Error Estimates for Adaptive Spectral Decompositions

TL;DR

This paper derives $L^2$-error estimates for adaptive spectral decompositions of piecewise constant functions, demonstrating their effectiveness in low-dimensional approximations for inverse medium problems, supported by numerical examples.

Contribution

The paper provides the first $L^2$-error estimates for truncated adaptive spectral decompositions of piecewise constant functions, applicable in both continuous and discrete finite element contexts.

Findings

Error estimates validate the accuracy of AS decompositions.

Numerical examples confirm effectiveness for media satisfying and not satisfying theoretical assumptions.

AS decompositions enable efficient low-dimensional representations in inverse problems.

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 $L^2$-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.