A Gauss-Newton Method with No Additional PDE Solves Beyond Gradient Evaluation for Large-Scale PDE-Constrained Inverse Problems

TL;DR

This paper introduces a Gauss-Newton method for large-scale PDE-constrained inverse problems that avoids additional PDE solves beyond gradient evaluations, significantly improving computational efficiency in applications like Full-Waveform Inversion.

Contribution

The paper presents a novel Gauss-Newton approach that eliminates extra PDE solves, enabling faster and more efficient large-scale PDE-constrained optimization.

Findings

Achieves efficiency comparable to gradient-based methods

Retains fast convergence of Gauss-Newton methods

Demonstrated effectiveness on Full-Waveform Inversion

Abstract

Partial Differential Equation (PDE)-constrained optimization problems often take the form of an optimization of an objective function given as a sum of loss terms. Each function or gradient evaluation requires one or more PDE solves, which render these problems computationally demanding. While Gauss-Newton methods are well-suited for large-scale PDE-constrained optimization, their application to settings such as Full-Waveform Inversion (FWI) is hindered by the need for additional PDE solves to compute Jacobian-vector products. This paper proposes a Gauss-Newton approach that eliminates the need for extra PDE solves beyond those required for gradient computation. Our numerical experiments on FWI demonstrate that the proposed method achieves the efficiency of gradient-based schemes while retaining the fast convergence of Gauss-Newton methods.