Adaptivity with relaxation for ill-posed problems and global convergence for a coefficient inverse problem

Abstract

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. Bibliography: 30 titles. Illustration: 2 figures.

A globally convergent numerical method for a coefficient inverse problem with backscattering data

“…Indeed, it is well known that a good first approximation for the solution is the true key for any locally convergent numerical method, and the first stage provides that approximation. Furthermore, it follows from results of [4] that the resulting two-stage procedure converges globally. So, when working with the backscattering data, we use the gradient method on the second stage.…”

“…As to the choice of V 1,1 , it was taken as V 1,1 ≡ 0 in [1]. In later publications [2,3,4,5,11,16,17] the initial tail V 1,1 was taken as the one, which corresponds to the case c (x) ≡ 1, where c (x) := 1 is the value of the unknown coefficient outside of the domain of interest Ω, see (2.5). An observation was that while both these choices give the same result, the second choice leads to a faster convergence and both choices satisfy conditions of the global convergence theorem.…”

“…The development of globally convergent numerical methods, which are based on the layer stripping procedure with respect to the positive parameter s > 0 of the Laplace transform of a hyperbolic PDE, has started in [1] and was continued in [2,3,4,5,11,16,17]. We call s pseudo frequency.…”

“…Since CIPs are extremely challenging to handle, it is inevitable that an approximation is made in the globally convergent method of [1,2,3,4,5,11,16,17]. Namely, the high value of the pseudo frequency s is truncated.…”

“…It was shown in the follow up publication [5] that not only refractive indices but shapes of those inclusions can also be reconstructed with an excellent accuracy. Therefore, publications [5,11] have provided an independent verification on real rather than synthetic data of both the globally convergent method of [1] and the subsequently developed two-stage numerical procedure of [2,3,4].…”