strategies for diseases with multiple intervention methods. It comes out that combining chemoprophylaxis and education, the burden of TB can be reduced by 80 \% in 10 years}, language = {en} } @misc{GoetschelWeiserMaierhoferetal., author = {G{\"o}tschel, Sebastian and Weiser, Martin and Maierhofer, Christiane and Richter, Regina}, title = {Data Enhancement for Active Thermography}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-15243}, abstract = {Pulse thermography is a non-destructive testing method based on infrared imaging of transient thermal patterns. Heating the surface of the structure under test for a short period of time generates a non-stationary temperature distribution and thus a thermal contrast between the defect and the sound material. Due to measurement noise, preprocessing of the experimental data is necessary, before reconstruction algorithms can be applied. We propose a decomposition of the measured temperature into Green's function solutions to eliminate noise.}, language = {en} } @misc{LubkollSchielaWeiser, author = {Lubkoll, Lars and Schiela, Anton and Weiser, Martin}, title = {An optimal control problem in polyconvex hyperelasticity}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-14745}, number = {12-08}, abstract = {We consider a shape implant design problem that arises in the context of facial surgery. We introduce a reformulation as an optimal control problem, where the control acts as a boundary force. The state is modelled as a minimizer of a polyconvex hyperelastic energy functional. We show existence of optimal solutions and derive - on a formal level - first order optimality conditions. Finally, preliminary numerical results are presented.}, language = {en} } @misc{GuentherLameckerWeiser, author = {G{\"u}nther, Andreas and Lamecker, Hans and Weiser, Martin}, title = {Direct LDDMM of Discrete Currents with Adaptive Finite Elements}, doi = {10.1007/s11263-012-0599-3}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-13090}, number = {11-22}, abstract = {We consider Large Deformation Diffeomorphic Metric Mapping of general \$m\$-currents. After stating an optimization algorithm in the function space of admissable morph generating velocity fields, two innovative aspects in this framework are presented and numerically investigated: First, we spatially discretize the velocity field with conforming adaptive finite elements and discuss advantages of this new approach. Second, we directly compute the temporal evolution of discrete \$m\$-current attributes.}, language = {en} } @misc{DeuflhardSchielaWeiser, author = {Deuflhard, Peter and Schiela, Anton and Weiser, Martin}, title = {Mathematical Cancer Therapy Planning in Deep Regional Hyperthermia}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-14204}, number = {11-39}, abstract = {This paper surveys the required mathematics for a typical challenging problem from computational medicine, the cancer therapy planning in deep regional hyperthermia. In the course of many years of close cooperation with clinics, the medical problem gave rise to quite a number of subtle mathematical problems, part of which had been unsolved when the common project started. Efficiency of numerical algorithms, i.e. computational speed and monitored reliability, play a decisive role for the medical treatment. Off-the-shelf software had turned out to be not sufficient to meet the requirements of medicine. Rather, new mathematical theory as well as new numerical algorithms had to be developed. In order to make our algorithms useful in the clinical environment, new visualization software, a virtual lab, including 3D geometry processing of individual virtual patients had to be designed and implemented. Moreover, before the problems could be attacked by numerical algorithms, careful mathematical modelling had to be done. Finally, parameter identification and constrained optimization for the PDEs had to be newly analyzed and realized over the individual patient's geometry. Our new techniques had an impact on the specificity of the individual patients' treatment and on the construction of an improved hyperthermia applicator.}, language = {en} } @misc{WeiserGoetschel, author = {Weiser, Martin and G{\"o}tschel, Sebastian}, title = {State Trajectory Compression for Optimal Control with Parabolic PDEs}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11676}, number = {10-05}, abstract = {In optimal control problems with nonlinear time-dependent 3D PDEs, full 4D discretizations are usually prohibitive due to the storage requirement. For this reason gradient and quasi-Newton methods working on the reduced functional are often employed. The computation of the reduced gradient requires one solve of the state equation forward in time, and one backward solve of the adjoint equation. The state enters into the adjoint equation, again requiring the storage of a full 4D data set. We propose a lossy compression algorithm using an inexact but cheap predictor for the state data, with additional entropy coding of prediction errors. As the data is used inside a discretized, iterative algorithm, lossy coding maintaining an error bound is sufficient.}, language = {en} } @misc{GoetschelWeiserSchiela, author = {G{\"o}tschel, Sebastian and Weiser, Martin and Schiela, Anton}, title = {Solving Optimal Control Problems with the Kaskade 7 Finite Element Toolbox}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11909}, number = {10-25}, abstract = {This paper presents concepts and implementation of the finite element toolbox Kaskade 7, a flexible C++ code for solving elliptic and parabolic PDE systems. Issues such as problem formulation, assembly and adaptivity are discussed at the example of optimal control problems. Trajectory compression for parabolic optimization problems is considered as a case study.