In this article we analyze the linear parabolic partial differential equation with a stochastic domain deformation. In particular, we concentrate on the problem of numerically approximating the statistical moments of a given Quantity of Interest (QoI). The geometry is assumed to be random. The parabolic problem is remapped to a fixed deterministic domain with random coefficients and shown to admit an extension on a well defined region embedded in the complex hyperplane. The stochastic moments of the QoI are computed by employing a collocation method in conjunction with an isotropic Smolyak sparse grid. Theoretical sub-exponential convergence rates as a function to the number of collocation interpolation knots are derived. Numerical experiments are performed and they confirm the theoretical error estimates.

在本文中，我们分析了具有随机域变形的线性抛物型偏微分方程。特别是，我们专注于数值近似给定感兴趣数量 (QoI) 的统计矩的问题。几何形状被假定为随机的。抛物线问题被重新映射到具有随机系数的固定确定域，并显示允许在嵌入复杂超平面的明确区域上进行扩展。QoI 的随机矩是通过采用搭配方法与各向同性 Smolyak 稀疏网格相结合来计算的。推导出作为搭配插值节点数量的函数的理论次指数收敛速度。进行了数值实验，它们证实了理论误差估计。