2020/06/23 by Gerhard Kirsten, Valeria Simoncini, Kirsten, Gerhard +1
Decision Sciences · Engineering · Physics and Astronomy · #15A24 #37M99 #65F30 #65N06 #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Power System Optimization and Stability #Probabilistic and Robust Engineering Design
paper · pdf · doi:10.48550/arxiv.2006.13289
openalex publication_date 2020/06/23 · openalex created_date 2021/06/07 · openalex updated_date 2026/07/28
We are interested in numerically approximating the solution \bf U(t) of the large dimensional semilinear matrix differential equation \bf U(t) = \bf A\bf U(t) + \bf U(t) \bf B + \cal F(\bf U,t), with appropriate starting and boundary conditions, and t ∈ [0, Tf]. 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. By maintaining a matrix-oriented reduction, we are able to employ first order exponential integrators at negligible costs. Numerical experiments on benchmark problems illustrate the effectiveness of the new setting.