Inexact Gauss-Newton methods with matrix approximation by sampling for nonlinear least-squares and systems

TL;DR

This paper introduces stochastic inexact Gauss-Newton methods that utilize sampling strategies to approximate matrices, providing theoretical analysis and numerical validation for solving nonlinear least-squares and systems efficiently.

Contribution

It proposes a novel stochastic inexact Gauss-Newton framework with sampling-based matrix approximations and analyzes their iteration complexity for nonlinear problems.

Findings

The methods achieve desired accuracy with probabilistic guarantees.

Numerical experiments validate the effectiveness of the proposed algorithms.

Theoretical bounds guide sampling strategies for efficient convergence.

Abstract

We develop and analyze stochastic inexact Gauss-Newton methods for nonlinear least-squares problems and for nonlinear systems ofequations. Random models are formed using suitable sampling strategies for the matrices involved in the deterministic models. The analysis of the expected number of iterations needed in the worst case to achieve a desired level of accuracy in the first-order optimality condition provides guidelines for applying sampling and enforcing, with \minor{a} fixed probability, a suitable accuracy in the random approximations. Results of the numerical validation of the algorithms are presented.