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Adaptivity with relaxation for ill-posed problems and global convergence for a coefficient inverse problem. (English. Russian original) Zbl 1286.65147
J. Math. Sci., New York 167, No. 3, 279-325 (2010); translation from Probl. Mat. Anal. 46, 3-44 (2010).
Summary: A new framework of the functional analysis is developed for the finite element adaptive method (adaptivity) for the Tikhonov regularization functional for some ill-posed problems. As a result, the relaxation property for adaptive mesh refinements is established. An application to a multidimensional coefficient inverse problem for a hyperbolic equation is discussed. This problem arises in the inverse scattering of acoustic and electromagnetic waves. First, a globally convergent numerical method provides a good approximation for the correct solution of this problem. Next, this approximation is enhanced via the subsequent application of the adaptivity. Analytical results are verified computationally.

65N21 Numerical methods for inverse problems for boundary value problems involving PDEs
65J22 Numerical solution to inverse problems in abstract spaces
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text: DOI
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