Locally-averaged McCormick relaxations for discretization-regularized inverse problems

TL;DR

This paper introduces a novel approach using locally-averaged McCormick relaxations to compute approximate dual bounds, improving the regularization and convergence of discretization-regularized inverse problems in PDEs.

Contribution

It proposes a new method combining local averaging with McCormick relaxations for better bounds in PDE coefficient inverse problems, ensuring convergence.

Findings

The method provides tighter bounds than traditional relaxations.

Discretization regularization ensures convergence of the inverse problem solution.

Numerical experiments validate the theoretical convergence and effectiveness.

Abstract

In this paper, by means of a standard model problem, we devise an approach to computing approximate dual bounds for use in global optimization of coefficient identification in partial differential equations (PDEs) by, e.g., (spatial) branch-and-bound methods. Linearization is achieved by a McCormick relaxation (that is, replacing the bilinear PDE term by a linear one and adding inequality constraints), combined with local averaging to reduce the number of inequalities. Optimization-based bound tightening allows us to tighten the relaxation and thus reduce the induced error. Combining this with a quantification of the discretization error and the propagated noise, we prove that the resulting discretization regularizes the inverse problem, thus leading to an overall convergent scheme. Numerical experiments illustrate the theoretical findings.