Approximation of Set-Valued Functions with images sets in $\mathbb{R}^d$

TL;DR

This paper explores methods for approximating continuous set-valued functions from finite samples, focusing on high-order algorithms for functions with complex topology changes, extending previous 1D results to higher dimensions.

Contribution

It extends previous 1D approximation algorithms for set-valued functions to higher dimensions, providing detailed methods for $d=2$ and beyond.

Findings

Developed high-order approximation algorithms for $d=2$.

Extended approximation techniques to higher dimensions.

Addressed topology change points in set-valued functions.

Abstract

Given a finite number of samples of a continuous set-valued function F, mapping an interval to non-empty compact subsets of $\mathbb{R}^d$, $F: [a,b] \to K(\mathbb{R}^d)$, 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 $Graph(F)=\{(t,y)\ | \ y\in F(t),\ t\in [a,b]\}\in K(\mathbb{R}^{d+1})$. 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.