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Multilevel lattice-based kernel approximation for elliptic PDEs with random coefficients

Alexander D. Gilbert, Michael B. Giles, Frances Y. Kuo, Ian H. Sloan, Abirami Srikumar

TL;DR

This work advances efficient approximation of elliptic PDEs with periodic random coefficients by coupling multilevel finite element discretization with lattice-based kernel interpolation over the parametric domain. The multilevel framework distributes work across a hierarchy of meshes and kernel interpolants, achieving the same $L^2$ accuracy as the single-level method but at reduced cost, with analytic error and cost bounds supported by regularity theory. A full parametric regularity analysis, along with an abstract complexity theorem and weight optimization, underpins the theoretical gains, while numerical experiments corroborate substantial speedups and accuracy improvements. The approach offers a practical pathway to scalable uncertainty quantification for high-dimensional parametric PDEs and sets the stage for potential adaptive extensions in the future.

Abstract

This paper introduces a multilevel kernel-based approximation method to estimate efficiently solutions to elliptic partial differential equations (PDEs) with periodic random coefficients. Building upon the work of Kaarnioja, Kazashi, Kuo, Nobile, Sloan (Numer. Math., 2022) on kernel interpolation with quasi-Monte Carlo (QMC) lattice point sets, we leverage multilevel techniques to enhance computational efficiency while maintaining a given level of accuracy. In the function space setting with product-type weight parameters, the single-level approximation can achieve an accuracy of $\varepsilon>0$ with cost $\mathcal{O}(\varepsilon^{-η-ν-θ})$ for positive constants $η, ν, θ$ depending on the rates of convergence associated with dimension truncation, kernel approximation, and finite element approximation, respectively. Our multilevel approximation can achieve the same $\varepsilon$ accuracy at a reduced cost $\mathcal{O}(\varepsilon^{-η-\max(ν,θ)})$. Full regularity theory and error analysis are provided, followed by numerical experiments that validate the efficacy of the proposed multilevel approximation in comparison to the single-level approach.

Multilevel lattice-based kernel approximation for elliptic PDEs with random coefficients

TL;DR

This work advances efficient approximation of elliptic PDEs with periodic random coefficients by coupling multilevel finite element discretization with lattice-based kernel interpolation over the parametric domain. The multilevel framework distributes work across a hierarchy of meshes and kernel interpolants, achieving the same accuracy as the single-level method but at reduced cost, with analytic error and cost bounds supported by regularity theory. A full parametric regularity analysis, along with an abstract complexity theorem and weight optimization, underpins the theoretical gains, while numerical experiments corroborate substantial speedups and accuracy improvements. The approach offers a practical pathway to scalable uncertainty quantification for high-dimensional parametric PDEs and sets the stage for potential adaptive extensions in the future.

Abstract

This paper introduces a multilevel kernel-based approximation method to estimate efficiently solutions to elliptic partial differential equations (PDEs) with periodic random coefficients. Building upon the work of Kaarnioja, Kazashi, Kuo, Nobile, Sloan (Numer. Math., 2022) on kernel interpolation with quasi-Monte Carlo (QMC) lattice point sets, we leverage multilevel techniques to enhance computational efficiency while maintaining a given level of accuracy. In the function space setting with product-type weight parameters, the single-level approximation can achieve an accuracy of with cost for positive constants depending on the rates of convergence associated with dimension truncation, kernel approximation, and finite element approximation, respectively. Our multilevel approximation can achieve the same accuracy at a reduced cost . Full regularity theory and error analysis are provided, followed by numerical experiments that validate the efficacy of the proposed multilevel approximation in comparison to the single-level approach.

Paper Structure

This paper contains 24 sections, 9 theorems, 101 equations, 5 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Notation
  4. Parametric variational formulation
  5. Dimension truncation
  6. Finite element methods
  7. Lattice-based kernel interpolation
  8. Single-level kernel interpolation for PDEs
  9. Multilevel kernel approximation
  10. Error decomposition for multilevel methods
  11. Cost analysis of multilevel methods
  12. Abstract complexity analysis
  13. Parametric regularity analysis
  14. Multilevel error analysis
  15. Estimating the multilevel FE error
  16. ...and 9 more sections

Key Result

Theorem 1

Given $h_0\in (0,1)$ and $d\ge1$, define $h_{\ell} \coloneq h_0\,2^{-\ell}$ for $\ell\geq 0$, and suppose there are positive constants $\beta,\kappa,\mu,\rho$, and $\tau$ such that Given $0<\varepsilon<\min(1,2h_0^\beta)$, and assuming $d\leq\tau \leq d + \frac{\beta}{\kappa}$, we may choose integers $L$ given by eq:cond_L, $s\simeq h_L^{-\frac{\beta}{\kappa}}$, and $N_0,\ldots,N_L$ given by eq:N

Figures (5)

  • Figure 1: Cost for the FE solve on the left. The FE error $\|u^s_{h^*}-u^s_h\|_{L^2(\Omega\times D)}$ on the right. They demonstrate $\beta = 2$ and $\tau\approx d = 2$ in Assumptions \ref{['asm:trun_error']} and \ref{['asm:cost']} of Theorem \ref{['thm:ml-complexity']}.
  • Figure 2: Comparing dimension truncation error with FE error.
  • Figure 3: Matching single-level kernel interpolation error and FE error.
  • Figure 4: Estimates of $\|(I-I_N)(u_\ell - u_{\ell-1})\|_{L^2(\Omega\times D)}$ with varying $N$ and $\ell$.
  • Figure 5: Computation time for the single-level and multilevel approximations against their errors.

Theorems & Definitions (16)

  • Theorem 1
  • proof
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 5
  • proof
  • Theorem 6
  • Theorem 7
  • proof
  • ...and 6 more