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Adaptive wavelet schemes for an elliptic control problem with Dirichlet boundary control

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Abstract

An adaptive algorithm based on wavelets is proposed for the fast numerical solution of control problems governed by elliptic boundary value problems with Dirichlet boundary control. A quadratic cost functional representing Sobolev norms of the state and a regularization in terms of the control is to be minimized subject to linear constraints in weak form. In particular, the constraints are formulated as a saddle point problem that allows to handle the varying boundary conditions explicitly. In the framework of (biorthogonal) wavelets, a representer for the functional is derived in terms of ℓ2-norms of wavelet expansion coefficients and the constraints are written in form of an ℓ2 automorphism. Standard techniques from optimization are then used to deduce the resulting first order necessary conditions as a (still infinite) system in ℓ2. Applying the machinery developed in [8,9] which has been extended to control problems in [14], an adaptive method is proposed which can be interpreted as an inexact gradient method for the control. In each iteration step, in turn the primal and the adjoint saddle point system are solved up to a prescribed accuracy by an adaptive iterative Uzawa algorithm for saddle point problems which has been proposed in [10]. Under these premises, it can be shown that the adaptive algorithm containing now three layers of iterations is asymptotically optimal. This means that the convergence rate achieved for computing the solution up to a desired target tolerance is asymptotically the same as the wavelet-best N-term approximation of the solution, and the total computational work is proportional to the number of computational unknowns.

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Authors and Affiliations

  1. Institut für Angewandte Mathematik and Institut für Numerische Simulation, Universität Bonn, Wegelerstr. 6, 53115, Bonn, Germany

    Angela Kunoth

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  1. Angela Kunoth
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Correspondence to Angela Kunoth.

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AMS subject classification

65K10, 65N99, 93B40

Angela Kunoth: This work has been supported partly by the Deutsche Forschungsgemeinschaft (SFB 611) at the Universität Bonn and by the European Community’s Human Potential Programme under contract HPRN-CT-2002-00286 ‘Breaking Complexity’.

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Kunoth, A. Adaptive wavelet schemes for an elliptic control problem with Dirichlet boundary control. Numer Algor 39, 199–220 (2005). https://doi.org/10.1007/s11075-004-3630-0

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