# Poly-Sinc Solution of Stochastic Elliptic Differential Equations

## TL;DR

This paper presents a novel numerical method combining polynomial chaos and Sinc point approximation to efficiently solve stochastic elliptic PDEs with high accuracy using few collocation points.

## Contribution

It introduces a new approach integrating polynomial chaos with Sinc point interpolation for solving high-dimensional stochastic PDEs efficiently.

## Key findings

- Accurate solutions achieved with few collocation points.
- Method effectively handles high-dimensional stochastic systems.
- Numerical examples demonstrate high accuracy with Legendre polynomials.

## 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. 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.