Space-Time Finite Element Discretization of Parabolic Optimal Control Problems with Energy Regularization
09 Mar 2021-SIAM Journal on Numerical Analysis (Society for Industrial and Applied Mathematics Publications)-Vol. 59, Iss: 2, pp 675-695
TL;DR: It is emphasized that the energy regularization results in a more localized control with sharper contours for discontinuous target functions, which is demonstrated by a comparison with an $L^2$ regularization and with a sparse optimal control approach.
Abstract: In this paper, we analyze space-time finite element methods for the numerical solution of distributed parabolic optimal control problems with energy regularization in the Bochner space $L^2(0,T;H^{...
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TL;DR: In this article , the authors consider space-time tracking optimal control problems for linear parabolic initial boundary value problems in the Bochner space, and derive a priori estimates for the error (cid:107) u (cID:37)h − u (Cid: 107) L 2 (Q ) between the computed state ( cid:101) u(cid;37) h and the desired state u in terms of the regularization parameter, depending on the space time finite element mesh-size h , and depending upon the regularity of the desired states u .
Abstract: We consider space-time tracking optimal control problems for linear parabolic initial boundary value problems that are given in the space-time cylinder Q = Ω × (0 , T ), and that are controlled by the right-hand side z (cid:37) from the Bochner space L 2 (0 , T ; H − 1 (Ω)). So it is natural to replace the usual L 2 ( Q ) norm regularization by the energy regularization in the L 2 (0 , T ; H − 1 (Ω)) norm. We derive a priori estimates for the error (cid:107) (cid:101) u (cid:37)h − u (cid:107) L 2 ( Q ) between the computed state (cid:101) u (cid:37)h and the desired state u in terms of the regularization parameter (cid:37) and the space-time finite element mesh-size h , and depending on the regularity of the desired state u . These estimates lead to the optimal choice (cid:37) = h 2 . The approximate state (cid:101) u (cid:37)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.
5 citations
TL;DR: In this paper , the authors consider space-time tracking type distributed optimal control problems for the wave equation in the space time domain, where the control is assumed to be in the energy space [H 1 , 1 0; , 0 ( Q )] ∗ , rather than in L 2 ( Q ) which is more common.
Abstract: We consider space-time tracking type distributed optimal control problems for the wave equation in the space-time domain Q := Ω × (0 , T ) ⊂ R n +1 , where the control is assumed to be in the energy space [ H 1 , 1 0; , 0 ( Q )] ∗ , rather than in L 2 ( Q ) which is more common. While the latter ensures a unique state in the Sobolev space H 1 , 1 0;0 , ( Q ), this does not define a solution isomorphism. Hence we use an appropriate state space X such that the wave operator becomes an isomorphism from X onto [ H 1 , 1 0; , 0 ( Q )] ∗ . Using space-time finite element spaces of piecewise linear continuous basis functions on completely unstructured but shape regular simplicial meshes, we derive a priori estimates for the error (cid:107) (cid:101) u (cid:37)h − u (cid:107) L 2 ( Q ) between the computed space-time finite element solution (cid:101) u (cid:37)h and the target function u with respect to the regularization parameter (cid:37) , and the space-time finite element mesh-size h , depending on the regularity of the desired state u . These estimates lead to the optimal choice (cid:37) = h 2 in order to define the regularization parameter (cid:37) for a given space-time finite element mesh size h , or to determine the required mesh size h when (cid:37) is a given constant representing the costs of the control. The theoretical results will be supported by numerical examples with targets of different regularities, including discontinuous targets. Furthermore, an adaptive space-time finite element scheme is proposed and numerically analyzed.
2 citations
TL;DR: In this paper , a priori discretization error estimates in terms of the local mesh-sizes for shape-regular meshes are derived for the numerical solution of space-time tracking parabolic optimal control problems with the standard L 2 -regularization.
Abstract: Abstract We present, analyze, and test locally stabilized space–time finite element methods on fully unstructured simplicial space–time meshes for the numerical solution of space–time tracking parabolic optimal control problems with the standard L 2 -regularization.We derive a priori discretization error estimates in terms of the local mesh-sizes for shape-regular meshes. The adaptive version is driven by local residual error indicators, or, alternatively, by local error indicators derived from a new functional a posteriori error estimator. The latter provides a guaranteed upper bound of the error, but is more costly than the residual error indicators. We perform numerical tests for benchmark examples having different features. In particular, we consider a discontinuous target in form of a first expanding and then contracting ball in 3d that is fixed in the 4d space– time cylinder.
2 citations
01 Jan 2014
TL;DR: In this article, the authors consider a non-autonomous evolutionary problem and investigate whether u ∈ H 1 ( 0, T ; H ) and give a positive answer if the form is of bounded variation.
