A comparative study of structural similarity and regularization for joint inverse problems governed by PDEs

TL;DR

This paper compares different regularization techniques for joint inverse problems governed by PDEs, emphasizing the effectiveness and computational efficiency of the vectorial total variation method in reconstructing correlated parameter fields.

Contribution

The study introduces and evaluates several joint regularizations, highlighting the superiority of vectorial total variation for scalable and accurate PDE-based inverse problem solutions.

Findings

Vectorial total variation outperforms other regularizations in reconstruction quality.

It enables scalable and efficient solvers for joint inverse problems.

Numerical experiments confirm its robustness across different inverse problem classes.

Abstract

Joint inversion refers to the simultaneous inference of multiple parameter fields from observations of systems governed by single or multiple forward models. In many cases these parameter fields reflect different attributes of a single medium and are thus spatially correlated or structurally similar. By imposing prior information on their spatial correlations via a joint regularization term, we seek to improve the reconstruction of the parameter fields relative to inversion for each field independently. One of the main challenges is to devise a joint regularization functional that conveys the spatial correlations or structural similarity between the fields while at the same time permitting scalable and efficient solvers for the joint inverse problem. We describe several joint regularizations that are motivated by these goals: a cross-gradient and a normalized cross-gradient structural similarity term, the vectorial total variation, and a joint regularization based on the nuclear norm of the gradients. Based on numerical results from three classes of inverse problems with piecewise-homogeneous parameter fields, we conclude that the vectorial total variation functional is preferable to the other methods considered. Besides resulting in good reconstructions in all experiments, it allows for scalable, efficient solvers for joint inverse problems governed by PDE forward models.