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Multivariate approximation by polynomial and generalised rational functions

R. Díaz Millán, V. Peiris, N. Sukhorukova, J. Ugon

TL;DR

This paper develops an optimization-based approach to multivariate Chebyshev (uniform) approximation on finite grids for two models: multivariate polynomials and multivariate generalised rational functions (ratios of linear forms with flexible bases). It proves that the polynomial case yields a convex (semi-infinite) linear program, while the generalised rational case produces a quasiconvex problem that can be tackled with a bisection method; convex feasibility subproblems arising in the bisection can be solved by linear programming on a finite grid, or via splitting methods in other settings. The authors provide a thorough numerical comparison on a nonsmooth test function, finding that generalised rational approximations can achieve lower maximum error than polynomials but at a higher computational cost, reflecting a trade-off between accuracy and speed. They also discuss future directions, including extending other univariate techniques to the multivariate domain and improving feasibility-solving techniques for larger problems.

Abstract

In this paper we develop an optimisation based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalised rational approximation. In the second case the approximations are ratios of linear forms and the basis functions are not limited to monomials. It is already known that in the case of multivariate polynomial approximation on a finite grid the corresponding optimisation problems can be reduced to solving a linear programming problem, while the area of multivariate rational approximation is not so well understood.In this paper we demonstrate that in the case of multivariate generalised rational approximation the corresponding optimisation problems are quasiconvex. This statement remains true even when the basis functions are not limited to monomials. Then we apply a bisection method, which is a general method for quasiconvex optimisation. This method converges to an optimal solution with given precision. We demonstrate that the convex feasibility problems appearing in the bisection method can be solved using linear programming. Finally, we compare the deviation error and computational time for multivariate polynomial and generalised rational approximation with the same number of decision variables.

Multivariate approximation by polynomial and generalised rational functions

TL;DR

This paper develops an optimization-based approach to multivariate Chebyshev (uniform) approximation on finite grids for two models: multivariate polynomials and multivariate generalised rational functions (ratios of linear forms with flexible bases). It proves that the polynomial case yields a convex (semi-infinite) linear program, while the generalised rational case produces a quasiconvex problem that can be tackled with a bisection method; convex feasibility subproblems arising in the bisection can be solved by linear programming on a finite grid, or via splitting methods in other settings. The authors provide a thorough numerical comparison on a nonsmooth test function, finding that generalised rational approximations can achieve lower maximum error than polynomials but at a higher computational cost, reflecting a trade-off between accuracy and speed. They also discuss future directions, including extending other univariate techniques to the multivariate domain and improving feasibility-solving techniques for larger problems.

Abstract

In this paper we develop an optimisation based approach to multivariate Chebyshev approximation on a finite grid. We consider two models: multivariate polynomial approximation and multivariate generalised rational approximation. In the second case the approximations are ratios of linear forms and the basis functions are not limited to monomials. It is already known that in the case of multivariate polynomial approximation on a finite grid the corresponding optimisation problems can be reduced to solving a linear programming problem, while the area of multivariate rational approximation is not so well understood.In this paper we demonstrate that in the case of multivariate generalised rational approximation the corresponding optimisation problems are quasiconvex. This statement remains true even when the basis functions are not limited to monomials. Then we apply a bisection method, which is a general method for quasiconvex optimisation. This method converges to an optimal solution with given precision. We demonstrate that the convex feasibility problems appearing in the bisection method can be solved using linear programming. Finally, we compare the deviation error and computational time for multivariate polynomial and generalised rational approximation with the same number of decision variables.

Paper Structure

This paper contains 13 sections, 1 theorem, 30 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and motivation
  3. Polynomial approximation
  4. Generalised rational approximation
  5. Optimisation problems
  6. Multivariate polynomial approximation
  7. Multivariate generalised rational approximation
  8. Bisection method for multivariate generalised rational approximation
  9. Solving the convex feasibility subproblem
  10. Linear Programming
  11. Splitting Method
  12. Numerical experiments
  13. Conclusions and future research directions

Key Result

Theorem 1

Function is quasiconvex on $X$.

Figures (3)

  • Figure 1: The original function $f(x,y)$.
  • Figure 2: Multivariate polynomial approximation $P({\bf A},x,y)$ of degree 4 and its error curve.
  • Figure 3: Multivariate generalised rational approximation $P({\bf A},x,y)$ of degree 4 and its error curve.

Theorems & Definitions (5)

  • Definition 1
  • Definition 2
  • Definition 3
  • Example 1
  • Theorem 1