论文标题
投影仪切割方案的稳定性属性，用于动态低等级近似的随机抛物线方程
Stability properties of a projector-splitting scheme for dynamical low rank approximation of random parabolic equations
论文作者
论文摘要
我们考虑随机抛物线方程的动力学低等级（DLR）近似，并提出一类完全离散的数值方案。与连续的DLR近似类似，我们的方案显示出满足离散的变异公式。通过利用此属性，我们建立了我们的方案的稳定性：我们表明，在抛物线型CFL条件下，我们的显式和半平整版本在有条件地稳定，这不取决于DLR解决方案的最小单数值；而我们的隐式方案是无条件的稳定。此外，我们表明，在某些情况下，如果系统中的随机性足够小，则可以无条件地稳定。此外，我们表明这些方案可以解释为投影仪拆分集成剂，并且与Lubich等人提出的计划密切相关。 [位num。数学，54：171-188，2014;暹罗J.肛门，53：917-941，2015]，我们的稳定性分析也适用。分析得到了数值结果的支持，显示获得的稳定性条件的清晰度。
We consider the Dynamical Low Rank (DLR) approximation of random parabolic equations and propose a class of fully discrete numerical schemes. Similarly to the continuous DLR approximation, our schemes are shown to satisfy a discrete variational formulation. By exploiting this property, we establish stability of our schemes: we show that our explicit and semi-implicit versions are conditionally stable under a parabolic type CFL condition which does not depend on the smallest singular value of the DLR solution; whereas our implicit scheme is unconditionally stable. Moreover, we show that, in certain cases, the semi-implicit scheme can be unconditionally stable if the randomness in the system is sufficiently small. Furthermore, we show that these schemes can be interpreted as projector-splitting integrators and are strongly related to the scheme proposed by Lubich et al. [BIT Num. Math., 54:171-188, 2014; SIAM J. on Num. Anal., 53:917-941, 2015], to which our stability analysis applies as well. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions.
