Compressive Hermite interpolation: sparse, high-dimensional approximation from gradient-augmented measurements

TL;DR

This paper extends sparse polynomial approximation methods to include gradient measurements, demonstrating improved error bounds and reduced dimensionality effects, supported by numerical experiments.

Contribution

It introduces a gradient-augmented approach to polynomial approximation, achieving stronger error bounds with similar sample complexity, reducing the curse of dimensionality.

Findings

Gradient-augmented measurements improve approximation error bounds.

Sample complexity remains algebraic in sparsity and logarithmic in dimension.

Numerical experiments confirm the benefits of gradient information.

Abstract

We consider the sparse polynomial approximation of a multivariate function on a tensor product domain from samples of both the function and its gradient. When only function samples are prescribed, weighted $\ell^1$ minimization has recently been shown to be an effective procedure for computing such approximations. We extend this work to the gradient-augmented case. Our main results show that for the same asymptotic sample complexity, gradient-augmented measurements achieve an approximation error bound in a stronger Sobolev norm, as opposed to the $L^2$-norm in the unaugmented case. For Chebyshev and Legendre polynomial approximations, this sample complexity estimate is algebraic in the sparsity $s$ and at most logarithmic in the dimension $d$, thus mitigating the curse of dimensionality to a substantial extent. We also present several experiments numerically illustrating the benefits of gradient information over an equivalent number of function samples only.