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

• Published in: Journal of computational physics Vol. 297; pp. 700 - 720
• Main Authors: Vidal-Codina, F., Nguyen, N.C., Giles, M.B., Peraire, J.
• Format: Journal Article
• Language: English
• Published: United States Elsevier Inc 15-09-2015
• Subjects: A posteriori error estimation; ALGORITHMS; Approximation; APPROXIMATIONS; CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Computation; COMPUTERIZED SIMULATION; CONVERGENCE; CORRELATION FUNCTIONS; Discretization; ERRORS; Hybridizable discontinuous Galerkin method; Mathematical models; MATHEMATICAL SOLUTIONS; Model reduction; MONTE CARLO METHOD; Multilevel Monte Carlo method; PARTIAL DIFFERENTIAL EQUATIONS; Reduced basis method; Reduction; STATISTICS; Stochastic elliptic PDEs; STOCHASTIC PROCESSES; Stochasticity; Variance; Variance reduction; Multilevel Monte Carlo method; A posteriori error estimation; Variance reduction; Stochastic elliptic PDEs; Model reduction; Reduced basis method; Hybridizable discontinuous Galerkin method
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2015.05.041

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 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.