SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page A2182-A2205, January 2020.
We consider the numerical solution of large-scale symmetric differential matrix Riccati equations. Under certain hypotheses on the data, reduced order methods have recently arisen as a promising class of solution strategies by forming low-rank approximations to the sought after solution at selected timesteps. We show that great computational and memory savings are obtained by a reduction process onto rational Krylov subspaces as opposed to current approaches. By specifically addressing the solution of the reduced differential equation and reliable stopping criteria, we are able to obtain accurate final approximations at low memory and computational requirements. This is obtained by employing a two-phase strategy that separately enhances the accuracy of the algebraic approximation and the time integration. The new method allows us to numerically solve much larger problems than in the current literature. Numerical experiments on benchmark problems illustrate the effectiveness of the procedure with respect to existing solvers.

中文翻译：

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解大型微分矩阵Riccati方程的降阶方法

SIAM科学计算杂志，第42卷，第4期，第A2182-A2205页，2020年1月。
我们考虑大型对称微分矩阵Riccati方程的数值解。在数据的某些假设下，降阶方法最近通过在选定的时间步长形成对所寻求的解决方案的低阶近似而成为一种有前途的解决方案策略。我们表明，与当前方法相比，通过对有理Krylov子空间进行归约处理可以节省大量计算和内存。通过专门解决简化的微分方程和可靠的停车准则的解决方案，我们能够在低内存和计算要求的情况下获得准确的最终近似值。这是通过采用两阶段策略来实现的，该策略分别提高了代数逼近的精度和时间积分。新方法使我们能够在数值上解决比当前文献更大的问题。基准问题的数值实验说明了该程序相对于现有求解器的有效性。