Parallel Multilevel Monte Carlo Algorithms for Elliptic PDEs with Random Coefficients

Zakharov, Petr E.; Iliev, Oleg; Möhring, Jan; Shegunov, Nikolay

doi:10.1007/978-3-030-41032-2_53

Cited by 3 publications

(1 citation statement)

References 24 publications

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“…The parallelization of MLMC has been considered in recent years [43,44,18,42,54,22,8] as multiple levels of spatial/temporal discretization introduce additional scheduling challenges to effectively utilize resources of a parallel environment with minimal idling. One can conceptually exploit three levels of parallelism in MLMC.…”

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A Massively Parallel Implementation of Multilevel Monte Carlo for Finite Element Models

Badia¹,

Hampton²,

Principe³

2021

Preprint

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A. The Multilevel Monte Carlo (MLMC) method has proven to be an effective variance-reduction statistical method for Uncertainty Quantification (UQ) in Partial Differential Equation (PDE) models, combining model computations at different levels to create an accurate estimate. Still, the computational complexity of the resulting method is extremely high, particularly for 3D models, which requires advanced algorithms for the efficient exploitation of High Performance Computing (HPC). In this article we present a new implementation of the MLMC in massively parallel computer architectures, exploiting parallelism within and between each level of the hierarchy. The numerical approximation of the PDE is performed using the finite element method but the algorithm is quite general and could be applied to other discretization methods as well, although the focus is on parallel sampling. The two key ingredients of an efficient parallel implementation are a good processor partition scheme together with a good scheduling algorithm to assign work to different processors. We introduce a multiple partition of the set of processors that permits the simultaneous execution of different levels and we develop a dynamic scheduling algorithm to exploit it. The problem of finding the optimal scheduling of distributed tasks in a parallel computer is an NP-complete problem. We propose and analyze a new greedy scheduling algorithm to assign samples and we show that it is a 2-approximation, which is the best that may be expected under general assumptions. On top of this result we design a distributed memory implementation using the Message Passing Interface (MPI) standard. Finally we present a set of numerical experiments illustrating its scalability properties.

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confidence: 99%

A Massively Parallel Implementation of Multilevel Monte Carlo for Finite Element Models

Badia¹,

Hampton²,

Principe³

2021

Preprint

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On Parallel MLMC for Stationary Single Phase Flow Problem

Iliev

Shegunov

Armyanov

et al. 2022

Lecture Notes in Computer Science

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On Dynamic Parallelization of Multilevel Monte Carlo Algorithm

Shegunov

Iliev

2020

Cybernetics and Information Technologies

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MultiLevel Monte Carlo (MLMC) attracts great interest for numerical simulations of Stochastic Partial Differential Equations (SPDEs), due to its superiority over the standard Monte Carlo (MC) approach. MLMC combines in a proper manner many cheap fast simulations with few slow and expensive ones, the variance is reduced, and a significant speed up is achieved. Simulations with MC/MLMC consist of three main components: generating random fields, solving deterministic problem and reduction of the variance. Each part is subject to a different degree of parallelism. Compared to the classical MC, MLMC introduces “levels” on which the sampling is done. These levels have different computational cost, thus, efficiently utilizing the parallel resources becomes a non-trivial problem. The main focus of this paper is the parallelization of the MLMC Algorithm.

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Parallel Multilevel Monte Carlo Algorithms for Elliptic PDEs with Random Coefficients

Cited by 3 publications

References 24 publications

A Massively Parallel Implementation of Multilevel Monte Carlo for Finite Element Models

A Massively Parallel Implementation of Multilevel Monte Carlo for Finite Element Models

On Parallel MLMC for Stationary Single Phase Flow Problem

On Dynamic Parallelization of Multilevel Monte Carlo Algorithm

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