A Preconditioned Riemannian Gauss-Newton Method for Least Squares Inverse Eigenvalue Problems
This work provides a more efficient optimization approach for the inverse eigenvalue problem, which is important for applications like parameter identification in Sturm-Liouville problems.
The paper develops a Riemannian inexact Gauss-Newton method with a preconditioned conjugate gradient solver for the least squares inverse eigenvalue problem, achieving efficient reconstruction of symmetric matrices from partial eigenvalues. Numerical tests, including an inverse Sturm-Liouville problem, demonstrate the method's efficiency.
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.