A Modified Levenberg-Marquardt Algorithm for Large-Scale Inverse Problems

Abstract

Distributed parameter estimation problems typically involve attempts to invert infinite dimensional nonlinear compact operators. In this case the derivative, or Jacobian, is a compact linear operator. Via the Hilbert-Schmidt Theorem one can construct, from a truncated spectral decomposition consisting of the largest eigenvalues and corresponding eigenfunctions, a uniformly convergent sequence of finite rank operator approximations to the Jacobian. This truncated spectral decomposition can be computed using a variety of iterative methods, including Subspace Iteration and the Lanczos method [5]. The approximate Jacobians can then be incorporated into a quasi-Newton scheme for solving the nonlinear problem. The purpose of this paper is to demonstrate that by combining Subspace Iteration with costate, or adjoint, ideas similar to those in [7], one can efficiently solve large-scale distributed parameter estimation problems.

Research was supported in part by NSF under Grant DMS-9106609.

Research was supported in part by the Air Force Office of Scientific Research under grant AFOSR-90-0091 and by the Department of Energy under contract #SK966-19. Part of this work was carried out while the second author was a visitor at the University of Southern California, Los Angeles, CA.