Abstract
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.
Highlights
In many applications the values of the parameters of the problem are not exactly known
If the physical system is described by a partial differential equation (PDE), the combination with the stochastic model results in a stochastic partial differential equation (SPDE)
The inner product in (5) can be defined for different types of weighting function W ; it is possible to prove that the optimal convergence rate of a Generalized Polynomial Chaos (gPC) model can be achieved when the weighting function W agrees to the joint probability density function (PDF) of the random variables considered in a standard form [9,18]
Summary
In many applications the values of the parameters of the problem are not exactly known. These uncertainties inherent in the model yield uncertainties in the results of numerical simulations. Stochastic methods are one way to model these uncertainties by using random fields [1]. If the physical system is described by a partial differential equation (PDE), the combination with the stochastic model results in a stochastic partial differential equation (SPDE). The solution of the SPDE is again a random field, describing both the expected response and quantifying its uncertainty. SPDEs can be interpreted mathematically in several ways
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
