[Paper Review] A matrix-oriented POD-DEIM algorithm applied to nonlinear differential matrix equations.

Gerhard Kirsten, Valeria Simoncini|arXiv (Cornell University)|Jun 23, 2020

Model Reduction and Neural Networks31 references4 citations

TL;DR

This paper proposes a matrix-oriented Proper Orthogonal Decomposition with a two-sided Discrete Empirical Interpolation Method (POD-DEIM) for solving large-scale nonlinear matrix differential equations. By exploiting matrix structure through left and right projection-based interpolation, it achieves efficient, structure-aware model order reduction, demonstrating superior performance on benchmark problems including coupled systems.

ABSTRACT

We are interested in approximating the numerical solution ${\bf U}(t)$ of the large dimensional nonlinear matrix differential equation $\dot{\bf U}(t) = A{\bf U}(t) + {\bf U}(t)B + {\cal F}({\bf U},t) + G$, with appropriate starting and boundary conditions, and $t \in [0, T_f]$. In the framework of the Proper Orthogonal Decomposition (POD) methodology and the Discrete Empirical Interpolation Method (DEIM), we derive a novel matrix-oriented reduction process leading to an effective, structure aware low order approximation of the original problem. The reduction of the nonlinear term is also performed by means of a fully matricial interpolation using left and right projections onto two distinct reduction spaces, giving rise to a new two-sided version of DEIM. Several numerical experiments based on typical benchmark problems illustrate the effectiveness of the new matrix-oriented setting, also for coupled systems of nonlinear matrix differential equations.

Motivation & Objective

• To develop a reduced-order model for large-scale nonlinear matrix differential equations while preserving matrix structure.
• To address the challenge of efficiently approximating the nonlinear term in matrix differential equations using a matricial, structure-aware approach.
• To extend the DEIM framework to a two-sided matrix formulation that operates on both left and right projection spaces.
• To demonstrate the effectiveness of the proposed method on benchmark problems, including coupled systems of matrix ODEs.
• To ensure computational efficiency and accuracy in long-time integration of nonlinear matrix systems.

Proposed method

• The method applies Proper Orthogonal Decomposition (POD) to extract spatial and temporal modes from snapshot matrices of the solution trajectory.
• It constructs a low-dimensional subspace using POD for both the solution matrix and the nonlinear term's structure.
• A two-sided DEIM is introduced, where interpolation is performed via separate left and right projection matrices onto distinct reduced subspaces.
• The nonlinear term is approximated using a matricial interpolation that respects the matrix structure of the original problem.
• The reduced system is derived by projecting the full matrix differential equation onto the POD subspace, with the nonlinear term reconstructed via the two-sided DEIM interpolation.
• The method preserves the matrix nature of the problem throughout the reduction, avoiding vectorization and maintaining structural fidelity.

Experimental results

Research questions

• RQ1 Can a matrix-oriented POD-DEIM framework effectively reduce the computational cost of large-scale nonlinear matrix differential equations?
• RQ2 How does the two-sided DEIM approach compare to standard DEIM in preserving accuracy for matrix-structured problems?
• RQ3 To what extent does the method maintain structure and accuracy in coupled systems of nonlinear matrix ODEs?
• RQ4 What is the impact of matrix-specific interpolation on the stability and convergence of the reduced-order model?
• RQ5 Can the proposed method achieve significant speedups while maintaining high accuracy in long-time simulations?

Key findings

• The matrix-oriented POD-DEIM method achieves significant computational savings by preserving the matrix structure throughout the reduction process.
• The two-sided DEIM formulation enables accurate approximation of the nonlinear term using only a small number of interpolation points, reducing online computational cost.
• Numerical experiments show that the method maintains high accuracy even for long-time integration, outperforming standard POD-DEIM in structure-aware problems.
• The approach is effective for coupled systems of nonlinear matrix differential equations, demonstrating robustness and scalability.
• The method reduces the dimensionality of the problem while preserving the intrinsic matrix structure, leading to more accurate and efficient reduced-order models.
• The results confirm that structure-aware model order reduction via matrix-oriented DEIM leads to improved approximation quality compared to vectorized approaches.