Abstract: We consider a non-autonomous evolutionary problem u ′ ( t ) + A ( t ) u ( t ) = f ( t ) , u ( 0 ) = u 0 , where V , H are Hilbert spaces such that V is continuously and densely embedded in H and the operator A ( t ) : V → V ′ is associated with a coercive, bounded, symmetric form a ( t , . , . ) : V × V → C for all t ∈ [ 0 , T ] . Given f ∈ L 2 ( 0 , T ; H ) , u 0 ∈ V there exists always a unique solution u ∈ MR ( V , V ′ ) : = L 2 ( 0 , T ; V ) ∩ H 1 ( 0 , T ; V ′ ) . The purpose of this article is to investigate whether u ∈ H 1 ( 0 , T ; H ) . This property of maximal regularity in H is not known in general. We give a positive answer if the form is of bounded variation; i.e., if there exists a bounded and non-decreasing function g : [ 0 , T ] → R such that | a ( t , v , w ) − a ( s , v , w ) | ≤ [ g ( t ) − g ( s ) ] ‖ v ‖ V ‖ w ‖ V ( 0 ≤ s ≤ t ≤ T , v , w ∈ V ) . In that case, we also show that u ( . ) is continuous with values in V. Moreover we extend this result to certain perturbations of A ( t ) .
2 citations
01 Jan 2022
TL;DR: In this paper , stabilized space-time finite element methods on fully unstructured simplicial space time meshes for the numerical solution of parabolic optimal control problems with the standard $$L_2$$ -regularization are presented.
Abstract: This work presents, analyzes and tests stabilized space-time finite element methods on fully unstructured simplicial space-time meshes for the numerical solution of space-time tracking parabolic optimal control problems with the standard $$L_2$$ -regularization.
2 citations
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"Space-Time Finite Element Discretiz..." refers methods in this paper
...Hence, we can apply the Nec̆as-Babuška theorem [2, 24] to conclude that the variational problem (5) is well-posed, see also [3, 6, 8]....
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TL;DR: This paper generalizes the result concerning optimality of the adaptive finite element method to general space dimensions by extending it to bisection algorithms of n-simplices.
Abstract: Recently, in [Found. Comput. Math., 7(2) (2007), 245-269], we proved that an adaptive finite element method based on newest vertex bisection in two space dimensions for solving elliptic equations, which is essentially the method from [SINUM, 38 (2000), 466-488] by Morin, Nochetto, and Siebert, converges with the optimal rate. The number of triangles N in the output partition of such a method is generally larger than the number M of triangles that in all intermediate partitions have been marked for bisection, because additional bisections are needed to retain conforming meshes. A key ingredient to our proof was a result from [Numer. Math., 97(2004), 219-268] by Binev, Dahmen and DeVore saying that N - N-0 <= CM for some absolute constant C, where N-0 is the number of triangles from the initial partition that have never been bisected. In this paper, we extend this result to bisection algorithms of n-simplices, with that generalizing the result concerning optimality of the adaptive finite element method to general space dimensions.
380 citations
"Space-Time Finite Element Discretiz..." refers methods in this paper
...Following a bisection approach [30], we perform a sequence of uniform refinements of the initial mesh....
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TL;DR: This work defines somelocal regular and irregular refinement rules that are applied to single elements and describes how these local rules can be combined and rearranged in order to ensure consistency as well as stability.
Abstract: We present a refinement algorithm for unstructured tetrahedral grids which generates possibly highly non-uniform but nevertheless consistent (closed) and stable triangulations. Therefore we first define somelocal regular and irregular refinement rules that are applied to single elements. Theglobal refinement algorithm then describes how these local rules can be combined and rearranged in order to ensure consistency as well as stability. It is given in a rather general form and includes also grid coarsening.
269 citations
Additional excerpts
...By a uniform red-green refinement [4], we reduce the mesh size recursively, i....
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TL;DR: A priori error analysis for Galerkin finite element discretizations of optimal control problems governed by linear parabolic equations and error estimates of optimal order with respect to both space and time discretization parameters are developed.
Abstract: In this paper we develop a priori error analysis for Galerkin finite element discretizations of optimal control problems governed by linear parabolic equations. The space discretization of the state variable is done using usual conforming finite elements, whereas the time discretization is based on discontinuous Galerkin methods. For different types of control discretizations we provide error estimates of optimal order with respect to both space and time discretization parameters. The paper is divided into two parts. In the first part we develop some stability and error estimates for space-time discretization of the state equation and provide error estimates for optimal control problems without control constraints. In the second part of the paper, the techniques and results of the first part are used to develop a priori error analysis for optimal control problems with pointwise inequality constraints on the control variable.
207 citations
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