A model reduction approach to numerical inversion for a parabolic partial differential equation

TL;DR

This paper introduces a model reduction-based numerical inversion algorithm for parabolic PDEs, improving efficiency and convergence in electromagnetic subsurface resistivity imaging by using rational Krylov subspaces and nonlinear preconditioning.

Contribution

It presents a novel inversion method combining rational interpolation and Krylov subspace projection to enhance stability and efficiency in solving inverse problems for parabolic PDEs.

Findings

The algorithm reduces computational cost by avoiding repeated time-domain simulations.

It improves convergence speed and reduces local minima trapping in inverse problems.

Numerical experiments demonstrate stability and effectiveness of the proposed method.

Abstract

We propose a novel numerical inversion algorithm for the coefficients of parabolic partial differential equations, based on model reduction. The study is motivated by the application of controlled source electromagnetic exploration, where the unknown is the subsurface electrical resistivity and the data are time resolved surface measurements of the magnetic field. The algorithm presented in this paper considers inversion in one and two dimensions. The reduced model is obtained with rational interpolation in the frequency (Laplace) domain and a rational Krylov subspace projection method. It amounts to a nonlinear mapping from the function space of the unknown resistivity to the small dimensional space of the parameters of the reduced model. We use this mapping as a nonlinear preconditioner for the Gauss-Newton iterative solution of the inverse problem. The advantage of the inversion algorithm is twofold. First, the nonlinear preconditioner resolves most of the nonlinearity of the problem. Thus the iterations are less likely to get stuck in local minima and the convergence is fast. Second, the inversion is computationally efficient because it avoids repeated accurate simulations of the time-domain response. We study the stability of the inversion algorithm for various rational Krylov subspaces, and assess its performance with numerical experiments.