Approximation of set-valued functions with images sets in ℝ d

Abstract Given a finite number of samples of a continuous set-valued function F, mapping an interval to nonempty compact subsets of R^d, F: a, b K (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 F (t), \ t a, b\ K (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, together with Q. Muzaffar, we presented an algorithm for approximating set-valued functions with one-dimensional sets (d=1) as images, achieving a high approximation order near points of topology change. Here, we build upon the results and algorithms for the case d=1, first in more detail for the important case d=2, and later for approximating set-valued functions and their graphs in higher dimensions.