A model and variance reduction method for computing statistical outputs of stochastic elliptic partial differential equations

TL;DR

This paper introduces a combined approach using HDG discretization, reduced basis methods, and multilevel variance reduction to efficiently compute statistical outputs of stochastic elliptic PDEs, with error control.

Contribution

The paper develops a novel multilevel variance reduction technique integrated with HDG and reduced basis methods for stochastic elliptic PDEs, enabling fast and reliable statistical computations.

Findings

Significant reduction in Monte Carlo simulation time.

High-order accurate solutions via HDG discretization.

Effective error estimation and adaptive algorithm.

Abstract

We present a model and variance reduction method for the fast and reliable computation of statistical outputs of stochastic elliptic partial differential equations. Our method consists of three main ingredients: (1) the hybridizable discontinuous Galerkin (HDG) discretization of elliptic partial differential equations (PDEs), which allows us to obtain high-order accurate solutions of the governing PDE; (2) the reduced basis method for a new HDG discretization of the underlying PDE to enable real-time solution of the parameterized PDE in the presence of stochastic parameters; and (3) a multilevel variance reduction method that exploits the statistical correlation among the different reduced basis approximations and the high-fidelity HDG discretization to accelerate the convergence of the Monte Carlo simulations. The multilevel variance reduction method provides efficient computation of the statistical outputs by shifting most of the computational burden from the high-fidelity HDG approximation to the reduced basis approximations. Furthermore, we develop \textit{a posteriori} error estimates for our approximations of the statistical outputs. Based on these error estimates, we propose an algorithm for optimally choosing both the dimensions of the reduced basis approximations and the sizes of Monte Carlo samples to achieve a given error tolerance. We provide numerical examples to demonstrate the performance of the proposed method.