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Title: A two-level stochastic collocation method for semilinear elliptic equations with random coefficients

Authors:
; ; ;
Publication Date:
Sponsoring Org.:
USDOE
OSTI Identifier:
1410838
Grant/Contract Number:
AC05-76RL01830
Resource Type:
Journal Article: Publisher's Accepted Manuscript
Journal Name:
Journal of Computational and Applied Mathematics
Additional Journal Information:
Journal Volume: 315; Journal Issue: C; Related Information: CHORUS Timestamp: 2017-12-01 07:51:05; Journal ID: ISSN 0377-0427
Publisher:
Elsevier
Country of Publication:
Belgium
Language:
English

Citation Formats

Chen, Luoping, Zheng, Bin, Lin, Guang, and Voulgarakis, Nikolaos. A two-level stochastic collocation method for semilinear elliptic equations with random coefficients. Belgium: N. p., 2017. Web. doi:10.1016/j.cam.2016.10.030.
Chen, Luoping, Zheng, Bin, Lin, Guang, & Voulgarakis, Nikolaos. A two-level stochastic collocation method for semilinear elliptic equations with random coefficients. Belgium. doi:10.1016/j.cam.2016.10.030.
Chen, Luoping, Zheng, Bin, Lin, Guang, and Voulgarakis, Nikolaos. Mon . "A two-level stochastic collocation method for semilinear elliptic equations with random coefficients". Belgium. doi:10.1016/j.cam.2016.10.030.
@article{osti_1410838,
title = {A two-level stochastic collocation method for semilinear elliptic equations with random coefficients},
author = {Chen, Luoping and Zheng, Bin and Lin, Guang and Voulgarakis, Nikolaos},
abstractNote = {},
doi = {10.1016/j.cam.2016.10.030},
journal = {Journal of Computational and Applied Mathematics},
number = C,
volume = 315,
place = {Belgium},
year = {Mon May 01 00:00:00 EDT 2017},
month = {Mon May 01 00:00:00 EDT 2017}
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record at 10.1016/j.cam.2016.10.030

Citation Metrics:
Cited by: 1work
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  • In this work, we propose a novel two-level discretization for solving semilinear elliptic equations with random coefficients. Motivated by the two-grid method for deterministic partial differential equations (PDEs) introduced by Xu, our two-level stochastic collocation method utilizes a two-grid finite element discretization in the physical space and a two-level collocation method in the random domain. In particular, we solve semilinear equations on a coarse meshmore » $$\mathcal{T}_H$$ with a low level stochastic collocation (corresponding to the polynomial space $$\mathcal{P}_{P}$$) and solve linearized equations on a fine mesh $$\mathcal{T}_h$$ using high level stochastic collocation (corresponding to the polynomial space $$\mathcal{P}_p$$). We prove that the approximated solution obtained from this method achieves the same order of accuracy as that from solving the original semilinear problem directly by stochastic collocation method with $$\mathcal{T}_h$$ and $$\mathcal{P}_p$$. The two-level method is computationally more efficient, especially for nonlinear problems with high random dimensions. Numerical experiments are also provided to verify the theoretical results.« less
  • n this paper we show how stochastic collocation method (SCM) could fail to con- verge for nonlinear differential equations with random coefficients. First, we consider Navier-Stokes equation with uncertain viscosity and derive error estimates for stochastic collocation discretization. Our analysis gives some indicators on how the nonlinearity negatively affects the accuracy of the method. The stochastic collocation method is then applied to noisy Lorenz system. Simulation re- sults demonstrate that the solution of a nonlinear equation could be highly irregular on the random data and in such cases, stochastic collocation method cannot capture the correct solution.
  • The aim of this article is to study the Dirichlet problem for second-order semilinear degenerate elliptic PDEs and the connections of these problems with stochastic exit time control problems.
  • In recent years, there has been intense interest in understanding various physical phenomena in random heterogeneous media. Any accurate description/simulation of a process in such media has to satisfactorily account for the twin issues of randomness as well as the multilength scale variations in the material properties. An accurate model of the material property variation in the system is an important prerequisite towards complete characterization of the system response. We propose a general methodology to construct a data-driven, reduced-order model to describe property variations in realistic heterogeneous media. This reduced-order model then serves as the input to the stochastic partialmore » differential equation describing thermal diffusion through random heterogeneous media. A decoupled scheme is used to tackle the problems of stochasticity and multilength scale variations in properties. A sparse-grid collocation strategy is utilized to reduce the solution of the stochastic partial differential equation to a set of deterministic problems. A variational multiscale method with explicit subgrid modeling is used to solve these deterministic problems. An illustrative example using experimental data is provided to showcase the effectiveness of the proposed methodology.« less