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

Non-dominated sorting arranges a set of points in n-dimensional 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 $$O(h^\frac{1}{n})$$O(h1n). In this paper, we introduce two new numerical schemes that have formal rates of O(h) and we prove the usual $$O(\sqrt{h})$$O(h) theoretical rates. We also present the results of numerical simulations illustrating the difference between the formal and theoretical rates.

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