A regularizing Levenberg - Marquardt scheme, with applications to inverse groundwater filtration problems

Abstract

The first part of this paper studies a Levenberg - Marquardt scheme for nonlinear inverse problems where the corresponding Lagrange (or regularization) parameter is chosen from an inexact Newton strategy. While the convergence analysis of standard implementations based on trust region strategies always requires the invertibility of the Fréchet derivative of the nonlinear operator at the exact solution, the new Levenberg - Marquardt scheme is suitable for ill-posed problems as long as the Taylor remainder is of second order in the interpolating metric between the range and domain topologies. Estimates of this type are established in the second part of the paper for ill-posed parameter identification problems arising in inverse groundwater hydrology. Both transient and steady-state data are investigated. Finally, the numerical performance of the new Levenberg - Marquardt scheme is studied and compared to a usual implementation on a realistic but synthetic two-dimensional model problem from the engineering literature.