论文标题
应用于半连接基质微分方程的矩阵式POD-DEIM算法
A matrix-oriented POD-DEIM algorithm applied to semilinear matrix differential equations
论文作者
论文摘要
我们有兴趣在数值上近似于大小半连续矩阵的$ {\ bf u}（t）$ f}（{\ bf u}，t）$，具有适当的启动和边界条件，$ t \ in [0，t_f] $。在适当的正交分解（POD）方法和离散的经验插值方法（DEIM）的框架中，我们得出了一种新颖的面向基质的减少过程，从而导致有效的，结构上意识到原始问题的低阶近似值。非线性项的还原也是通过使用左右右图在两个不同的还原空间上进行完全临床插值执行的，从而产生了Deim的新两面版本。通过保持面向矩阵的减少，我们可以以微不足道的成本使用一阶指数集成商。 基准问题的数值实验说明了新设置的有效性。
We are interested in numerically approximating the solution ${\bf U}(t)$ of the large dimensional semilinear matrix differential equation $\dot{\bf U}(t) = { \bf A}{\bf U}(t) + {\bf U}(t){ \bf B} + {\cal F}({\bf U},t)$, 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.
