Multiscale scattered data analysis in samplet coordinates

TL;DR

This paper introduces a multiscale data interpolation method using samplet coordinates for radial basis functions, providing bounded condition numbers, error estimates, and efficient computational complexity for large datasets.

Contribution

The paper develops a multiscale interpolation framework with samplet coordinate representation, ensuring bounded condition numbers and efficient computation for large scattered data sets.

Findings

Bounded condition numbers for multiscale system blocks.

Error estimates for numerical approximation of the system.

Computational complexity of O(N log^2 N) for large data sets.

Abstract

We study multiscale scattered data interpolation schemes for globally supported radial basis functions with focus on the Mat\'ern class. The multiscale approximation is constructed through a sequence of residual corrections, where radial basis functions with different lengthscale parameters are combined to capture varying levels of detail. We prove that the condition numbers of the the diagonal blocks of the corresponding multiscale system remain bounded independently of the particular level, allowing us to use an iterative solver with a bounded number of iterations for the numerical solution. Employing an appropriate diagonal scaling, the multiscale system becomes well conditioned. We exploit this fact to derive a general error estimate bounding the consistency error issuing from a numerical approximation of the multiscale system. To apply the multiscale approach to large data sets, we suggest to represent each level of the multiscale system in samplet coordinates. Samplets are localized, discrete signed measures exhibiting vanishing moments and allow for the sparse approximation of generalized Vandermonde matrices issuing from a vast class of radial basis functions. Given a quasi-uniform set of $N$ data sites, and local approximation spaces with exponentially decreasing dimension, the samplet compressed multiscale system can be assembled with cost $\mathcal{O}(N \log^2 N)$. The overall cost of the proposed approach is $\mathcal{O}(N \log^2 N)$. The theoretical findings are accompanied by extensive numerical studies in two and three spatial dimensions.