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

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.

💡 Research Summary

The paper introduces a novel reduced‑order modeling framework for parametric dynamical systems, particularly time‑dependent parabolic PDEs, by exploiting a frequency‑domain formulation combined with a variable‑separation (VS) based domain decomposition (DD‑VS). The authors first apply a temporal Fourier transform to the original time‑dependent problem, converting it into a family of complex‑valued elliptic equations that depend on the frequency ω and the original parameters ξ. In this formulation, ω is treated as an additional one‑dimensional parameter, so the problem becomes a “spatial‑parameter” PDE defined over a combined parameter space μ = (ω, ξ).

A central contribution is the extension of the VS method, originally designed for real‑valued problems, to complex‑valued elliptic equations. By separating the complex solution into its real and imaginary components, the authors rewrite the bilinear and linear forms as affine functions of μ (equations (3.2)). This affine decomposition enables an offline‑online split: the μ‑independent matrices (stiffness, mass, and load vectors) are assembled once, while only scalar coefficient functions of μ need to be evaluated during the online stage. The VS‑greedy algorithm iteratively selects parameter samples μ_k that maximize the residual norm, solves the corresponding full‑order system, extracts the error vector, and adds a new separated term ζ_k(μ) c_k to the reduced representation. After N iterations, the solution is approximated as a sum of N separated terms, each being a product of a μ‑dependent scalar function and a μ‑independent vector.

To further reduce computational effort, the authors embed the VS approach into a non‑overlapping domain decomposition framework. The computational domain is partitioned into subdomains; interior degrees of freedom and interface degrees of freedom are distinguished. By block Gaussian elimination, the global system is reduced to a Schur complement defined only on the interfaces. The interface problem and each subdomain interior problem are independently reduced using the VS‑greedy procedure, yielding surrogate models for both. Because the original equations are complex, the authors treat the real and imaginary parts together, forming a coupled real system that can be processed with the same VS machinery.

The offline phase therefore consists of (i) constructing affine decompositions of the operators, (ii) generating a training set of parameter samples, (iii) applying the VS‑greedy algorithm separately to the interface and each subdomain to obtain N separated terms, and (iv) storing the resulting basis vectors and scalar coefficient functions. The online phase is remarkably cheap: given a new parameter ξ, the algorithm forms μ = (ω, ξ) for each quadrature frequency, evaluates the pre‑computed scalar functions ζ_j(μ), forms the linear combination of the stored vectors to obtain the frequency‑domain solution ˆu(x, ω; ξ), and finally reconstructs the time‑domain solution u(x, t; ξ) via an inverse Fourier transform using Legendre‑Gauss‑Lobatto quadrature points and weights. Crucially, the online cost does not depend on the size of the finite‑element mesh; it scales only with the number of separated terms N and the number of frequency quadrature points.

Three numerical experiments validate the methodology. The first example solves a 2‑D diffusion problem with a parametric diffusion coefficient and multiple frequencies, demonstrating that the proposed DD‑VS approach achieves accuracy comparable to traditional reduced‑basis or POD methods while delivering online speedups of two to three orders of magnitude. The second example tackles a nonlinear reaction‑diffusion system; after linearization and applying a discrete empirical interpolation method (DEIM) to recover an affine structure, the DD‑VS method still provides substantial computational savings. The third example addresses a 3‑D electromagnetic scattering problem with high frequencies, showing that the complex‑valued DD‑VS can handle large‑scale problems, reduce memory consumption, and exhibit good parallel scalability.

In conclusion, the authors present a highly efficient offline‑online reduced‑order modeling strategy for parametric dynamical systems that leverages the decoupling properties of the Fourier transform and the locality of domain decomposition. While the offline stage requires solving several full‑order problems, this cost is incurred only once. The online stage is independent of spatial discretization, making the method suitable for real‑time simulation, many‑query contexts such as design optimization, and uncertainty quantification. Future work is suggested on extending the framework to non‑affine or strongly nonlinear parameter dependencies, adaptive subdomain partitioning, and GPU‑accelerated implementations.