arrow
Volume 36, Issue 2
A Multiscale Multilevel Monte Carlo Method for Multiscale Elliptic PDEs with Random Coefficients

Junlong Lyu & Zhiwen Zhang

Commun. Math. Res., 36 (2020), pp. 154-192.

Published online: 2020-05

Export citation
  • Abstract

We propose a multiscale multilevel Monte Carlo (MsMLMC) method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the offline stage, we construct a small number of reduced basis functions within each coarse grid block, which can then be used to approximate the multiscale finite element basis functions. In the online stage, we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis. The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid, without requiring the coarsest grid to resolve the smallest-scale of the solution. We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients. Moreover, we provide convergence analysis of the proposed method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.

  • Keywords

Random partial differential equations (RPDEs), uncertainty quantification (UQ), multiscale finite element method (MsFEM), multilevel Monte Carlo (MLMC), reduced basis, convergence analysis.

  • AMS Subject Headings

35J15, 65C05, 65N12, 65N30, 65Y20

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address
  • BibTex
  • RIS
  • TXT
@Article{CMR-36-154, author = {Junlong and Lyu and and 7672 and and Junlong Lyu and Zhiwen and Zhang and and 7673 and and Zhiwen Zhang}, title = {A Multiscale Multilevel Monte Carlo Method for Multiscale Elliptic PDEs with Random Coefficients}, journal = {Communications in Mathematical Research }, year = {2020}, volume = {36}, number = {2}, pages = {154--192}, abstract = {

We propose a multiscale multilevel Monte Carlo (MsMLMC) method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the offline stage, we construct a small number of reduced basis functions within each coarse grid block, which can then be used to approximate the multiscale finite element basis functions. In the online stage, we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis. The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid, without requiring the coarsest grid to resolve the smallest-scale of the solution. We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients. Moreover, we provide convergence analysis of the proposed method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.

}, issn = {2707-8523}, doi = {https://doi.org/10.4208/cmr.2020-0009}, url = {http://global-sci.org/intro/article_detail/cmr/16927.html} }
TY - JOUR T1 - A Multiscale Multilevel Monte Carlo Method for Multiscale Elliptic PDEs with Random Coefficients AU - Lyu , Junlong AU - Zhang , Zhiwen JO - Communications in Mathematical Research VL - 2 SP - 154 EP - 192 PY - 2020 DA - 2020/05 SN - 36 DO - http://doi.org/10.4208/cmr.2020-0009 UR - https://global-sci.org/intro/article_detail/cmr/16927.html KW - Random partial differential equations (RPDEs), uncertainty quantification (UQ), multiscale finite element method (MsFEM), multilevel Monte Carlo (MLMC), reduced basis, convergence analysis. AB -

We propose a multiscale multilevel Monte Carlo (MsMLMC) method to solve multiscale elliptic PDEs with random coefficients in the multi-query setting. Our method consists of offline and online stages. In the offline stage, we construct a small number of reduced basis functions within each coarse grid block, which can then be used to approximate the multiscale finite element basis functions. In the online stage, we can obtain the multiscale finite element basis very efficiently on a coarse grid by using the pre-computed multiscale basis. The MsMLMC method can be applied to multiscale RPDE starting with a relatively coarse grid, without requiring the coarsest grid to resolve the smallest-scale of the solution. We have performed complexity analysis and shown that the MsMLMC offers considerable savings in solving multiscale elliptic PDEs with random coefficients. Moreover, we provide convergence analysis of the proposed method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method for several multiscale stochastic problems without scale separation.

Junlong Lyu & Zhiwen Zhang. (2020). A Multiscale Multilevel Monte Carlo Method for Multiscale Elliptic PDEs with Random Coefficients. Communications in Mathematical Research . 36 (2). 154-192. doi:10.4208/cmr.2020-0009
Copy to clipboard
The citation has been copied to your clipboard