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Low-rank cross approximation of function-valued tensors for reduced-order modeling of parametric PDEs

Stanislav Budzinskiy, Vladimir Kazeev, Maxim Olshanskii

TL;DR

The paper develops a data-driven, non-intrusive reduced-order modeling framework for parametric PDEs by extending low-rank tensor approximations to function-valued tensors with entries in a Hilbert space. It introduces Tucker-based decompositions, Tucker-cross approximations, and higher-order SVD in this function-valued setting, along with an adaptive cross-approximation algorithm (TuckerABC) that selects informative samples to build compact encoder-decoder representations of parameter-to-solution maps. The proposed approach yields an interpolatory ROM that is exact on a chosen subgrid and can be integrated with or without projection-based steps, providing a scalable, physics-informed alternative to POD-Galerkin and neural network-based surrogates. Numerical experiments on nonlinear parametric Stokes and Monge–Ampère equations demonstrate the method’s accuracy and robustness under mesh refinement and different inner-product choices, highlighting its potential for efficient, non-intrusive model order reduction of nonlinear PDEs.

Abstract

The paper considers function-valued tensors, viewed as multidimensional arrays with entries in an abstract Hilbert space. Despite the absence of the algebraic structure of a field, the geometric inner-product structure suffices to introduce the Tucker rank, higher-order SVD, and Tucker-cross decomposition for function-valued tensors. An adaptive cross-approximation algorithm is developed to compute low-rank approximations of such tensors. The framework is motivated by, and applied to, model order reduction of the parameter-to-solution map for a parametric PDE. The resulting reduced-order model can be interpreted as an encoder-decoder scheme with a nonlinear encoder and a multilinear decoder. The performance of the proposed non-intrusive approximation method is demonstrated in numerical examples for two nonlinear parametric PDE systems.

Low-rank cross approximation of function-valued tensors for reduced-order modeling of parametric PDEs

TL;DR

The paper develops a data-driven, non-intrusive reduced-order modeling framework for parametric PDEs by extending low-rank tensor approximations to function-valued tensors with entries in a Hilbert space. It introduces Tucker-based decompositions, Tucker-cross approximations, and higher-order SVD in this function-valued setting, along with an adaptive cross-approximation algorithm (TuckerABC) that selects informative samples to build compact encoder-decoder representations of parameter-to-solution maps. The proposed approach yields an interpolatory ROM that is exact on a chosen subgrid and can be integrated with or without projection-based steps, providing a scalable, physics-informed alternative to POD-Galerkin and neural network-based surrogates. Numerical experiments on nonlinear parametric Stokes and Monge–Ampère equations demonstrate the method’s accuracy and robustness under mesh refinement and different inner-product choices, highlighting its potential for efficient, non-intrusive model order reduction of nonlinear PDEs.

Abstract

The paper considers function-valued tensors, viewed as multidimensional arrays with entries in an abstract Hilbert space. Despite the absence of the algebraic structure of a field, the geometric inner-product structure suffices to introduce the Tucker rank, higher-order SVD, and Tucker-cross decomposition for function-valued tensors. An adaptive cross-approximation algorithm is developed to compute low-rank approximations of such tensors. The framework is motivated by, and applied to, model order reduction of the parameter-to-solution map for a parametric PDE. The resulting reduced-order model can be interpreted as an encoder-decoder scheme with a nonlinear encoder and a multilinear decoder. The performance of the proposed non-intrusive approximation method is demonstrated in numerical examples for two nonlinear parametric PDE systems.

Paper Structure

This paper contains 21 sections, 5 theorems, 44 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Outline
  3. Related work on function-valued tensors
  4. Related work on tensor methods for parametric PDEs
  5. Low-rank cross approximation as reduced-order modeling
  6. Matrix cross approximation
  7. Bi-parametric PDE with vector- and function-valued QoI
  8. Function-valued matrix cross approximation
  9. Reduced-order modeling
  10. Encoder-decoder scheme
  11. Interpolation on a subgrid
  12. Comparison with the reduced-basis method
  13. Cross approximation of function-valued tensors
  14. Tucker decomposition
  15. Tucker-cross approximation
  16. ...and 6 more sections

Key Result

Lemma 3.1

Let $\bm{\mathsf{A}} \in \mathrm{H}^{m \times n}$ and $r = \mathrm{rank_{c}} ( \bm{\mathsf{A}} )$, and let $I \subseteq [m]$ and $J \subseteq [n]$.

Figures (3)

  • Figure 1: Numerical finite-element solution to eq:stokeseq:stokes_bcs obtained with FEniCS.
  • Figure 2: Relative $\ell_2(\mathbb{R}^{100})$-norm approximation errors of TuckerABC as applied to the function-valued tensor $\bm{{\mathscr{A}}} \in (\mathbb{R}^{100})^{64 \times 64 \times 64}$ of reduced-basis solutions to eq:stokeseq:stokes_bcs for different hyperparameters of the algorithm.
  • Figure 3: Numerical finite-element solution to \ref{['eq:monge_ampere_vmm']} obtained with FEniCS.

Theorems & Definitions (11)

  • Remark 2.1
  • Lemma 3.1
  • Proof 1
  • Proposition 3.2
  • Proof 2
  • Proposition 3.3
  • Proof 3
  • Theorem 3.4
  • Proof 4
  • Theorem 3.5
  • ...and 1 more