In this paper, we introduce a numerical solution of a stochastic partial differential equation (SPDE) of elliptic type using polynomial chaos along side with polynomial approximation at Sinc points. These Sinc points are defined by a conformal map and when mixed with the polynomial interpolation, it yields an accurate approximation. The first step to solve SPDE is to use stochastic Galerkin method in conjunction with polynomial chaos, which implies a system of deterministic partial differential equations to be solved. The main difficulty is the higher dimensionality of the resulting system of partial differential equations. The idea here is to solve this system using a small number of collocation points in space. This collocation technique is called Poly-Sinc and is used for the first time to solve high-dimensional systems of partial differential equations. Two examples are presented, mainly using Legendre polynomials for stochastic variables. These examples illustrate that we require to sample at few points to get a representation of a model that is sufficiently accurate.

在本文中，我们介绍了利用多项式混沌和Sinc点处的多项式逼近的椭圆型随机偏微分方程（SPDE）的数值解。这些Sinc点由共形图定义，当与多项式插值混合时，可以得出精确的近似值。求解SPDE的第一步是将随机Galerkin方法与多项式混沌结合使用，这意味着需要求解确定性偏微分方程组。主要困难是偏微分方程组所得系统的较高维数。这里的想法是使用空间中的少量配置点来解决该系统。这种搭配技术称为Poly-Sinc，首次用于求解偏微分方程的高维系统。给出了两个例子，主要使用勒让德多项式作为随机变量。这些示例说明，我们需要在几个点上进行采样以得到足够准确的模型表示。