Minimum Sobolev norm interpolation of derivative data

TL;DR

This paper investigates the problem of reconstructing functions on manifolds using data on values and derivatives, demonstrating that high-degree polynomial solutions always exist under mild conditions and converge to the target function as sampling density increases.

Contribution

It establishes the existence of solutions for derivative data interpolation on manifolds with high-degree polynomials and provides explicit constructions and convergence estimates.

Findings

Solutions exist for sufficiently high-degree polynomials given derivative data.

The constructed polynomials converge to the target function with dense sampling.

Numerical experiments show the method's stability and high-order accuracy.

Abstract

We study the problem of reconstructing a function on a manifold satisfying some mild conditions, given data on the values and some derivatives of the function at arbitrary points on the manifold. While the problem of finding a polynomial of two variables with total degree $\le n$ given the values of the polynomial and some of its derivatives at exactly the same number of points as the dimension of the polynomial space is sometimes impossible, we show that such a problem always has a solution in a very general situation if the degree of the polynomials is sufficiently large. We give estimates on how large the degree should be, and give explicit constructions for such a polynomial even in a far more general case. As the number of sampling points at which the data is available increases, our polynomials converge to the target function on the set where the sampling points are dense. Numerical examples in single and double precision show that this method is stable and of high-order.