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28
Directional sparsity in optimal control of partial differential equations
 SIAM J. Control Optim
, 2012
"... Abstract. We study optimal control problems in which controls with certain sparsity patterns are preferred. For timedependent problems the approach can be used to find locations for control devices that allow controlling the system in an optimal way over the entire time interval. The approach uses ..."
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Abstract. We study optimal control problems in which controls with certain sparsity patterns are preferred. For timedependent problems the approach can be used to find locations for control devices that allow controlling the system in an optimal way over the entire time interval. The approach uses a nondifferentiable cost functional to implement the sparsity requirements; additionally, bound constraints for the optimal controls can be included. We study the resulting problem in appropriate function spaces and present two solution methods of Newton type, based on different formulations of the optimality system. Using elliptic and parabolic test problems we research the sparsity properties of the optimal controls and analyze the behavior of the proposed solution algorithms.
Regularizationrobust preconditioners for timedependent PDE constrained optimization problems
, 2011
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http://www.am.unierlangen.de/home/spp1253 CrankNicolson schemes for optimal control problems
, 2010
"... with evolution equations ..."
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Optimal control with stochastic PDE constraints and uncertain controls
 Comput. Methods Appl. Mech. Engrg
"... The optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed. The Lagrangian that defines the problem is postulated in terms of stochastic functions, with the control function possibly decomposed into an unknown d ..."
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The optimal control of problems that are constrained by partial differential equations with uncertainties and with uncertain controls is addressed. The Lagrangian that defines the problem is postulated in terms of stochastic functions, with the control function possibly decomposed into an unknown deterministic component and a known zeromean stochastic component. The extra freedom provided by the stochastic dimension in defining cost functionals is explored, demonstrating the scope for controlling statistical aspects of the system response. Oneshot stochastic finite element methods are used to find approximate solutions to control problems. It is shown that applying the stochastic collocation finite element method to the formulated problem leads to a coupling between stochastic collocation points when a deterministic optimal control is considered or when moments are included in the cost functional, thereby forgoing the primary advantage of the collocation method over the stochastic Galerkin method for the considered problem. The application of the presented methods is demonstrated through a number of numerical examples. The presented framework is sufficiently general to also consider a class of inverse problems, and numerical examples of this type are also presented.
EFFICIENT PRECONDITIONERS FOR OPTIMALITY SYSTEMS ARISING IN CONNECTION WITH INVERSE PROBLEMS
, 2010
"... This paper is devoted to the numerical treatment of linear optimality systems (OS) arising in connection with inverse problems for partial differential equations. If such inverse problems are regularized by Tikhonov regularization, then it follows from standard theory that the associated OS is wel ..."
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Cited by 4 (3 self)
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This paper is devoted to the numerical treatment of linear optimality systems (OS) arising in connection with inverse problems for partial differential equations. If such inverse problems are regularized by Tikhonov regularization, then it follows from standard theory that the associated OS is wellposed, provided that the regularization parameter α is positive and that the involved state equation satisfies suitable assumptions. We explain and analyze how certain mapping properties of the operators appearing in the OS can be employed to define efficient preconditioners for finite element (FE) approximations of such systems. The key feature of the scheme is that the numberof iterations needed to solve the preconditioned problem by the minimal residual method is bounded independentlyof the mesh parameter h, used in the FE discretization, and only increases moderately as α → 0. More specifically, if the stopping criterion for the iteration process is defined in terms of the associated energy norm, then the number of iterations required (in the severely illposed case) cannot grow faster than O((ln(α)) 2). Our analysis is based on a careful study of the involved operators which yields the distribution of the eigenvalues of the preconditioned OS. Finally, the theoretical results are illuminated by a number of numerical experiments addressing both a model problem studied by Borzi, Kunisch and Kwak [14] and an inverse problem arising in connection with electrocardiography [41].
FAST ITERATIVE SOLUTION OF REACTIONDIFFUSION CONTROL PROBLEMS ARISING FROM CHEMICAL PROCESSES
, 2012
"... PDEconstrained optimization problems, and the development of preconditioned iterative methods for the efficient solution of the arising matrix system, is a field of numerical analysis that has recently been attracting much attention. In this paper, we analyze and develop preconditioners for matri ..."
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Cited by 3 (2 self)
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PDEconstrained optimization problems, and the development of preconditioned iterative methods for the efficient solution of the arising matrix system, is a field of numerical analysis that has recently been attracting much attention. In this paper, we analyze and develop preconditioners for matrix systems that arise from the optimal control of reactiondiffusion equations, which themselves result from chemical processes. Important aspects in our solvers are saddle point theory, mass matrix representation and effective Schur complement approximation, as well as the outer (Newton) iteration to take account of the nonlinearity of the underlying PDEs.
Multigrid preconditioning of linear systems for semismooth Newton methods
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ANALYSIS OF THE MINIMAL RESIDUAL METHOD APPLIED TO ILLPOSED OPTIMALITY SYSTEMS
, 2012
"... We analyze the performance of the Minimal Residual Method applied to linear KarushKuhnTucker systems arising in connection with inverse problems. Such optimality systems typically have a saddle point structure and have unique solutions for all α> 0, where α is the parameter employed in the Tik ..."
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We analyze the performance of the Minimal Residual Method applied to linear KarushKuhnTucker systems arising in connection with inverse problems. Such optimality systems typically have a saddle point structure and have unique solutions for all α> 0, where α is the parameter employed in the Tikhonov regularization. Unfortunately, the associated spectral condition number is very large for small values of α, which strongly indicates that their numerical treatment is difficult. Our main result shows that a broad range of linear ill posed optimality systems can be solved with a number of iterations of order O(ln(α −1)). More precisely, in the severely ill posed case the number of iterations needed by the Minimal Residual Method cannot grow faster than O(ln(α −1)) as α → 0. This result is obtained by carefully analyzing the spectrum of the associated saddle point operator: Except for a few isolated eigenvalues, the spectrum consists of bounded intervals. Krylov subspace methods handle such problems very well. We illuminate our theoretical findings with some numerical results for inverse problems involving partial differential equations. Our investigation is inspired by Prof. H. Egger’s discussion of similar results valid for the conjugate gradient algorithm applied to the normal equations.
Operator preconditioning for a class of inequality constrained optimal control problems
"... Abstract We propose and analyze two strategies for preconditioning linear operator equations that arise in PDE constrained optimal control in the framework of conjugate gradient methods. Our particular focus is on control or state constrained problems, where we consider the question of robustness w ..."
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Abstract We propose and analyze two strategies for preconditioning linear operator equations that arise in PDE constrained optimal control in the framework of conjugate gradient methods. Our particular focus is on control or state constrained problems, where we consider the question of robustness with respect to critical parameters. We construct a preconditioner that yields favorable robustness properties with respect to critical parameters.