Fitting a Sobolev function to data

TL;DR

This paper presents an efficient algorithm to extend data defined on a finite set to a Sobolev space function with minimal norm, with applications in data fitting and approximation.

Contribution

The paper introduces a novel algorithm for Sobolev extension that is computationally efficient, running in near-linear time relative to data size.

Findings

Algorithm computes minimal-norm Sobolev extension efficiently

Runtime is at most proportional to N log N for N data points

Provides both the extension and the norm's order of magnitude

Abstract

We exhibit an algorithm to solve the following extension problem: Given a finite set $E \subset \mathbb{R}^n$ and a function $f: E \rightarrow \mathbb{R}$, compute an extension $F$ in the Sobolev space $L^{m,p}(\mathbb{R}^n)$, $p>n$, with norm having the smallest possible order of magnitude, and secondly, compute the order of magnitude of the norm of $F$. Here, $L^{m,p}(\mathbb{R}^n)$ denotes the Sobolev space consisting of functions on $\mathbb{R}^n$ whose $m$th order partial derivatives belong to $L^p(\mathbb{R}^n)$. The running time of our algorithm is at most $C N \log N$, where $N$ denotes the cardinality of $E$, and $C$ is a constant depending only on $m$,$n$, and $p$.