Adaptive spectral decompositions for inverse medium problems

Abstract

Inverse medium problems involve the reconstruction of a spatially varying unknown medium from available observations by exploring a restricted search space of possible solutions. Standard grid-based representations are very general but all too often computationally prohibitive due to the high dimension of the search space. Adaptive spectral decompositions instead expand the unknown medium in a basis of eigenfunctions of a judicious elliptic operator, which depends itself on the medium. Here the AS decomposition is combined with a standard inexact Newton-type method for the solution of time-harmonic scattering problems governed by the Helmholtz equation. By repeatedly adapting both the eigenfunction basis and its dimension, the resulting adaptive spectral inversion (ASI) method substantially reduces the dimension of the search space during the nonlinear optimization. Rigorous estimates of the AS decomposition are proved for a general piecewise constant medium. Numerical results illustrate the accuracy and efficiency of the ASI method for time-harmonic inverse scattering problems, including a salt dome model from geophysics.

Error Estimates for Adaptive Spectral Decompositions

“…Adaptive spectral (AS) decompositions have been proposed as low-dimensional search spaces during the iterative solution of inverse medium problems [1][2][3][4][5]. For piecewise constant media, in particular, AS decompositions have proved remarkably efficient and accurate.…”

“…So far, however, their remarkable approximation properties are only supported by numerical evidence. Here, starting from [5], we derive L 2 -error estimates for AS approximations of piecewise constant functions.…”

“…In [11], AS decompositions were used for solving 2-D and 3-D seismic inverse problems for the Helmholtz equation. First theoretical estimates for AS decompositions together with an approach for adapting the dimension of the search space were derived in [5].…”

“…When u consists of a sum of K characteristic functions χ A k of sets A k , each compactly contained in , the expansion (1.1) in the spectral basis of L ε [u δ ] truncated after K terms has proved remarkably accurate, as it essentially recovers u and in fact decomposes u into the characteristic functions comprising it. In [5], it is shown that the gradients of the first K eigenfunctions of L ε [u δ ] are small away from the discontinuities of u. Thus, in regions where u is constant, ϕ 1 , .…”

“…Starting from [5], we derive L 2 error estimates for the projection of any v ∈ u +span{χ A k } onto the appropriate affine subspace. In our main result, given by Theorem 3.6, we prove that the L 2 -projection error of v (in particular of u itself) is bounded by O( √ ε + δ).…”