Adaptive Space-Time Finite Element Methods for Optimization Problems Governed by Nonlinear Parabolic Systems

Subject of this work is the development of concepts for the efficient numerical solution of optimization problems governed by parabolic partial differential equations. Optimization problems of this type arise for instance from the optimal control of physical processes and from the identification of unknown parameters in mathematical models describing such processes. For their numerical treatment, these generically infinite-dimensional optimal control and parameter estimation problems have to be discretized by finite-dimensional approximations. This discretization process causes errors which have to be taken into account to obtain reliable numerical results. Focal point of the thesis at hand is the assessment of these discretization errors by a priori and especially a posteriori error analyses. Thereby, we consider Galerkin finite element discretizations of the state and the control variable in space and time. For the a priori analysis, we concentrate on the case of linear-quadratic optimal control problems. In this configuration, we prove error estimates of optimal order with respect to all involved discretization parameters. The a posteriori error estimation techniques are developed for a general class of nonlinear optimization problems. They provide separated and evaluable estimates for the errors caused by the different parts of the discretization and yield refinement indicators, which can be used for the automatic choice of suitable discrete spaces. The usage of adaptive refinement techniques within a strategy for balancing the several error contributions leads to efficient discretizations for the continuous problems. The presented results and developed concepts are substantiated by various numerical examples including large scale optimization problems motivated by concrete applications from engineering and chemistry

A residual based snapshot location strategy for POD in distributed optimal control of linear parabolic equations

In this paper we study the approximation of a distributed optimal control problem for linear para\-bolic PDEs with model order reduction based on Proper Orthogonal Decomposition (POD-MOR). POD-MOR is a Galerkin approach where the basis functions are obtained upon information contained in time snapshots of the parabolic PDE related to given input data. In the present work we show that for POD-MOR in optimal control of parabolic equations it is important to have knowledge about the controlled system at the right time instances. For the determination of the time instances (snapshot locations) we propose an a-posteriori error control concept which is based on a reformulation of the optimality system of the underlying optimal control problem as a second order in time and fourth order in space elliptic system which is approximated by a space-time finite element method. Finally, we present numerical tests to illustrate our approach and to show the effectiveness of the method in comparison to existing approaches

Dual weighted residual method for laser surface hardening of steel problem

Abstract. The main focus of this article is on the development of Adaptive Finite Element Method (AFEM) for the optimal control problem of laser surface hardening of steel governed by a dynamical system consisting of a semi-linear parabolic equation and an ordinary differential equation using Dual Weighted Residual Method (DWR). A posteriori error estimators using DWR method have been developed when a continuous piecewise linear discretization has been used for the finite element approximation of space variables and a discontinuous Galerkin method has been used for time and control discretizations. Further numerical results obtained are presented are compared with residual method numerical results. Key Words. Laser surface of steel problem, Adaptive finite element methods, Dual weighted residual methods, a posteriori error estimates. 1

Analytic Regularity and GPC Approximation for Control Problems Constrained by Linear Parametric Elliptic and Parabolic PDEs

This paper deals with linear-quadratic optimal control problems constrained by a parametric or stochastic elliptic or parabolic PDE. We address the (difficult) case that the state equation depends on a countable number of parameters i.e., on $\sigma_j$ with $j\in\N$, and that the PDE operator may depend non-affinely on the parameters. We consider tracking-type functionals and distributed as well as boundary controls. Building on recent results in [CDS1, CDS2], we show that the state and the control are analytic as functions depending on these parameters $\sigma_j$. We establish sparsity of generalized polynomial chaos (gpc) expansions of both, state and control, in terms of the stochastic coordinate sequence $\sigma = (\sigma_j)_{j\ge 1}$ of the random inputs, and prove convergence rates of best $N$-term truncations of these expansions. Such truncations are the key for subsequent computations since they do {\em not} assume that the stochastic input data has a finite expansion. In the follow-up paper [KS2], we explain two methods how such best $N$-term truncations can practically be computed, by greedy-type algorithms as in [SG, Gi1], or by multilevel Monte-Carlo methods as in [KSS]. The sparsity result allows in conjunction with adaptive wavelet Galerkin schemes for sparse, adaptive tensor discretizations of control problems constrained by linear elliptic and parabolic PDEs developed in [DK, GK, K], see [KS2]

An adaptive finite element method for laser surface hardening of steel problem

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