A Preconditioned Riemannian Gauss-Newton Method for Least Squares Inverse Eigenvalue Problems

TL;DR

This paper introduces a Riemannian inexact Gauss-Newton method with preconditioning for solving least squares inverse eigenvalue problems, demonstrating its efficiency through convergence analysis and numerical tests.

Contribution

It develops a novel Riemannian Gauss-Newton algorithm with preconditioning specifically for inverse eigenvalue problems, including convergence analysis and practical numerical validation.

Findings

The method converges globally and locally.

Numerical tests show high efficiency and accuracy.

Application to inverse Sturm-Liouville problem demonstrates practical utility.

Abstract

This paper is concerned with the least squares inverse eigenvalue problem of reconstructing a linear parameterized real symmetric matrix from the prescribed partial eigenvalues in the sense of least squares, which was originally proposed by Chen and Chu [SIAM J. Numer. Anal., 33 (1996), pp. 2417--2430]. We provide a Riemannian inexact Gausss-Newton method for solving the least squares inverse eigenvalue problem. The global and local convergence analysis of the proposed method is discussed. Also, a preconditioned conjugate gradient method with an efficient preconditioner is proposed for solving the Riemannian Gauss-Newton equation. Finally, some numerical tests, including an application in the inverse Sturm-Liouville problem, are reported to illustrate the efficiency of the proposed method.