Function approximation using gradient information with application to parametric and stochastic differential equations

TL;DR

This paper enhances multivariate function approximation by incorporating derivative information into the least squares method, reducing function evaluations while maintaining accuracy, with applications to polynomial basis and differential equations.

Contribution

It introduces a derivative-augmented least squares method for efficient multivariate function approximation, including techniques for fast derivative computation.

Findings

Reduced number of function evaluations needed for accurate approximation

Demonstrated effectiveness through numerical examples

Improved computational efficiency in polynomial and differential equation applications

Abstract

In the paper we consider the problem of multivariate function approximation in polynomial basis. In order to solve this problem, we adjust the least squares method (LSM) by adding information about derivatives of the function. This modification allows reducing the number of evaluations of approximating function while keeping the accuracy at the appropriate level. We propose several techniques for time-efficient calculation of derivatives in various applications. Numerical examples are given for comparison between the standard LSM and the proposed approach.