Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Multiscale scattered data analysis in samplet coordinates

Sara Avesani, Rüdiger Kempf, Michael Multerer, Holger Wendland

TL;DR

The paper develops a multiscale interpolation framework for scattered data using globally supported radial basis functions of Matérn type, augmented by samplet-based compression to enable large-scale computations. By proving uniform conditioning of the scaled diagonal blocks and deriving comprehensive error bounds, the authors show that the resulting system remains well-conditioned and tractable even for millions of data sites, with an overall cost of $\mathcal{O}(N\log^2 N)$. The method combines a hierarchical residual-correction scheme across levels with a samplet transform that yields a sparse, compressible representation of the generalized Vandermonde matrices, enabling efficient solution via preconditioned conjugate gradients. Numerical experiments in 2D and 3D demonstrate exponential convergence with level, stable performance across smooth and non-smooth targets, and competitiveness with compactly supported approaches, thereby enabling practical large-scale scattered-data interpolation using globally supported Matérn kernels.

Abstract

We study multiscale scattered data interpolation schemes for globally supported radial basis functions with focus on the Matérn 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.

Multiscale scattered data analysis in samplet coordinates

TL;DR

The paper develops a multiscale interpolation framework for scattered data using globally supported radial basis functions of Matérn type, augmented by samplet-based compression to enable large-scale computations. By proving uniform conditioning of the scaled diagonal blocks and deriving comprehensive error bounds, the authors show that the resulting system remains well-conditioned and tractable even for millions of data sites, with an overall cost of . The method combines a hierarchical residual-correction scheme across levels with a samplet transform that yields a sparse, compressible representation of the generalized Vandermonde matrices, enabling efficient solution via preconditioned conjugate gradients. Numerical experiments in 2D and 3D demonstrate exponential convergence with level, stable performance across smooth and non-smooth targets, and competitiveness with compactly supported approaches, thereby enabling practical large-scale scattered-data interpolation using globally supported Matérn kernels.

Abstract

We study multiscale scattered data interpolation schemes for globally supported radial basis functions with focus on the Matérn 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 data sites, and local approximation spaces with exponentially decreasing dimension, the samplet compressed multiscale system can be assembled with cost . The overall cost of the proposed approach is . The theoretical findings are accompanied by extensive numerical studies in two and three spatial dimensions.
Paper Structure (11 sections, 14 theorems, 67 equations, 5 figures, 10 tables, 1 algorithm)

This paper contains 11 sections, 14 theorems, 67 equations, 5 figures, 10 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Classical multiscale method
  3. Condition number bound for generalized Vandermonde matrices
  4. Numerical approximation of the multiscale system
  5. Samplets for scattered data approximation
  6. Multiscale algorithm in samplet coordinates
  7. Numerical results
  8. Franke's function in two spatial dimensions
  9. Non-smooth function
  10. Function on a point cloud in three spatial dimensions
  11. Conclusion

Key Result

Theorem 2.1

Let $\Omega \subseteq \mathbb{R}^d$ be a bounded domain with Lipschitz boundary. Let $X_1, X_2, \dots$ be a sequence of point sets in $\Omega$ with fill-distances $h_1, h_2, \dots$ satisfying for $\ell = 1,2, \dots$ with fixed $\mu \in (0,1)$, $c \in (0,1]$ and $h_1$ sufficiently small. Further, let $\Phi\colon \mathbb{R}^d \to \mathbb{R}$ be a reproducing kernel for $H^\theta(\mathbb{R}^d)$, i

Figures (5)

  • Figure 1: Example of the sparsity patterns of the matrices $\boldsymbol K^{\Sigma,\rho}_{\ell, \ell '}$ for $\ell, \ell ' \in \{1,2,3\}$ of the compressed linear system from \ref{['eq:InterpolationSystemSamplets']}.
  • Figure 2: Comparison of the overall assembly time (blue) with theoretical growth rates $N \log^{\alpha} N$ for $\alpha = 1,2,3$.
  • Figure 3: Solution (left) and corresponding residuals (right) for the interpolation of Franke's function for levels $\ell=1, 4, 7, 10$.
  • Figure 4: Solution (left) and corresponding residuals (right) for the interpolation of the solution to the Laplace equation on the L-shaped domain for levels $\ell=2, 4, 6, 8$.
  • Figure 5: Interpolant of the test function on the Lucy point cloud for levels $\ell=1, \ldots, 5$ using the Matérn-1/2 RBF.

Theorems & Definitions (22)

  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • proof
  • ...and 12 more