Nesterov's Accelerated Gradient Method for Nonlinear Ill-Posed Problems with a Locally Convex Residual Functional

TL;DR

This paper analyzes the convergence of Nesterov's Accelerated Gradient method for nonlinear ill-posed problems with locally convex residuals, demonstrating its effectiveness through numerical examples.

Contribution

It provides a convergence analysis of Nesterov's method for ill-posed problems under local convexity assumptions, extending its applicability beyond well-posed cases.

Findings

Convergence established for ill-posed problems with locally convex residuals

Numerical examples show the method's practical effectiveness

Applicable to nonlinear inverse problems like auto-convolution

Abstract

In this paper, we consider Nesterov's Accelerated Gradient method for solving Nonlinear Inverse and Ill-Posed Problems. Known to be a fast gradient-based iterative method for solving well-posed convex optimization problems, this method also leads to promising results for ill-posed problems. Here, we provide a convergence analysis for ill-posed problems of this method based on the assumption of a locally convex residual functional. Furthermore, we demonstrate the usefulness of the method on a number of numerical examples based on a nonlinear diagonal operator and on an inverse problem in auto-convolution.