Consensus ADMM for Inverse Problems Governed by Multiple PDE Models

TL;DR

This paper introduces a consensus ADMM framework for efficiently solving large-scale inverse problems involving multiple PDE models, combining second-order optimization and proximal methods, with applications in medical imaging.

Contribution

The paper develops a novel consensus ADMM approach tailored for inverse problems with multiple PDEs, incorporating adaptations for infinite-dimensional consistency and demonstrating effectiveness in imaging applications.

Findings

Effective handling of large-scale inverse PDE problems.

Successful application to electrical impedance tomography.

Improved computational efficiency with proposed methods.

Abstract

The Alternating Direction Method of Multipliers (ADMM) provides a natural way of solving inverse problems with multiple partial differential equations (PDE) forward models and nonsmooth regularization. ADMM allows splitting these large-scale inverse problems into smaller, simpler sub-problems, for which computationally efficient solvers are available. In particular, we apply large-scale second-order optimization methods to solve the fully-decoupled Tikhonov regularized inverse problems stemming from each PDE forward model. We use fast proximal methods to handle the nonsmooth regularization term. In this work, we discuss several adaptations (such as the choice of the consensus norm) needed to maintain consistency with the underlining infinite-dimensional problem. We present two imaging applications inspired by electrical impedance tomography and quantitative photoacoustic tomography to demonstrate the proposed method's effectiveness.