Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Kernel interpolation on generalized sparse grids

Michael Griebel, Helmut Harbrecht, Michael Multerer

TL;DR

This work develops kernel interpolation on generalized sparse grids for product regions of varying dimension by leveraging reproducing kernel Hilbert spaces and product kernels. It derives improved error estimates via duality, and implements an efficient sparse grid algorithm based on the combination technique combined with samplet-based compression to handle nonlocal kernels. The approach enables solving extremely large problems with billions of interpolation points and demonstrates strong scalability across up to 18 dimensions in mixed-dimension geometries. Numerical experiments with Matérn kernels validate the theory and illustrate practical performance gains for high-dimensional scattered data interpolation.

Abstract

We consider scattered data approximation on product regions of equal and different dimensionality. On each of these regions, we assume quasi-uniform but unstructured data sites and construct optimal sparse grids for scattered data interpolation on the product region. For this, we derive new improved error estimates for the respective kernel interpolation error by invoking duality arguments. An efficient algorithm to solve the underlying linear system of equations is proposed. The algorithm is based on the sparse grid combination technique, where a sparse direct solver is used for the elementary anisotropic tensor product kernel interpolation problems. The application of the sparse direct solver is facilitated by applying a samplet matrix compression to each univariate kernel matrix, resulting in an essentially sparse representation of the latter. In this way, we obtain a method that is able to deal with large problems up to billions of interpolation points, especially in case of reproducing kernels of nonlocal nature. Numerical results are presented to qualify and quantify the approach.

Kernel interpolation on generalized sparse grids

TL;DR

This work develops kernel interpolation on generalized sparse grids for product regions of varying dimension by leveraging reproducing kernel Hilbert spaces and product kernels. It derives improved error estimates via duality, and implements an efficient sparse grid algorithm based on the combination technique combined with samplet-based compression to handle nonlocal kernels. The approach enables solving extremely large problems with billions of interpolation points and demonstrates strong scalability across up to 18 dimensions in mixed-dimension geometries. Numerical experiments with Matérn kernels validate the theory and illustrate practical performance gains for high-dimensional scattered data interpolation.

Abstract

We consider scattered data approximation on product regions of equal and different dimensionality. On each of these regions, we assume quasi-uniform but unstructured data sites and construct optimal sparse grids for scattered data interpolation on the product region. For this, we derive new improved error estimates for the respective kernel interpolation error by invoking duality arguments. An efficient algorithm to solve the underlying linear system of equations is proposed. The algorithm is based on the sparse grid combination technique, where a sparse direct solver is used for the elementary anisotropic tensor product kernel interpolation problems. The application of the sparse direct solver is facilitated by applying a samplet matrix compression to each univariate kernel matrix, resulting in an essentially sparse representation of the latter. In this way, we obtain a method that is able to deal with large problems up to billions of interpolation points, especially in case of reproducing kernels of nonlocal nature. Numerical results are presented to qualify and quantify the approach.
Paper Structure (22 sections, 3 theorems, 74 equations, 10 figures, 2 tables, 4 algorithms)

This paper contains 22 sections, 3 theorems, 74 equations, 10 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reproducing kernel Hilbert spaces
  4. Error estimates
  5. Multilevel sequences
  6. Multivariate setting
  7. Tensor product spaces
  8. Sparse tensor product spaces
  9. Error estimates
  10. Comparison of sampling rates
  11. Sparse grid combination technique
  12. Implementation
  13. Construction of nested point sets
  14. Computing the sparse grid kernel interpolant
  15. Fast solution of the linear system of equations
  16. ...and 7 more sections

Key Result

Theorem 3.1

Let ${\boldsymbol 0}\le{\boldsymbol t}<{\boldsymbol s}<{\boldsymbol t}'\le 2{\boldsymbol s}$ and $f\in {\boldsymbol H}^{{\boldsymbol t}'}(\boldsymbol\Omega)$. Then, there holds the error estimate Here, $P\in \mathbb{N}$ counts how often the minimum is attained in the exponent.

Figures (10)

  • Figure 1: Visualization of the subsampling procedure starting from a set of 1000 uniformly chosen random points on $[0,1]^2$.
  • Figure 2: Sparsity patterns of the samplet compressed exponential kernel on the unit square (left) for $300\,000$ data sites, the nested dissection reordering (middle), and the Cholesky factor (right). Each dot represents a matrix block of size $300\times 300$. The number of entries per block is color coded, where lighter blocks have less entries.
  • Figure 3: Computation times for the canonical sparse grid on the unit hypercube $(0,1)^m$ and $m=3,6,9,12,15,18$.
  • Figure 4: Convergence of the kernel interpolant on the canonical sparse grid in $(0,1)^m$.
  • Figure 5: Sketch of the regular grid points (blue) and the evaluation points (red) on the unit interval, the unit square, and the unit cube.
  • ...and 5 more figures

Theorems & Definitions (7)

  • Definition 2.1
  • Theorem 3.1: Convergence
  • proof
  • Remark 3.2
  • Theorem 3.3: Complexity
  • proof
  • Theorem 3.4: Cost-complexity rate