2025/01/24 by Nira Dyn, David Levin, Dyn, Nira +1
Computer Science · Mathematics · #Approximation Theory and Sequence Spaces #FOS: Mathematics #Mathematical Approximation and Integration #Numerical Analysis (math.NA) #Optimization and Variational Analysis
paper · pdf · doi:10.48550/arxiv.2501.14591
openalex publication_date 2025/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a finite number of samples of a continuous set-valued function F, mapping an interval to non-empty compact subsets of ℝd, F: [a,b] → K(ℝd), we discuss the problem of computing good approximations of F. We also discuss algorithms for a direct high-order evaluation of the graph of F, namely, the set Graph(F)=\(t,y) | y∈ F(t), t∈ [a,b]\∈ K(ℝd+1). A set-valued function can be continuous and yet have points where the topology of the image sets changes. The main challenge in set-valued function approximation is to derive high-order approximations near these points. In a previous paper, we presented with Q. Muzaffar, an algorithm for approximating set-valued functions with 1D sets (d=1) as images, achieving high approximation order near points of topology change. Here we build upon the results and algorithms in the d=1 case, first in more detail for the important case d=2, and later for approximating set-valued functions and their graphs in higher dimensions.