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

Abstract

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 1 n ). In this paper, we introduce two new numerical schemes that have formal rates of O(h) and we prove the usual O( √ h) theoretical rates. We also present the results of numerical simulations illustrating the difference between the formal and theoretical rates.

High-order Filtered Schemes for the Hamilton-Jacobi Continuum Limit of Nondominated Sorting

“…Given the stability result in Theorem 2.4, it is now a standard application of the Barles-Souganidis framework [1] to prove convergence of the filtered scheme (2.10) to the viscosity solution of (2.4). In fact, the proof of convergence of the filtered schemes is very similar to the results in [5]. Let us mention, however, that these convergence results do not establish any convergence rate.…”

“…Indeed, in the special case that f = 1 the viscosity solution is u = n(x 1 · · · x n ) 1/n . Following [5] we first perform a singularity factorization before solving the Hamilton-Jacobi equation. In particular, let u be the viscosity solution of (1.1) and define…”

“…Furthermore, this scheme can be solved with a (linear time) fast sweeping algorithm visiting each grid point exactly once. For more details on this scheme, we refer the reader to [5]. We define the k th -order upwind finite difference scheme approximating the left-hand side of (2.4) by…”

“…We run simulations on both backward difference schemes and filtered schemes of orders 1, 2, 3, 5, 8, and 13 with two probability density functions f 1 and f 2 that were introduced before in [5]. The function f 1 is defined as follows:…”

“…Nondominated sorting is fundamental in multi-objective optimization problems, which are ubiquitous in science and engineering, and more recently in machine learning. For more details on the connection to nondominated sorting and applications, we refer the reader to [3,[5][6][7][8][9][11][12][13][14]. The Hamilton-Jacobi equation (1.1) has a unique non-decreasing viscosity solution.…”