Total variation regularization of multi-material topology optimization

TL;DR

This paper introduces a total variation regularization approach for identifying diffusion coefficients in a domain, addressing the mathematical challenges with a new reparametrization technique and demonstrating its effectiveness through numerical examples.

Contribution

It presents a novel reparametrization method for total variation regularized inverse problems in diffusion coefficient identification, enhancing the solvability of the optimization problem.

Findings

Effective numerical reconstruction of diffusion coefficients.

Reparametrization improves handling of nonsmooth optimization.

Numerical examples validate the proposed approach.

Abstract

This work is concerned with the determination of the diffusion coefficient from distributed data of the state. This problem is related to homogenization theory on the one hand and to regularization theory on the other hand. An approach is proposed which involves total variation regularization combined with a suitably chosen cost functional that promotes the diffusion coefficient assuming prespecified values at each point of the domain. The main difficulty lies in the delicate functional-analytic structure of the resulting nondifferentiable optimization problem with pointwise constraints for functions of bounded variation, which makes the derivation of useful pointwise optimality conditions challenging. To cope with this difficulty, a novel reparametrization technique is introduced. Numerical examples using a regularized semismooth Newton method illustrate the structure of the obtained diffusion coefficient. ~