We are interested in approximating the numerical solution {\bf U}(t) of the large dimensional nonlinear matrix differential equation \bf U(t) = A { \bf  \bf U}(t) + {\bf U}(t) B { \bf  + {\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 .
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 \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. 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 .
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approximating the numerical
numerically approximating the
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nonlinear
semilinear
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Several numerical experiments based on typical
By maintaining a matrix-oriented reduction, we are able to employ first order exponential integrators at negligible costs. Numerical experiments on
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matrix-oriented setting, also for coupled systems of nonlinear matrix differential equations
setting