PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem

Lili  Chang; Wei  Gong; Guiquan  Sun; Ningning  Yan

doi:10.3934/ipi.2015.9.791

Article Contents

2015, Volume 9, Issue 3: 791-814. Doi: 10.3934/ipi.2015.9.791

This issue Previous Article Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Stochastic extension of the boundary map Next Article Nomonotone spectral gradient method for sparse recovery

PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem

1.
Complex Systems Research Center, Shanxi University, Taiyuan 030006, Shan'xi
2.
LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, CAS, Beijing 100190
3.
LSEC, NCMIS, Institute of Systems Science, Academy of Mathematics and Systems Science, CAS, Beijing 100190

Received: November 2014

Revised: March 2015

Published: August 2015

Abstract / Introduction Related Papers Cited by

Abstract

This paper concerns the approximation of a Cauchy problem for the elliptic equation. The inverse problem is transformed into a PDE-constrained optimal control problem and these two problems are equivalent under some assumptions. Different from the existing literature which is also based on the optimal control theory, we consider the state equation in the sense of very weak solution defined by the transposition technique. In this way, it does not need to impose any regularity requirement on the given data. Moreover, this method can yield theoretical analysis simply and numerical computation conveniently. To deal with the ill-posedness of the control problem, Tikhonov regularization term is introduced. The regularized problem is well-posed and its solution converges to the non-regularized counterpart as the regularization parameter approaches zero. We establish the finite element approximation to the regularized control problem and the convergence of the discrete problem is also investigated. Then we discuss the first order optimality condition of the control problem further and obtain an efficient numerical scheme for the Cauchy problem via the adjoint state equation. The paper is ended with numerical experiments.

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Mathematics Subject Classification: Primary: 49J20, 65N21; Secondary: 65N30.

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