Computing the quasiconvex envelope using a nonlocal line solver

TL;DR

This paper introduces a nonlocal line solver algorithm for computing the quasiconvex envelope of functions, leveraging a level set operator and explicit formulas for one-dimensional cases, advancing the numerical methods for solving related PDEs.

Contribution

The paper presents a novel nonlocal line solver algorithm for efficiently computing the quasiconvex envelope of functions, based on explicit formulas and level set methods.

Findings

The algorithm effectively computes quasiconvex envelopes of functions.

Explicit formulas for the QCE on a line are derived.

The method provides a new approach to solving related PDEs.

Abstract

Recently in a series of articles, Barron, Goebel, and Jensen \cite{barron2012functions} \cite{barron2012quasiconvex} \cite{barron2013quasiconvex} \cite{barron2013uniqueness} have studied second order degenerate elliptic PDE and first order nonlocal PDEs for the quasiconvex envelope. Quasiconvex functions are functions whose level sets are convex. The PDE is difficult to solve. In this article we present an algorithm for computing the quasiconvex envelope (QCE) of a given function. The QCE operator is a level set operator, so this algorithm gives a method to compute convex hull of sets represented by a level set functions. We present a nonlocal line solver for the quasiconvex envelope (QCE), based on solving the one dimensional problem on lines. We find an explicit formula for the QCE of a function defined on a line.