Abstract
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 in [29, 30], to which our stability analysis applies as well. The analysis is supported by numerical results showing the sharpness of the obtained stability conditions.
Highlights
Many physical and engineering applications are modeled by time-dependent partial differential equations (PDEs) with input data often subject to uncertainty due to measurement errors or insufficient knowledge
In this work we proposed and analyzed three types of discretization schemes, namely explicit, implicit and semi-implicit, to obtain a numerical solution of the Dynamical Low Rank (DLR) system of evolution equations for the deterministic and stochastic modes
Such discrete DLR solution was obtained by projecting the discretized dynamics on the tangent space of the low-rank manifold at an intermediate point
Summary
Many physical and engineering applications are modeled by time-dependent partial differential equations (PDEs) with input data often subject to uncertainty due to measurement errors or insufficient knowledge. In this work we propose a class of numerical schemes to approximate the evolution equations for the mean, the deterministic basis and the stochastic basis, which can be of explicit, semi-implicit or implicit type. The stability properties of both the discrete and the continuous DLR solutions do not depend on the size of their singular values, even without any ε-approximability condition on f − L. The sharpness of the obtained stability conditions on the time step and spatial discretization is supported by the numerical results provided in the last section. We dedicate a subsection to show that a rank-deficient solution obtained by our scheme still satisfies a suitable discrete variational formulation and has the same stability properties as the full-rank case.
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