Error Estimates for Adaptive Spectral Decompositions

Abstract

Adaptive spectral (AS) decompositions associated with a piecewise constant function, u, yield small subspaces where the characteristic functions comprising u are well approximated. When combined with Newton-like optimization methods for the solution of inverse medium problems, AS decompositions have proved remarkably efficient in providing at each nonlinear iteration a low-dimensional search space. Here, we derive $$L^2$$ L 2 -error estimates for the AS decomposition of u, truncated after K terms, when u is piecewise constant and consists of K characteristic functions over Lipschitz domains and a background. Our estimates apply both to the continuous and the discrete Galerkin finite element setting. Numerical examples illustrate the accuracy of the AS decomposition for media that either do, or do not, satisfy the assumptions of the theory.

Adaptive Spectral Inversion for inverse medium problems

“…From theorem 3.1 we conclude that the first K eigenfunctions of L ε [u h ] are almost piecewise constant in D h , that is away from any discontinuities, when u is piecewise constant with K inclusions. This indicates that the AS decomposition (3.3) is able to approximate u well throughout Ω, which was proved in [1]: Theorem 3.2. Let u be given by (3.5), u h ∈ V h be its FE interpolant, φ k , k ⩾ 1 the eigenfunctions of the operator L ε [u h ] and Π K [u h ]u be its projection.…”

“…) is well defined and for h small even uniformly elliptic in Ω [1]; in practice, we always set ε = 10 −8 . Hence, there exists a non-decreasing sequence of strictly positive eigenvalues {λ k } k⩾1 with corresponding eigenfunctions {φ k } k⩾1 satisfying…”

“…In [1,2], rigorous L 2 -error bounds were derived for the AS decomposition. More precisely, consider a piecewise constant function with zero boundary values of the form…”

“…Moreover, the connection between L ε and TV-regularization motivated the use of different elliptic operators depending on the expected smoothness in the target medium [18]. Recently, a first step was taken in developing a mathematical theory underpinning the remarkable accuracy of that decomposition for the approximation of piecewise constant functions [2], which led to rigorous L 2 -error estimates in [1].…”

“…Here we identify a key angle condition, which yields convergence of the Adaptive Inversion iteration, and also prove that it is a genuine regularization method. In section 3, we present the Adaptive Spectral (AS) decomposition and recall its approximation properties [1,2]. By combining the Adaptive Inversion iteration together with the AS decomposition, we propose in section 4 the Adaptive Spectral Inversion (ASI) Algorithm, which incorporates the new angle condition from section 2 by including the sensitivities of the gradient into the construction of the search space at the subsequent iteration.…”