LSSC 2019: Large-Scale Scientific Computing pp 463-472

# Parallel Multilevel Monte Carlo Algorithms for Elliptic PDEs with Random Coefficients

• Petr Zakharov
• Oleg Iliev
• Jan Mohring
• Nikolay Shegunov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11958)

## Abstract

In this work, we developed and investigated Monte Carlo algorithms for elliptic PDEs with random coefficients. We considered groundwater flow as a model problem, where a permeability field represents random coefficients. The computational complexity is the main challenge in uncertainty quantification methods. The computation contains generating of a random coefficient and solving of partial differential equations. The permeability field was generated using the circulant embedding method. Multilevel Monte Carlo (MLMC) simulation can be based on different approximations of partial differential equations. We developed three MLMC algorithms based on finite volume, finite volume with renormalization and renormalization approximation. We compared numerical simulations and parallel performance of MLMC algorithms for 2D and 3D problems.

## Keywords

Monte Carlo method Stochastic PDE Renormalization

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