多项式混沌求解随机椭圆型微分方程
Poly-Sinc Solution of Stochastic Elliptic Differential Equations
AI总结：
本文提出了一种利用多项式混沌和Sinc点多项式逼近求解随机椭圆型偏微分方程的方法，通过将Sinc点与多项式插值结合，实现高精度近似，并通过两个例子展示了如何通过少量采样点获得足够准确的模型表示。
摘要：
在本文中，我们介绍了一种利用多项式混沌和Sinc点多项式逼近来求解随机偏微分方程（SPDE）的方法。这些Sinc点由共形映射定义，当与多项式插值结合时，可以得到高精度的近似。解决SPDE的第一步是利用随机伽辽金方法与多项式混沌相结合，这会得到一组需要求解的确定性偏微分方程。主要困难在于所得到的偏微分方程组的高维性。本文的思路是使用少量的插值点来求解该组方程。两个例子主要使用Legendre多项式来处理随机变量。这些例子表明，我们只需要在少量点上采样，就可以得到足够准确的模型表示。
英文摘要：
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. 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
