A data scalable augmented Lagrangian KKT preconditioner for large scale inverse problems

TL;DR

This paper introduces a scalable augmented Lagrangian preconditioner for large-scale inverse problems, improving convergence and accuracy over traditional methods by effectively handling highly informative data.

Contribution

It proposes a novel block diagonal augmented Lagrangian preconditioner that simplifies the solution of KKT systems in large inverse problems, outperforming regularization-based approaches.

Findings

Preconditioner effectively accelerates convergence in large inverse problems.

Numerical results show fewer iterations needed for accurate reconstructions.

Preconditioner outperforms traditional regularization methods in a Poisson source inversion example.

Abstract

Current state of the art preconditioners for the reduced Hessian and the Karush-Kuhn-Tucker (KKT) operator for large scale inverse problems are typically based on approximating the reduced Hessian with the regularization operator. However, the quality of this approximation degrades with increasingly informative observations or data. Thus the best case scenario from a scientific standpoint (fully informative data) is the worse case scenario from a computational perspective. In this paper we present an augmented Lagrangian-type preconditioner based on a block diagonal approximation of the augmented upper left block of the KKT operator. The preconditioner requires solvers for two linear subproblems that arise in the augmented KKT operator, which we expect to be much easier to precondition than the reduced Hessian. Analysis of the spectrum of the preconditioned KKT operator indicates that the preconditioner is effective when the regularization is chosen appropriately. In particular, it is effective when the regularization does not over-penalize highly informed parameter modes and does not under-penalize uninformed modes. Finally, we present a numerical study for a large data/low noise Poisson source inversion problem, demonstrating the effectiveness of the preconditioner. In this example, three MINRES iterations on the KKT system with our preconditioner results in a reconstruction with better accuracy than 50 iterations of CG on the reduced Hessian system with regularization preconditioning.