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

Abstract

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 mesh $$\mathcal{T}_H$$ 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.

Authors:
; ; ;
Publication Date:
Research Org.:
Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
Sponsoring Org.:
USDOE
OSTI Identifier:
1339789
Report Number(s):
PNNL-SA-103740
Journal ID: ISSN 0377-0427; KJ0401000
DOE Contract Number:
AC05-76RL01830
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Computational and Applied Mathematics; Journal Volume: 315
Country of Publication:
United States
Language:
English
Subject:
Semilinear problems; random coefficients; two-grid; stochastic collocation

Citation Formats

Chen, Luoping, Zheng, Bin, Lin, Guang, and Voulgarakis, Nikolaos. A two-level stochastic collocation method for semilinear elliptic equations with random coefficients. United States: 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. United States. 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". United States. doi:10.1016/j.cam.2016.10.030.
@article{osti_1339789,
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 = {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 mesh $\mathcal{T}_H$ 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.},
doi = {10.1016/j.cam.2016.10.030},
journal = {Journal of Computational and Applied Mathematics},
number = ,
volume = 315,
place = {United States},
year = {Mon May 01 00:00:00 EDT 2017},
month = {Mon May 01 00:00:00 EDT 2017}
}