Multilevel Optimization for Inverse Problems

TL;DR

This paper introduces a multilevel optimization framework that significantly reduces computational costs in inverse problems involving expensive forward models, applicable to various optimization methods.

Contribution

The paper presents a unifying multilevel optimization framework that is theoretically proven to lower computational costs across multiple inverse problem-solving techniques.

Findings

Framework reduces evaluation costs of forward models

Applicable to gradient descent, ensemble Kalman inversion, Langevin sampling

Numerical experiments confirm theoretical advantages

Abstract

Inverse problems occur in a variety of parameter identification tasks in engineering. Such problems are challenging in practice, as they require repeated evaluation of computationally expensive forward models. We introduce a unifying framework of multilevel optimization that can be applied to a wide range of optimization-based solvers. Our framework provably reduces the computational cost associated with evaluating the expensive forward maps stemming from various physical models. To demonstrate the versatility of our analysis, we discuss its implications for various methodologies including multilevel (accelerated, stochastic) gradient descent, a multilevel ensemble Kalman inversion and a multilevel Langevin sampler. We also provide numerical experiments to verify our theoretical findings.