A Damped Newton Algorithm for Generated Jacobian Equations

TL;DR

This paper introduces a damped Newton algorithm tailored for solving generated Jacobian equations in a semi-discrete setting, with applications demonstrated in near-field parallel refractor problems.

Contribution

It presents a novel damped Newton method specifically designed for generated Jacobian equations and applies it to a practical non-imaging optics problem.

Findings

Algorithm effectively solves semi-discrete generated Jacobian equations.

Numerical experiments demonstrate convergence and practical utility.

Application to near-field parallel refractor problem shows real-world relevance.

Abstract

Generated Jacobian Equations have been introduced by Trudinger [Disc. cont. dyn. sys (2014), pp. 1663-1681] as a generalization of Monge-Amp{\`e}re equations arising in optimal transport. In this paper, we introduce and study a damped Newton algorithm for solving these equations in the semi-discrete setting, meaning that one of the two measures involved in the problem is finitely supported and the other one is absolutely continuous. We also present a numerical application of this algorithm to the near-field parallel refractor problem arising in non-imaging problems.