Efficient solution of parameter identification problems with $H^1$ regularization

TL;DR

This paper presents an efficient method for solving $H^1$ regularized parameter identification problems by combining algebraic multigrid, Sherman-Morrison-Woodbury formula, and fast Poisson solvers, demonstrated on electrical resistivity tomography.

Contribution

It introduces a scalable preconditioning approach for linearized problems in $H^1$ regularization using low-rank modifications and fast solvers, applicable in functional and discrete settings.

Findings

Method exhibits excellent scaling with problem size.

Preconditioner significantly accelerates convergence.

Validated on electrical resistivity tomography problem.

Abstract

We consider the identification of spatially distributed parameters under $H^1$ regularization. Solving the associated minimization problem by Gauss-Newton iteration results in linearized problems to be solved in each step that can be cast as boundary value problems involving a low-rank modification of the Laplacian. Using algebraic multigrid as a fast Laplace solver, the Sherman-Morrison-Woodbury formula can be employed to construct a preconditioner for these linear problems which exhibits excellent scaling w.r.t. the relevant problem parameters. We first develop this approach in the functional setting, thus obtaining a consistent methodology for selecting boundary conditions that arise from the $H^1$ regularization. We then construct a method for solving the discrete linear systems based on combining any fast Poisson solver with the Woodbury formula. The efficacy of this method is then demonstrated with scaling experiments. These are carried out for a common nonlinear parameter identification problem arising in electrical resistivity tomography.