An adaptive finite element method for distributed elliptic optimal control problems with variable energy regularization

Langer, Ulrich; Löscher, Richard; Steinbach, Olaf; Yang, Huidong

doi:10.1016/j.camwa.2024.02.006

Cited by 4 publications

References 12 publications

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Mass‐lumping discretization and solvers for distributed elliptic optimal control problems

Langer,

Löscher,

Steinbach

et al. 2024

Numerical Linear Algebra App

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The purpose of this article is to investigate the effects of the use of mass‐lumping in the finite element discretization with mesh size of the reduced first‐order optimality system arising from a standard tracking‐type, distributed elliptic optimal control problem with regularization, involving a regularization (cost) parameter on which the solution depends. We show that mass‐lumping will not affect the error between the desired state and the computed finite element state , but will lead to a Schur‐complement system that allows for a fast matrix‐by‐vector multiplication. We show that the use of the Schur‐complement preconditioned conjugate gradient method in a nested iteration setting leads to an asymptotically optimal solver with respect to the complexity. While the proposed approach is applicable independently of the regularity of the given target, our particular interest is in discontinuous desired states that do not belong to the state space. However, the corresponding control belongs to whereas the cost as . This motivates to use in order to balance the error and the maximal costs we are willing to accept. This can be embedded into a nested iteration process on a sequence of refined finite element meshes in order to control the error and the cost simultaneously.

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Mass‐lumping discretization and solvers for distributed elliptic optimal control problems

Langer,

Löscher,

Steinbach

et al. 2024

Numerical Linear Algebra App

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Robust Iterative Solvers for Algebraic Systems Arising from Elliptic Optimal Control Problems

Langer,

Löscher,

Steinbach

et al. 2024

Lecture Notes in Computer Science

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Robust space-time finite element methods for parabolic distributed optimal control problems with energy regularization

Langer,

Steinbach,

Yang

2024

Adv Comput Math

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As in our previous work (SINUM 59(2):660–674, 2021) we consider space-time tracking optimal control problems for linear parabolic initial boundary value problems that are given in the space-time cylinder $$Q = \Omega \times (0,T)$$ Q = Ω × ( 0 , T ) , and that are controlled by the right-hand side $$z_\varrho $$ z ϱ from the Bochner space $$L^2(0,T;H^{-1}(\Omega ))$$ L 2 ( 0 , T ; H - 1 ( Ω ) ) . So it is natural to replace the usual $$L^2(Q)$$ L 2 ( Q ) norm regularization by the energy regularization in the $$L^2(0,T;H^{-1}(\Omega ))$$ L 2 ( 0 , T ; H - 1 ( Ω ) ) norm. We derive new a priori estimates for the error $$\Vert \widetilde{u}_{\varrho h} - \overline{u}\Vert _{L^2(Q)}$$ ‖ u ~ ϱ h - u ¯ ‖ L 2 ( Q ) between the computed state $$\widetilde{u}_{\varrho h}$$ u ~ ϱ h and the desired state $$\overline{u}$$ u ¯ in terms of the regularization parameter $$\varrho $$ ϱ and the space-time finite element mesh size h, and depending on the regularity of the desired state $$\overline{u}$$ u ¯ . These new estimates lead to the optimal choice $$\varrho = h^2$$ ϱ = h 2 . The approximate state $$\widetilde{u}_{\varrho h}$$ u ~ ϱ h is computed by means of a space-time finite element method using piecewise linear and continuous basis functions on completely unstructured simplicial meshes for Q. The theoretical results are quantitatively illustrated by a series of numerical examples in two and three space dimensions. We also provide performance studies for different solvers.

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An adaptive finite element method for distributed elliptic optimal control problems with variable energy regularization

Cited by 4 publications

References 12 publications

Mass‐lumping discretization and solvers for distributed elliptic optimal control problems

Mass‐lumping discretization and solvers for distributed elliptic optimal control problems

Robust Iterative Solvers for Algebraic Systems Arising from Elliptic Optimal Control Problems

Robust space-time finite element methods for parabolic distributed optimal control problems with energy regularization

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