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Model order reduction of parametric dynamical systems by slice sampling tensor completion

Alexander V. Mamonov, Maxim A. Olshanskii

TL;DR

This work proposes a sparse sampling of the parameter domain, followed by a low-rank tensor completion, and introduces a low-rank tensor format called the hybrid tensor train, which is integrated into an interpolatory tensorial ROM.

Abstract

Recent studies have demonstrated the great potential of reduced order modeling for parametric dynamical systems using low-rank tensor decompositions (LRTD). In particular, within the framework of interpolatory tensorial reduced order models (ROM), LRTD is computed for tensors composed of snapshots of the system's solutions, where each parameter corresponds to a distinct tensor mode. This approach requires full sampling of the parameter domain on a tensor product grid, which suffers from the curse of dimensionality, making it practical only for systems with a small number of parameters. To overcome this limitation, we propose a sparse sampling of the parameter domain, followed by a low-rank tensor completion. The resulting specialized tensor completion problem is formulated for a tensor of order $C + D$, where $C$ fully sampled modes correspond to the snapshot degrees of freedom, and $D$ partially sampled modes correspond to the system's parameters. To address this non-standard tensor completion problem, we introduce a low-rank tensor format called the hybrid tensor train. Completion in this format is then integrated into an interpolatory tensorial ROM. We demonstrate the effectiveness of both the completion method and the ROM on several examples of dynamical systems derived from finite element discretizations of parabolic partial differential equations with parameter-dependent coefficients or boundary conditions.

Model order reduction of parametric dynamical systems by slice sampling tensor completion

TL;DR

This work proposes a sparse sampling of the parameter domain, followed by a low-rank tensor completion, and introduces a low-rank tensor format called the hybrid tensor train, which is integrated into an interpolatory tensorial ROM.

Abstract

Recent studies have demonstrated the great potential of reduced order modeling for parametric dynamical systems using low-rank tensor decompositions (LRTD). In particular, within the framework of interpolatory tensorial reduced order models (ROM), LRTD is computed for tensors composed of snapshots of the system's solutions, where each parameter corresponds to a distinct tensor mode. This approach requires full sampling of the parameter domain on a tensor product grid, which suffers from the curse of dimensionality, making it practical only for systems with a small number of parameters. To overcome this limitation, we propose a sparse sampling of the parameter domain, followed by a low-rank tensor completion. The resulting specialized tensor completion problem is formulated for a tensor of order , where fully sampled modes correspond to the snapshot degrees of freedom, and partially sampled modes correspond to the system's parameters. To address this non-standard tensor completion problem, we introduce a low-rank tensor format called the hybrid tensor train. Completion in this format is then integrated into an interpolatory tensorial ROM. We demonstrate the effectiveness of both the completion method and the ROM on several examples of dynamical systems derived from finite element discretizations of parabolic partial differential equations with parameter-dependent coefficients or boundary conditions.

Paper Structure

This paper contains 17 sections, 2 theorems, 67 equations, 5 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Slice sampling tensor completion
  3. Reduced basis for fully sampled space--time modes
  4. Tensor-train completion for partially sampled modes
  5. Algorithm for slice sampling tensor completion
  6. An estimate for the $\{C,D\}$-ranks
  7. Accuracy of HTT completion
  8. Adaptive completion.
  9. ROM for parametric dynamical systems
  10. Projection TROM
  11. In-tensor interpolation and finding the local basis
  12. Summary of the completion based TROM
  13. Complexity of CTROM
  14. Numerical examples
  15. Parameterized heat equation
  16. ...and 2 more sections

Key Result

Proposition 1

Tensor $\boldsymbol{\Phi}$ can be represented in the HTT format with $C$-ranks $\{q_1,\dots,q_{C}\}$ and $D$-ranks $\{r_0^{(\mathbf i)},\dots,r_{D}^{(\mathbf i)}\}$, ${\mathbf i\in \Omega_{C}^q}$, so that it holds for $i=1,\dots,\text{\small $C$}$, $j=1,\dots,\text{\small $D$}$, and for all $\mathbf i \in \Omega_{C}^q$.

Figures (5)

  • Figure 1: Domain $\Omega$ and the solution $u(\mathbf x, T, \boldsymbol{\alpha})$ of \ref{['eqn:heat']}--\ref{['eqn:bcins']} corresponding to $\boldsymbol{\alpha} = (0.5, 0, 0, 0.9)^T$.
  • Figure 2: CTROM solution error versus completion accuracy.
  • Figure 3: Completion statistics versus completion accuracy for parameter space dimensions $D=6,9,12$. Left panel: Percent of the observed tensor elements; Central panel: Compression factor; Right panel: Total number of degrees of freedom for representation in HTT format.
  • Figure 4: Completion statistics versus completion accuracy for various spatial resolutions. Left panel: Percent of the observed tensor elements; Central panel: Compression factor; Right panel: Total number of degrees of freedom for representation in HTT format.
  • Figure 5: Completion statistics versus completion accuracy for various resolutions of the parameter domain. Left panel: Percent of the observed tensor elements; Central panel: Compression factor as a ratio in HTT format; Right panel: A rescaled total number of degrees of freedom for representation in HTT format.

Theorems & Definitions (4)

  • Proposition 1
  • proof
  • Proposition 2
  • proof