Fictitious null spaces for improving the solution of injective inverse problems

TL;DR

This paper introduces a novel weighting approach using fictitious null spaces to enhance solutions of injective inverse problems, demonstrating improved results over traditional methods in PDE-based applications.

Contribution

It develops a new weighting technique based on fictitious null spaces for injective operators, improving inverse problem solutions beyond classical regularization methods.

Findings

The weighting method improves solution accuracy in PDE inverse problems.

Numerical examples show better results with the proposed approach.

The approach is applicable to sparsity regularization in inverse problems.

Abstract

For linear ill-posed problems with nontrivial null spaces, Tikhonov regularization and truncated singular value decomposition (TSVD) typically yield solutions that are close to the minimum norm solution. Such a bias is not always desirable, and we have therefore in a series of papers developed a weighting procedure which produces solutions with a different and controlled bias. This methodology can also conveniently be invoked when sparsity regularization is employed. The purpose of the present work is to study the potential use of this weighting applied to injective operators. The image under a compact operator of the singular vectors/functions associated with very small singular values will be almost zero. Consequently, one may regard these singular vectors/functions to constitute a basis for a fictitious null space that allows us to mimic the previous weighting procedure. It turns out that this regularization by weighting can improve the solution of injective inverse problems compared with more traditional approaches. We present some analysis of this methodology and exemplify it numerically, using sparsity regularization, for three PDE-driven inverse problems: the inverse heat conduction problem, the Cauchy problem for Laplace's equation, and the (linearized) Electrical Impedance Tomography problem with experimental data.