2020/03/31 by Ulrich Langer, Langer, Ulrich, Olaf Steinbach +5
Computer Science · Engineering · Mathematics · #35K20 #49J20 #65M15 #65M50 #65M60 #65Y05 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2003.14141
openalex publication_date 2020/03/31 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28
We consider a space-time finite element method on fully unstructured simplicial meshes for optimal sparse control of semilinear parabolic equations. The objective is a combination of a standard quadratic tracking-type functional including a Tikhonov regularization term and of the L1-norm of the control that accounts for its spatio-temporal sparsity. We use a space-time Petrov-Galerkin finite element discretization for the first-order necessary optimality system of the associated discrete optimal sparse control problem. The discretization is based on a variational formulation that employs piecewise linear finite elements simultaneously in space and time. Finally, the discrete nonlinear optimality system that consists of coupled forward-backward state and adjoint state equations is solved by a semismooth Newton method.