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The dimension weighted fast multipole method for scattered data approximation

Helmut Harbrecht, Michael Multerer, Jacopo Quizi

TL;DR

The paper addresses high-dimensional scattered data interpolation in the presence of anisotropic dimension weights by developing a dimension-weighted black-box fast multipole method (DW-FMM) that uses degenerate kernel approximations based on anisotropic polynomial interpolation. It establishes analytic extendability and dimension-free error bounds for the transported kernel, and derives a comprehensive theory for weighted total degree polynomial approximation with mild dimension dependence. The method combines hierarchical clustering, farfield degenerate-kernel expansions with transported Legendre polynomials, and approximate Fekete points to achieve dimension-robust accuracy with controllable costs, validated by numerical experiments including shape uncertainty quantification. The results demonstrate substantial reduction in interpolation points and robust accuracy in up to 20 dimensions, highlighting practical impact for high-dimensional uncertainty quantification and related applications.

Abstract

The present article is concerned scattered data approximation for higher dimensional data sets which exhibit an anisotropic behavior in the different dimensions. Tailoring sparse polynomial interpolation to this specific situation, we derive very efficient degenerate kernel approximations which we then use in a dimension weighted fast multipole method. This dimension weighted fast multipole method enables to deal with many more dimensions than the standard black-box fast multipole method based on interpolation. A thorough analysis of the method is provided including rigorous error estimates. The accuracy and the cost of the approach are validated by extensive numerical results. As a relevant application, we apply the approach to a shape uncertainty quantification problem.

The dimension weighted fast multipole method for scattered data approximation

TL;DR

The paper addresses high-dimensional scattered data interpolation in the presence of anisotropic dimension weights by developing a dimension-weighted black-box fast multipole method (DW-FMM) that uses degenerate kernel approximations based on anisotropic polynomial interpolation. It establishes analytic extendability and dimension-free error bounds for the transported kernel, and derives a comprehensive theory for weighted total degree polynomial approximation with mild dimension dependence. The method combines hierarchical clustering, farfield degenerate-kernel expansions with transported Legendre polynomials, and approximate Fekete points to achieve dimension-robust accuracy with controllable costs, validated by numerical experiments including shape uncertainty quantification. The results demonstrate substantial reduction in interpolation points and robust accuracy in up to 20 dimensions, highlighting practical impact for high-dimensional uncertainty quantification and related applications.

Abstract

The present article is concerned scattered data approximation for higher dimensional data sets which exhibit an anisotropic behavior in the different dimensions. Tailoring sparse polynomial interpolation to this specific situation, we derive very efficient degenerate kernel approximations which we then use in a dimension weighted fast multipole method. This dimension weighted fast multipole method enables to deal with many more dimensions than the standard black-box fast multipole method based on interpolation. A thorough analysis of the method is provided including rigorous error estimates. The accuracy and the cost of the approach are validated by extensive numerical results. As a relevant application, we apply the approach to a shape uncertainty quantification problem.
Paper Structure (14 sections, 14 theorems, 96 equations, 10 figures)

This paper contains 14 sections, 14 theorems, 96 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Problem statement
  3. Kernels on anisotropic data sets
  4. Weighted total degree polynomial approximation
  5. The dimension weighted fast multipole method
  6. Clustering
  7. Farfield approximation
  8. Error analysis
  9. Approximate Fekete points
  10. Numerical experiments
  11. Comparison between interpolation techniques
  12. Accuracy of the kernel interpolation
  13. Shape uncertainty quantification
  14. Conclusion

Key Result

Lemma 3.1

The derivatives of the transported kernel satisfy the bound with ${{\boldsymbol b}}\mathrel{\mathrel{\mathop:}=}[b_1\ldots,b_d]$.

Figures (10)

  • Figure 1: Considering quasi-uniform data sites in an anisotropic bounding box yields isotropic clusters. As a consequence, the nearfield becomes feasible.
  • Figure 2: Number of interpolation points required for TDI, TPI, and WTDI based on a quadratic increase of the sequence $\{\tau_k\}$ of the convergence radii.
  • Figure 3: Comparison of the number of points used by WTDI based on the different rates of increase $\tau_k = k^{r}$ for $r=2,3,4$.
  • Figure 4: Average prediction error (left) and average compression error (right) for the benchmark study in case of $r=2$.
  • Figure 5: Average prediction error (left) and average compression error (right) for the benchmark study in case of $r=3$.
  • ...and 5 more figures

Theorems & Definitions (32)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Corollary 3.3
  • proof
  • Lemma 4.1
  • ...and 22 more