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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.

MSC:
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
Software:
TIGRA
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References:
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