Structure Preserving Approximation of Semiconcave Functions

TL;DR

This paper develops a method for smooth, structure-preserving approximation of semiconcave functions, crucial in variational problems, by representing them as infima of smooth functions and analyzing their active sets and gradients.

Contribution

It introduces a novel approximation technique that maintains semiconcavity, leveraging infimum representations and smoothing, with detailed analysis of active sets and gradient distributions.

Findings

Approximation in both C(ar fOmega) and W^{1,p}(fOmega) spaces.

Gradients of approximating functions form a probability distribution.

Numerical results demonstrate the effectiveness of the approach.

Abstract

This article addresses structure-preserving smooth approximation of semiconcave functions. semiconcave functions are of particular interest because they naturally arise in a variety of variational problems, including {optimal feedback control, game theory, and optimal transport}. We leverage the fact that any semiconcave function can be represented as the {infimum of a countable family of \(C^2\) functions}. This infimum is expressed in a form that allows {approximation by finitely many functions}, combined with {smoothing operations}, such that each element of the approximating sequence remains semiconcave. The {active sets of indices} contributing to the representation of the semiconcave function and its approximations are analyzed in detail. Moreover, we show that the {gradients of the elements in the expansion of the approximating functions form a probability distribution}, a property of particular interest for the {value function in optimal control}. Approximation results are established in \(C(\bar \Omega)\) and in \(W^{1,p}(\Omega)\) for \(p \in [1,\infty)\) and \(p = \infty\). Finally, {numerical results} are presented to illustrate the approach on a test example.