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Parallel Multilevel Monte Carlo Algorithms for Elliptic PDEs with Random Coefficients

: Zakharov, P.; Iliev, O.; Mohring, J.; Shegunov, N.


Lirkov, I.:
Large-Scale Scientific Computing. 12th International Conference, LSSC 2019 : Sozopol, Bulgaria, June 10-14, 2019, Revised Selected Papers
Cham: Springer Nature, 2020 (Lecture Notes in Computer Science 11958)
ISBN: 978-3-030-41031-5 (Print)
ISBN: 978-3-030-41032-2 (Online)
ISBN: 978-3-030-41033-9
International Conference on Large-Scale Scientific Computations (LSSC) <12, 2019, Sozopol>
Fraunhofer ITWM ()

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.