Multivariate rational approximation of functions with curves of singularities
File(s)
Accepted version
Author(s)
Boulle, Nicolas
Herremans, Astrid
Huybrechs, Daan
Type
Journal Article
Abstract
Functions with singularities are notoriously difficult to approximate with conventional approximation schemes. In computational applications, they are often resolved with low-order piecewise polynomials, multilevel schemes, or other types of grading strategies. Rational functions are an exception to this rule: for univariate functions with point singularities, such as branch points, rational approximations exist with root-exponential convergence in the rational degree. This is typically enabled by the clustering of poles near the singularity. Both the theory and computational practice of rational functions for function approximation have focused on the univariate case, with extensions to two dimensions via identification with the complex plane. Multivariate rational functions, i.e., quotients of polynomials of several variables, are relatively unexplored in comparison. Yet, apart from a steep increase in theoretical complexity, they also offer a wealth of opportunities. A first observation is that singularities of multivariate rational functions may be continuous curves of poles, rather than isolated ones. By generalizing the clustering of poles from points to curves, we explore constructions of multivariate rational approximations to functions with curves of singularities.
Date Issued
2024-12-01
Date Acceptance
2024-07-26
Citation
SIAM Journal on Scientific Computing, 2024, 46 (6), pp.A3401-A3426
ISSN
1064-8275
Publisher
Society for Industrial and Applied Mathematics
Start Page
A3401
End Page
A3426
Journal / Book Title
SIAM Journal on Scientific Computing
Volume
46
Issue
6
Copyright Statement
Copyright © 2024 Society for Industrial and Applied Mathematics. This is the author’s accepted manuscript made available under a CC-BY licence in accordance with Imperial’s Research Publications Open Access policy (www.imperial.ac.uk/oa-policy)
License URL
Subjects
ALGORITHM
FRAMES
least-squares
Mathematics
Mathematics, Applied
multivariate functions
Physical Sciences
rational approximation
Science & Technology
singularity
SYSTEMS
Publication Status
Published
Date Publish Online
2024-11-04