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Numerical Aspects of the Tensor Product Multilevel Method for High-dimensional, Kernel-based Reconstruction on Sparse Grids

Markus Büttner, Rüdiger Kempf, Holger Wendland

TL;DR

The paper addresses high-dimensional function reconstruction on Cartesian product domains by integrating Smolyak’s sparse-grid framework with a kernel-based multilevel residual-correction scheme (TPML). It presents two practical modifications to turn the theoretically derived TPML into a feasible method: (i) moving substantial computations offline to reduce online cost, and (ii) a nodal representation that exploits nested site sets to evaluate each data point only once and enable GPU-friendly implementations. Theoretical developments are complemented by numerical demonstrations on a 2D space-time tidal-flow interpolation and a 7D cantilever-beam QOI, showing favorable accuracy (e.g., $95\%$ of points achieving $\lesssim 10^{-2}$ error) and convergence behavior across multiple levels. Collectively, the work advances scalable high-dimensional scattered-data approximation on irregular domains with compactly supported kernels, expanding TPML’s applicability to real-world, high-dimensional problems. The proposed offline and nodal strategies have practical impact for efficient, parallelizable reconstruction in physics, engineering, and uncertainty quantification contexts.

Abstract

This paper investigates the approximation of functions with finite smoothness defined on domains with a Cartesian product structure. The recently proposed tensor product multilevel method (TPML) combines Smolyak's sparse grid method with a kernel-based residual correction technique. The contributions of this paper are twofold. First, we present two improvements on the TPML that reduce the computational cost of point evaluations compared to a naive implementation. Second, we provide numerical examples that demonstrate the effectiveness and innovation of the TPML.

Numerical Aspects of the Tensor Product Multilevel Method for High-dimensional, Kernel-based Reconstruction on Sparse Grids

TL;DR

The paper addresses high-dimensional function reconstruction on Cartesian product domains by integrating Smolyak’s sparse-grid framework with a kernel-based multilevel residual-correction scheme (TPML). It presents two practical modifications to turn the theoretically derived TPML into a feasible method: (i) moving substantial computations offline to reduce online cost, and (ii) a nodal representation that exploits nested site sets to evaluate each data point only once and enable GPU-friendly implementations. Theoretical developments are complemented by numerical demonstrations on a 2D space-time tidal-flow interpolation and a 7D cantilever-beam QOI, showing favorable accuracy (e.g., of points achieving error) and convergence behavior across multiple levels. Collectively, the work advances scalable high-dimensional scattered-data approximation on irregular domains with compactly supported kernels, expanding TPML’s applicability to real-world, high-dimensional problems. The proposed offline and nodal strategies have practical impact for efficient, parallelizable reconstruction in physics, engineering, and uncertainty quantification contexts.

Abstract

This paper investigates the approximation of functions with finite smoothness defined on domains with a Cartesian product structure. The recently proposed tensor product multilevel method (TPML) combines Smolyak's sparse grid method with a kernel-based residual correction technique. The contributions of this paper are twofold. First, we present two improvements on the TPML that reduce the computational cost of point evaluations compared to a naive implementation. Second, we provide numerical examples that demonstrate the effectiveness and innovation of the TPML.

Paper Structure

This paper contains 13 sections, 8 theorems, 49 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Tensor Product Multilevel Method
  3. Smolyak's Method
  4. Kernel Multilevel Method using Lagrange Functions
  5. The Tensor Product Multilevel Method
  6. Modifications of the Tensor Product Multilevel Method
  7. Towards more efficient point-evaluations
  8. A Nodal Representation
  9. Numerical Examples
  10. Tidal Flow at the Bight of Abaco
  11. Cantilever beam
  12. Proofs of the statements of \ref{['sec:Efficient\nEvaluations']}
  13. Proofs of the statements of \ref{['subsec:NodalRepresentation']}

Key Result

Theorem 2.5

Let $\mathfrak{u} = \{u_1, \dots, u_{\# \mathfrak{u}} \}$ be an ordered set. Define the combined operator$\mathcal{I}_{\mathfrak{u}}: C(\Omega) \to W_{\# \mathfrak{u}}$ by where the coefficients are given by $a(\mathfrak{u}, \boldsymbol{k}) = 1$, if $\# \mathfrak{u} = 1$ and for $\# \mathfrak{u} > 1$. Then the multilevel approximation operator $A_L: C(\Omega) \to \bigoplus_{i=1}^L W_i$ at level

Figures (6)

  • Figure 1: Bathymetry for the Bight of Abaco. Horizontal extends of the domain are 70 km in $x$ direction and 100 km in $y$ direction. Four virtual recording stations report the variables $\xi$, $U$ and $V$ at the positions marked by 1 -- 4.
  • Figure 2: Absolute pointwise difference for $U$ on the whole mesh (left), simulation and interpolation result for $U$ near the island (right).
  • Figure 3: Average pointwise error for the Bahamas example with different number of levels and variables.
  • Figure 4: Depth-integrated velocity in $x$ direction at recording station 2.
  • Figure 5: Cantilever beam from migliorati:ApproximationQOI.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Definition 2.6
  • Remark 2.7
  • Lemma 3.1
  • Remark 3.2
  • Theorem 3.3
  • ...and 13 more