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Multilevel Sparse Tensor Approximation for High-Dimensional Parametric PDEs

Martin Eigel, Philipp Trunschke, Dana Wrischnig

Abstract

In this paper the efficiency of multilevel sparse tensor approximation methods for high-dimensional affine parametric diffusion equations is investigated. Methodologically, the recently presented Sparse Alternating Least Squares (SALS) algorithm is employed to construct adaptive tensor train (TT) approximations of quantities of interest (QoI). By combining this tensor-based approach with a multilevel Galerkin discretization strategy, the solution's regularity can be exploited to significantly reduce computational costs by level-adapted sample sizes. A rigorous theoretical analysis is derived, demonstrating that the work overhead for the proposed multilevel method remains independent of the discretization level, which stands in stark contrast to the exponential growth observed in single-level approaches. The presented analysis is quite general and not constrained to the sparse TT format but uses a generic framework that can be extended to other model classes. Numerical experiments validate the predicted efficiency gains in high-dimensional settings.

Multilevel Sparse Tensor Approximation for High-Dimensional Parametric PDEs

Abstract

In this paper the efficiency of multilevel sparse tensor approximation methods for high-dimensional affine parametric diffusion equations is investigated. Methodologically, the recently presented Sparse Alternating Least Squares (SALS) algorithm is employed to construct adaptive tensor train (TT) approximations of quantities of interest (QoI). By combining this tensor-based approach with a multilevel Galerkin discretization strategy, the solution's regularity can be exploited to significantly reduce computational costs by level-adapted sample sizes. A rigorous theoretical analysis is derived, demonstrating that the work overhead for the proposed multilevel method remains independent of the discretization level, which stands in stark contrast to the exponential growth observed in single-level approaches. The presented analysis is quite general and not constrained to the sparse TT format but uses a generic framework that can be extended to other model classes. Numerical experiments validate the predicted efficiency gains in high-dimensional settings.
Paper Structure (26 sections, 15 theorems, 147 equations, 1 figure, 2 algorithms)

This paper contains 26 sections, 15 theorems, 147 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related work and contributions
  4. Setting
  5. Work estimates for general multilevel methods
  6. Single-level approximation
  7. Multi-level approximation
  8. A multilevel method using sparse Tensor Train approximation
  9. General definitions
  10. Definitions of the model classes
  11. Proof of assumption \ref{['asn:level_error']}
  12. Proof of assumption \ref{['asn:approx_error_eps']}
  13. Proof of assumption \ref{['asn:emp_approx_error']}
  14. Proof of assumption \ref{['asn:work_bound']}
  15. M-independent boundedness of all constants
  16. ...and 11 more sections

Key Result

Theorem 1

The single-level approximation satisfies the error bound with probability $1-p$ and a work overhead bounded by $\omega^{\mathrm{SL}}_L \lesssim 2^{\alpha\beta L} \log(p^{-1})$.

Figures (1)

  • Figure 1: Median RMSE vs. required work for SALS and SSALS with weaker (top) and stronger (bottom) decaying weights. Both methods converge similarly, stronger decay improves accuracy, and increasing the number of levels $L$ enhances the multilevel efficiency. Dashed lines indicate reference rates.

Theorems & Definitions (26)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 3
  • Definition 4
  • Theorem 5
  • Theorem 6
  • Theorem 7: Theorem 32.2 in Ern2021FEII.
  • Theorem 8
  • ...and 16 more