A dynamically adaptive wavelet approach to stochastic computations based on polynomial chaos - capturing all scales of random modes on independent grids

Journal Article · Wed Aug 10 04:00:00 UTC 2011 · Journal of Computational Physics

OSTI ID:21592605

Xiaoan, Ren; Wenquan, Wu ^[1]

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1. University of Shanghai for Science and Technology, Shanghai 200093 (China)

Highlights: {yields} New approach for stochastic computations based on polynomial chaos. {yields} Development of dynamically adaptive wavelet multiscale solver using space refinement. {yields} Accurate capture of steep gradients and multiscale features in stochastic problems. {yields} All scales of each random mode are captured on independent grids. {yields} Numerical examples demonstrate the need for different space resolutions per mode. - Abstract: In stochastic computations, or uncertainty quantification methods, the spectral approach based on the polynomial chaos expansion in random space leads to a coupled system of deterministic equations for the coefficients of the expansion. The size of this system increases drastically when the number of independent random variables and/or order of polynomial chaos expansions increases. This is invariably the case for large scale simulations and/or problems involving steep gradients and other multiscale features; such features are variously reflected on each solution component or random/uncertainty mode requiring the development of adaptive methods for their accurate resolution. In this paper we propose a new approach for treating such problems based on a dynamically adaptive wavelet methodology involving space-refinement on physical space that allows all scales of each solution component to be refined independently of the rest. We exemplify this using the convection-diffusion model with random input data and present three numerical examples demonstrating the salient features of the proposed method. Thus we establish a new, elegant and flexible approach for stochastic problems with steep gradients and multiscale features based on polynomial chaos expansions.

• OSTI ID: 21592605
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: 19 Vol. 230; ISSN JCTPAH; ISSN 0021-9991
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
CALCULATION METHODS
CHAOS THEORY
COMPUTERIZED SIMULATION
EQUATIONS
FUNCTIONS
MATHEMATICAL SOLUTIONS
MATHEMATICAL SPACE
MATHEMATICS
POLYNOMIALS
RANDOMNESS
RESOLUTION
SIMULATION
SPACE
STOCHASTIC PROCESSES