Computational polyconvexification of isotropic functions

TL;DR

This paper introduces a computational method for efficiently approximating the polyconvex envelope of isotropic functions by reducing the problem to a lower-dimensional convex envelope computation, significantly speeding up numerical calculations.

Contribution

The authors present a novel algorithm that simplifies the approximation of polyconvex envelopes by leveraging signed singular value representations and dimensional reduction.

Findings

The algorithm achieves significant speedup in numerical experiments.

Dimensional reduction from $d^2$ to $d$ simplifies computations.

The method effectively approximates the polyconvex envelope of isotropic functions.

Abstract

Based on the characterization of the polyconvex envelope of isotropic functions by their signed singular value representations, we propose a simple algorithm for the numerical approximation of the polyconvex envelope. Instead of operating on the $d^2$-dimensional space of matrices, the algorithm requires only the computation of the convex envelope of a function on a $d$-dimensional manifold, which is easily realized by standard algorithms. The significant speedup associated with the dimensional reduction from $d^2$ to $d$ is demonstrated in a series of numerical experiments.