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

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

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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

Functional a posteriori error estimates for time-periodic parabolic optimal control problems

This paper is devoted to the a posteriori error analysis of multiharmonic finite element approximations to distributed optimal control problems with time-periodic state equations of parabolic type. We derive a posteriori estimates of functional type, which are easily computable and provide guaranteed upper bounds for the state and co-state errors as well as for the cost functional. These theoretical results are confirmed by several numerical tests that show high efficiency of the a posteriori error bounds

Optimal Control of Convective FitzHugh-Nagumo Equation

We investigate smooth and sparse optimal control problems for convective FitzHugh-Nagumo equation with travelling wave solutions in moving excitable media. The cost function includes distributed space-time and terminal observations or targets. The state and adjoint equations are discretized in space by symmetric interior point Galerkin (SIPG) method and by backward Euler method in time. Several numerical results are presented for the control of the travelling waves. We also show numerically the validity of the second order optimality conditions for the local solutions of the sparse optimal control problem for vanishing Tikhonov regularization parameter. Further, we estimate the distance between the discrete control and associated local optima numerically by the help of the perturbation method and the smallest eigenvalue of the reduced Hessian

Reduced Order Optimal Control of the Convective FitzHugh-Nagumo Equation

In this paper, we compare three model order reduction methods: the proper orthogonal decomposition (POD), discrete empirical interpolation method (DEIM) and dynamic mode decomposition (DMD) for the optimal control of the convective FitzHugh-Nagumo (FHN) equations. The convective FHN equations consists of the semi-linear activator and the linear inhibitor equations, modeling blood coagulation in moving excitable media. The semilinear activator equation leads to a non-convex optimal control problem (OCP). The most commonly used method in reduced optimal control is POD. We use DEIM and DMD to approximate efficiently the nonlinear terms in reduced order models. We compare the accuracy and computational times of three reduced-order optimal control solutions with the full order discontinuous Galerkin finite element solution of the convection dominated FHN equations with terminal controls. Numerical results show that POD is the most accurate whereas POD-DMD is the fastest