A Modified Split Bregman Algorithm for Computing Microstructures Through Young Measures

TL;DR

This paper introduces a modified split Bregman algorithm combined with control systems theory to compute and analyze microstructures arising from nonconvex energy minimization problems, providing new numerical and analytical tools.

Contribution

It presents a novel numerical scheme integrating convex splitting and a modified split Bregman algorithm for microstructure computation, alongside a semi-analytical control systems approach.

Findings

Effective computation of microstructures from nonconvex energies.

Enhanced understanding of oscillatory microstructures in minimizing sequences.

New numerical method improves convergence and accuracy.

Abstract

The goal of this paper is to describe the oscillatory microstructure that can emerge from minimizing sequences for nonconvex energies. We consider integral functionals that are defined on real valued (scalar) functions $u(x)$ which are nonconvex in the gradient $\nabla u$ and possibly also in $u$. To characterize the microstructures for these nonconvex energies, we minimize the associated relaxed energy using two novel approaches: i) a semi-analytical method based on control systems theory, ii) and a numerical scheme that combines convex splitting together with a modified version of the split Bregman algorithm. These solutions are then used to gain information about minimizing sequences of the original problem and the spatial distribution of microstructure.