Poly-Sinc Solution of Stochastic Elliptic Differential Equations
This work addresses the computational challenge of high dimensionality in stochastic Galerkin methods for engineers and scientists solving SPDEs, but the improvement is incremental.
The paper introduces a numerical method for solving stochastic elliptic PDEs by combining polynomial chaos with Sinc-point interpolation, achieving accurate solutions with fewer collocation points. Two examples demonstrate that only a small number of sampling points are needed for sufficient accuracy.
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.