We extend stochastic basis adaptation and spatial domain decomposition methods to solve time varying stochastic partial differential equations (SPDEs) with a large number of input random parameters. Stochastic basis adaptation allows the determination of a low dimensional stochastic basis representation of a quantity of interest (QoI). Extending basis adaptation to time-dependent problems is challenging because small errors introduced in the previous time steps of the low dimensional approximate solution accumulate over time and cause divergence from the true solution. To address this issue we have introduced an approach where the basis adaptation varies at every time step so that the low dimensional basis is adapted to the QoI at that time step. We have coupled the time-dependent basis adaptation with domain decomposition to further increase the accuracy in the representation of the QoI. To illustrate the construction, we present numerical results for one-dimensional time varying linear and nonlinear diffusion equations with random space-dependent diffusion coefficients. Stochastic dimension reduction techniques proposed in the literature have mainly focused on quantifying the uncertainty in time independent and scalar QoI. To the best of our knowledge, this is the first attempt to extend dimensional reduction techniques to time varying and spatially dependent quantities such as the solution of SPDEs.
翻译:为了解决这一问题,我们引入了一种方法,即每个步骤的基础适应都不同,以便低维基调整能够适应QoI的阶段。我们将基于时间的基础调整与基于时间的分解结合起来,以进一步提高QoI代表的准确性。为了说明其构造,我们用随机的依靠空间的传播系数来展示一维时间不同线性和非线性扩散方程式的数值结果。