A twolevel stochastic collocation method for semilinear elliptic equations with random coefficients
Abstract
In this work, we propose a novel twolevel discretization for solving semilinear elliptic equations with random coefficients. Motivated by the twogrid method for deterministic partial differential equations (PDEs) introduced by Xu, our twolevel stochastic collocation method utilizes a twogrid finite element discretization in the physical space and a twolevel 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 twolevel 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):
 PNNLSA103740
Journal ID: ISSN 03770427; KJ0401000
 DOE Contract Number:
 AC0576RL01830
 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; twogrid; stochastic collocation
Citation Formats
Chen, Luoping, Zheng, Bin, Lin, Guang, and Voulgarakis, Nikolaos. A twolevel 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 twolevel 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. 2017.
"A twolevel 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 twolevel 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 twolevel discretization for solving semilinear elliptic equations with random coefficients. Motivated by the twogrid method for deterministic partial differential equations (PDEs) introduced by Xu, our twolevel stochastic collocation method utilizes a twogrid finite element discretization in the physical space and a twolevel 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 twolevel 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 = 2017,
month = 5
}

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