A matrix-oriented POD-DEIM algorithm applied to nonlinear differential matrix equations
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We are interested in approximating the numerical solution U(t) of the large dimensional nonlinear matrix differential equation U̇(t) = A U(t) + U(t)B + F( U,t) + G, with appropriate starting and boundary conditions, and t ∈ [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.
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