Numerical schemes and rates of convergence for the Hamilton-Jacobi equation continuum limit of nondominated sorting

TL;DR

This paper introduces two new numerical schemes for the Hamilton-Jacobi equation arising from nondominated sorting, achieving improved convergence rates and providing both theoretical analysis and numerical validation.

Contribution

It presents two novel numerical schemes with formal O(h) convergence for the PDE related to nondominated sorting, along with proof of their theoretical rates.

Findings

New schemes achieve formal O(h) convergence rate.

Numerical simulations compare formal and theoretical convergence rates.

The schemes outperform the standard method in convergence speed.

Abstract

Nondominated sorting arranges a set of points in Euclidean space into layers by repeatedly removing the coordinatewise minimal elements. It was recently shown that nondominated sorting of random points has a Hamilton-Jacobi equation continuum limit. The obvious numerical scheme for this PDE has a slow convergence rate of O(h^1/n) for a grid of spacing h>0 in dimension n. In this paper, we introduce two new numerical schemes that have formal rates of O(h) and we prove the usual O(h^1/2) theoretical rates. We also present the results of numerical simulations illustrating the difference between the formal and theoretical rates.