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A Reduced Order Modeling Method with Variable-Separation-Based Domain Decomposition for Parametric Dynamical Systems

Yuming Ba, Liang Chen, Yaru Chen, Qiuqi Li

TL;DR

This work addresses efficient simulation of parametric dynamical systems under uncertainty by transforming time dependence into a set of frequency-domain, complex-valued elliptic problems via a time Fourier transform. It develops a reduced-order framework built on a variable-separation-based domain decomposition (DD-VS) that constructs affine surrogate models for both the interface problem and subdomain interiors, enabling an online stage whose cost is independent of spatial discretization. The method leverages a complex-valued extension of VS and a Schur-complement interface formulation to produce a highly parallelizable offline-online pipeline, with the online phase dominated by evaluating parametric coefficients and performing an inverse Fourier transform. Numerical experiments on heat and reaction-diffusion equations demonstrate accurate approximations with substantial speedups compared to FEM-BE, validating the approach and its potential for large-scale parametric studies. Limitations of the Fourier approach motivate future work on direct time-domain DD and extensions to non-affine and multi-physical settings.

Abstract

This paper proposes a model order reduction method for a class of parametric dynamical systems. Using a temporal Fourier transform, we reformulate these systems into complex-valued elliptic equations in the frequency domain, containing frequency variables and parameters inherited from the original model. To reduce the computational cost of the frequency-variable elliptic equations, we extend the variable-separation-based domain decomposition method to the complex-valued context, resulting in an offline-online procedure for solving the parametric dynamical systems. At the offline stage, separate representations of the solutions for the interface problem and the subproblems are constructed. At the online stage, the solutions of the parametric dynamical systems for new parameter values can be directly derived by utilizing the separate representations and implementing the inverse Fourier transform. The proposed approach is capable of being highly efficient because the online stage is independent of the spatial discretization. Finally, we present three specific instances of parametric dynamical systems to demonstrate the effectiveness of the proposed method.

A Reduced Order Modeling Method with Variable-Separation-Based Domain Decomposition for Parametric Dynamical Systems

TL;DR

This work addresses efficient simulation of parametric dynamical systems under uncertainty by transforming time dependence into a set of frequency-domain, complex-valued elliptic problems via a time Fourier transform. It develops a reduced-order framework built on a variable-separation-based domain decomposition (DD-VS) that constructs affine surrogate models for both the interface problem and subdomain interiors, enabling an online stage whose cost is independent of spatial discretization. The method leverages a complex-valued extension of VS and a Schur-complement interface formulation to produce a highly parallelizable offline-online pipeline, with the online phase dominated by evaluating parametric coefficients and performing an inverse Fourier transform. Numerical experiments on heat and reaction-diffusion equations demonstrate accurate approximations with substantial speedups compared to FEM-BE, validating the approach and its potential for large-scale parametric studies. Limitations of the Fourier approach motivate future work on direct time-domain DD and extensions to non-affine and multi-physical settings.

Abstract

This paper proposes a model order reduction method for a class of parametric dynamical systems. Using a temporal Fourier transform, we reformulate these systems into complex-valued elliptic equations in the frequency domain, containing frequency variables and parameters inherited from the original model. To reduce the computational cost of the frequency-variable elliptic equations, we extend the variable-separation-based domain decomposition method to the complex-valued context, resulting in an offline-online procedure for solving the parametric dynamical systems. At the offline stage, separate representations of the solutions for the interface problem and the subproblems are constructed. At the online stage, the solutions of the parametric dynamical systems for new parameter values can be directly derived by utilizing the separate representations and implementing the inverse Fourier transform. The proposed approach is capable of being highly efficient because the online stage is independent of the spatial discretization. Finally, we present three specific instances of parametric dynamical systems to demonstrate the effectiveness of the proposed method.

Paper Structure

This paper contains 16 sections, 56 equations, 9 figures, 3 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The frequency-domain method
  4. VS and Domain Decomposition Methods
  5. VS method for complex-valued problems
  6. Domain decomposition method for complex-valued problems
  7. Domain Decomposition Based on Variable-Separation
  8. Affine representations of $S$ and $F$
  9. Reduced model for the interface problem
  10. Reduced model for subdomain problems
  11. Numerical Experiments
  12. Heat equation
  13. Reaction diffusion equation
  14. The first scenario
  15. The second scenario
  16. ...and 1 more sections

Figures (9)

  • Figure 1: Average relative error versus the number of separate terms $N_{S_1}$.
  • Figure 2: Average relative error versus the number of separate terms.
  • Figure 3: Comparison of the errors.
  • Figure 4: Comparison of the solutions in subdomains for the FEM-BE and FT-DD-VS methods.
  • Figure 5: Average relative error versus the number of separate terms.
  • ...and 4 more figures

Theorems & Definitions (1)

  • Remark 4.1