2017/12/04 by Warut Thawinrak, Thawinrak, Warut, Jeff Calder +1
Computer Science · Mathematics · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Biology Tumor Growth #Numerical Analysis (math.NA) #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.1712.01452
openalex publication_date 2017/12/04 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
We investigate high-order finite difference schemes for the Hamilton-Jacobi\nequation continuum limit of nondominated sorting. Nondominated sorting is an\nalgorithm for sorting points in Euclidean space into layers by repeatedly\nremoving minimal elements. It is widely used in multi-objective optimization,\nwhich finds applications in many scientific and engineering contexts, including\nmachine learning. In this paper, we show how to construct filtered schemes,\nwhich combine high order possibly unstable schemes with first order monotone\nschemes in a way that guarantees stability and convergence while enjoying the\nadditional accuracy of the higher order scheme in regions where the solution is\nsmooth. We prove that our filtered schemes are stable and converge to the\nviscosity solution of the Hamilton-Jacobi equation, and we provide numerical\nsimulations to investigate the rate of convergence of the new schemes.\n