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

Journal Article · Tue Sep 15 00:00:00 EDT 2015 · Journal of Computational Physics

DOI:https://doi.org/10.1016/J.JCP.2015.05.041· OSTI ID:22465658

Vidal-Codina, F., E-mail: [e-mail] ^[1]; Nguyen, N.C., E-mail: [e-mail] ^[1]; Giles, M.B., E-mail: [e-mail] ^[2]; Peraire, J., E-mail: [e-mail] ^[1]

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1. Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA 02139 (United States)
2. Mathematical Institute, University of Oxford, Oxford (United Kingdom)

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.

• OSTI ID: 22465658
• Journal Information: Journal of Computational Physics, Vol. 297; Other Information: Copyright (c) 2015 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA); ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ALGORITHMS
APPROXIMATIONS
COMPUTERIZED SIMULATION
CONVERGENCE
CORRELATION FUNCTIONS
ERRORS
MATHEMATICAL SOLUTIONS
MONTE CARLO METHOD
PARTIAL DIFFERENTIAL EQUATIONS
STATISTICS
STOCHASTIC PROCESSES