A matrix-oriented POD-DEIM algorithm applied to nonlinear differential matrix equations

by   Gerhard Kirsten, et al.

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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