Adaptive FEM with relaxation for a hyperbolic coefficient inverse problem

Recent research of publications (Beilina and Johnson, Numerical Mathematics and Advanced Applications: ENUMATH 2001, Springer, Berlin, 2001; Beilina, Applied and Computational Mathematics 1, 158-174, 2002; Beilina and Johnson, Mathematical Models and Methods in Applied Sciences 15, 23-37, 2005; Beilina and Clason, SIAM Journal on Scientific Computing 28, 382-402, 2006; Beilina, Applicable Analysis 90, 1461-1479, 2011; Beilina and Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012; Beilina and Klibanov, Journal of Inverse and Ill-posed Problems 18, 85-132, 2010; Beilina and Klibanov, Inverse Problems 26, 045012, 2010; Beilina and Klibanov, Inverse Problems 26, 125009, 2010; Beilina et al., Journal of Mathematical Sciences 167, 279-325, 2010) have shown that adaptive finite element method presents a useful tool for solution of hyperbolic coefficient inverse problems. In the above publications improvement in the image reconstruction is achieved by local mesh refinements using a posteriori error estimate in the Tikhonov functional and in the reconstructed coefficient. In this paper we apply results of the above publications and present the relaxation property for the mesh refinements and a posteriori error estimate for the reconstructed coefficient for a hyperbolic CIP, formulate an adaptive algorithm, and apply it to the reconstruction of the coefficient in hyperbolic PDE. Our numerical examples present performance of the two-step numerical procedure on the computationally simulated data where on the first step we obtain good approximation of the exact coefficient using approximate globally convergent method of Beilina and Klibanov (Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012), and on the second step we take this solution for further improvement via adaptive mesh refinements