}, language = {en} } @misc{Weiser, author = {Weiser, Martin}, title = {On goal-oriented adaptivity for elliptic optimal control problems}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11192}, number = {09-08}, abstract = {The paper proposes goal-oriented error estimation and mesh refinement for optimal control problems with elliptic PDE constraints using the value of the reduced cost functional as quantity of interest. Error representation, hierarchical error estimators, and greedy-style error indicators are derived and compared to their counterparts when using the all-at-once cost functional as quantity of interest. Finally, the efficiency of the error estimator and generated meshes are demonstrated on numerical examples.}, language = {en} } @misc{SchielaWeiser, author = {Schiela, Anton and Weiser, Martin}, title = {Barrier methods for a control problem from hyperthermia treatment planning}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-11567}, number = {09-36}, abstract = {We consider an optimal control problem from hyperthermia treatment planning and its barrier regularization. We derive basic results, which lay the groundwork for the computation of optimal solutions via an interior point path-following method. Further, we report on a numerical implementation of such a method and its performance at an example problem.}, language = {en} } @misc{Weiser, author = {Weiser, Martin}, title = {Optimization and Identification in Regional Hyperthermia}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10916}, number = {08-40}, abstract = {Regional hyperthermia is a cancer therapy aiming at heating tumors using phased array applicators. This article provides an overview over current mathematical challenges of delivering individually optimal treatments. The focus is on therapy planning and identification of technical as well as physiological quantities from MR thermometry measurements.}, language = {en} } @misc{WeiserRoelligArndtetal., author = {Weiser, Martin and R{\"o}llig, Mathias and Arndt, Ralf and Erdmann, Bodo}, title = {Development and test of a numerical model for pulse thermography in civil engineering}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10980}, number = {08-45}, abstract = {Pulse thermography of concrete structures is used in civil engineering for detecting voids, honeycombing and delamination. The physical situation is readily modeled by Fourier's law. Despite the simplicity of the PDE structure, quantitatively realistic numerical 3D simulation faces two major obstacles. First, the short heating pulse induces a thin boundary layer at the heated surface which encapsulates all information and therefore has to be resolved faithfully. Even with adaptive mesh refinement techniques, obtaining useful accuracies requires an unsatisfactorily fine discretization. Second, bulk material parameters and boundary conditions are barely known exactly. We address both issues by a semi-analytic reformulation of the heat transport problem and by parameter identification. Numerical results are compared with measurements of test specimens.}, language = {en} } @misc{Weiser, author = {Weiser, Martin}, title = {Pointwise Nonlinear Scaling for Reaction-Diffusion-Equations}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10493}, number = {07-45}, abstract = {Parabolic reaction-diffusion systems may develop sharp moving reaction fronts which pose a challenge even for adaptive finite element methods. We propose a method to transform the equation into an equivalent form that usually exhibits solutions which are easier to discretize, giving higher accuracy for a given number of degrees of freedom. The transformation is realized as an efficiently computable pointwise nonlinear scaling that is optimized for prototypical planar travelling wave solutions of the underlying reaction-diffusion equation. The gain in either performance or accuracy is demonstrated on different numerical examples.}, language = {en} } @misc{SchenkWaechterWeiser, author = {Schenk, Olaf and W{\"a}chter, Andreas and Weiser, Martin}, title = {Inertia Revealing Preconditioning For Large-Scale Nonconvex Constrained Optimization}, issn = {1438-0064}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-10314}, number = {07-32}, abstract = {Fast nonlinear programming methods following the all-at-once approach usually employ Newton's method for solving linearized Karush-Kuhn-Tucker (KKT) systems. In nonconvex problems, the Newton direction is only guaranteed to be a descent direction if the Hessian of the Lagrange function is positive definite on the nullspace of the active constraints, otherwise some modifications to Newton's method are necessary. This condition can be verified using the signs of the KKT's eigenvalues (inertia), which are usually available from direct solvers for the arising linear saddle point problems. Iterative solvers are mandatory for very large-scale problems, but in general do not provide the inertia. Here we present a preconditioner based on a multilevel incomplete \$LBL^T\$ factorization, from which an approximation of the inertia can be obtained. The suitability of the heuristics for application in optimization methods is verified on an interior point method applied to the CUTE and COPS test problems, on large-scale 3D PDE-constrained optimal control problems, as well as 3D PDE-constrained optimization in biomedical cancer hyperthermia treatment planning. The efficiency of the preconditioner is demonstrated on convex and nonconvex problems with \$150^3\$ state variables and \$150^2\$ control variables, both subject to bound constraints.}, language = {en} }