Robust preconditioners for PDE-constrained optimization with limited\n observations

2015/06/22 by Kent‐André Mardal, Mardal, Kent-André, Bjørn Fredrik Nielsen +3 · 3 citations
Computer Science · Engineering · Mathematics · #65F08 #65K10 #65N21 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.1506.06494

openalex publication_date 2015/06/22 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28

Abstract

Regularization robust preconditioners for PDE-constrained optimization\nproblems have been successfully developed. These methods, however, typically\nassume that observation data is available throughout the entire domain of the\nstate equation. For many inverse problems, this is an unrealistic assumption.\nIn this paper we propose and analyze preconditioners for PDE-constrained\noptimization problems with limited observation data, e.g. observations are only\navailable at the boundary of the solution domain. Our methods are robust with\nrespect to both the regularization parameter and the mesh size. That is, the\ncondition number of the preconditioned optimality system is uniformly bounded,\nindependently of the size of these two parameters. We first consider a\nprototypical elliptic control problem and thereafter more general\nPDE-constrained optimization problems. Our theoretical findings are illuminated\nby several numerical results.\n

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• An Isogeometric Tearing and Interconnecting method for conforming discretizations of the biharmonic problem
• Numerical solution of elliptic distributed optimal control problems with boundary value tracking
• State-based approach to the numerical solution of Dirichlet boundary optimal control problems for the Laplace equation

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