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A tensor-train reduced basis solver for parameterized partial differential equations on Cartesian grids

Nicholas Mueller, Yiran Zhao, Santiago Badia, Tiangang Cui

TL;DR

The paper develops a tensor-train reduced basis (TT-RB) framework for parameterized PDEs, exploiting a split-axes representation of space, time, and parameters to form TT decompositions of snapshot data. By combining TT-SVD basis construction, TT-MDEIM hyper-reduction, and a Galerkin projection, TT-RB achieves online-efficient reduced systems with offline costs significantly lower than traditional TPOD-based RB methods, especially on high-dimensional Cartesian grids. The authors derive a posteriori error estimates and demonstrate substantial offline speedups and competitive online accuracy across Poisson, heat, and transient elasticity problems in both 2D and 3D, highlighting the method’s potential for scalable, high-fidelity ROMs. The work also discusses extensions to unfitted finite elements for non-Cartesian geometries and outlines future directions including saddle-point and nonlinear problems.

Abstract

In this manuscript, we introduce the tensor-train reduced basis method, a novel projection-based reduced-order model designed for the efficient solution of parameterized partial differential equations. While reduced-order models are widely used for their computational efficiency compared to full-order models, they often involve significant offline computational costs. Our proposed approach mitigates this limitation by leveraging the tensor train format to efficiently represent high-dimensional finite element quantities. This method offers several advantages, including a reduced number of operations for constructing the reduced subspaces, a cost-effective hyper-reduction strategy for assembling the PDE residual and Jacobian, and a lower dimensionality of the projection subspaces for a given accuracy. We provide a posteriori error estimates to validate the accuracy of the method and evaluate its computational performance on benchmark problems, including the Poisson equation, heat equation, and transient linear elasticity in two- and three-dimensional domains. Although the current framework is restricted to problems defined on Cartesian grids, we anticipate that it can be extended to arbitrary shapes by integrating the tensor-train reduced basis method with unfitted finite element techniques.

A tensor-train reduced basis solver for parameterized partial differential equations on Cartesian grids

TL;DR

The paper develops a tensor-train reduced basis (TT-RB) framework for parameterized PDEs, exploiting a split-axes representation of space, time, and parameters to form TT decompositions of snapshot data. By combining TT-SVD basis construction, TT-MDEIM hyper-reduction, and a Galerkin projection, TT-RB achieves online-efficient reduced systems with offline costs significantly lower than traditional TPOD-based RB methods, especially on high-dimensional Cartesian grids. The authors derive a posteriori error estimates and demonstrate substantial offline speedups and competitive online accuracy across Poisson, heat, and transient elasticity problems in both 2D and 3D, highlighting the method’s potential for scalable, high-fidelity ROMs. The work also discusses extensions to unfitted finite elements for non-Cartesian geometries and outlines future directions including saddle-point and nonlinear problems.

Abstract

In this manuscript, we introduce the tensor-train reduced basis method, a novel projection-based reduced-order model designed for the efficient solution of parameterized partial differential equations. While reduced-order models are widely used for their computational efficiency compared to full-order models, they often involve significant offline computational costs. Our proposed approach mitigates this limitation by leveraging the tensor train format to efficiently represent high-dimensional finite element quantities. This method offers several advantages, including a reduced number of operations for constructing the reduced subspaces, a cost-effective hyper-reduction strategy for assembling the PDE residual and Jacobian, and a lower dimensionality of the projection subspaces for a given accuracy. We provide a posteriori error estimates to validate the accuracy of the method and evaluate its computational performance on benchmark problems, including the Poisson equation, heat equation, and transient linear elasticity in two- and three-dimensional domains. Although the current framework is restricted to problems defined on Cartesian grids, we anticipate that it can be extended to arbitrary shapes by integrating the tensor-train reduced basis method with unfitted finite element techniques.

Paper Structure

This paper contains 18 sections, 4 theorems, 128 equations, 3 figures, 8 tables, 6 algorithms.

Table of Contents

  1. Introduction
  2. Notation
  3. Reduced basis method in space time
  4. Full order model
  5. Space-time reduced-basis method
  6. A novel TT-RB solver
  7. Basis construction: a priori estimates
  8. Basis construction: the case of a rank-1 norm matrix
  9. Basis construction: the case of a rank-$K$ norm matrix
  10. Empirical interpolation method
  11. Approximation of the Jacobians
  12. Galerkin projection
  13. A posteriori error estimate
  14. Numerical results
  15. Poisson equation
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose the unfolding matrices $\bm{T}_{\widehat{i-1} i,i+1 : \mu}$ admit a low-rank approximation with relative errors $\varepsilon_i$ for given ranks $r_i$ : Then, the projection operator $\bm{\Phi}_{st,\widehat{t}}$ in eq: ttrb basis satisfies: where $\varepsilon = \sup_i \varepsilon_i$.

Figures (3)

  • Figure 1: Illustration of how the entry $\bm{U}_{1,2,t}^{\mu_{\star}}\left[i_1,i_2,i_t\right]$ is approximated using the tt cores. First, the vector-matrix product is performed between the first and second cores, evaluated at the specified indices. Next, the resulting row vector is multiplied by the column vector from the third core, also evaluated at the corresponding index. The resulting scalar represents the approximated tensor entry.
  • Figure 2: ttmdeim forward sweep, case $d=2$.
  • Figure 3: ttmdeim backward sweep, case $d=2$.

Theorems & Definitions (14)

  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Remark 3
  • Remark 4
  • Remark 5
  • Theorem 3
  • ...and 4 more