Total-variation minimization with bound constraints

TL;DR

This paper introduces an efficient algorithm for solving constrained optimization problems with total variation regularization, combining augmented Lagrangian methods with splitting and projection techniques, applicable to geomechanical and other inverse problems.

Contribution

The paper presents a novel algorithm that effectively handles bound and equality constraints in total variation regularized optimization, improving upon previous methods.

Findings

Successfully applied to geomechanical inverse problem

Effectively handles bound and equality constraints

Demonstrates robustness with noisy data

Abstract

We present a powerful and easy-to-implement algorithm for solving constrained optimization problems that involve $L_1$/total-variation regularization terms, and both equality and inequality constraints. We discuss the relationship of our method to earlier works of Goldstein and Osher (2009) and Chartrand and Wohlberg (2010), and demonstrate that our approach is a combination of the augmented Lagrangian method with splitting and model projection. We test the method on a geomechanical problem and invert highly compartmentalized pressure change from noisy surface uplift observations. We conclude the paper with a discussion of possible extension to a wide class of regularized optimization problems with bound and equality constraints.