scholarly journals An adaptive multilevel Monte Carlo algorithm for the stochastic drift–diffusion–Poisson system

2020 ◽  
Vol 368 ◽  
pp. 113163 ◽  
Author(s):  
Amirreza Khodadadian ◽  
Maryam Parvizi ◽  
Clemens Heitzinger
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Algorithm ◽  
Poisson System ◽  
Multilevel Monte Carlo ◽  
Drift Diffusion

scholarly journals A multilevel Monte Carlo finite difference method for random scalar degenerate convection–diffusion equations

2017 ◽  
Vol 14 (03) ◽  
pp. 415-454 ◽  
Author(s):  
Ujjwal Koley ◽  
Nils Henrik Risebro ◽  
Christoph Schwab ◽  
Franziska Weber
Keyword(s):  
Monte Carlo ◽  
Finite Difference ◽  
Convergence Rate ◽  
Finite Volume ◽  
Monte Carlo Algorithm ◽  
Diffusion Equations ◽  
Entropy Solutions ◽  
Finite Volume Scheme ◽  
Multilevel Monte Carlo ◽  
Convection Diffusion

This paper proposes a finite difference multilevel Monte Carlo algorithm for degenerate parabolic convection–diffusion equations where the convective and diffusive fluxes are allowed to be random. We establish a notion of stochastic entropy solutions to these equations. Our chief goal is to efficiently compute approximations to statistical moments of these stochastic entropy solutions. To this end, we design a multilevel Monte Carlo method based on a finite volume scheme for each sample. We present a novel convergence rate analysis of the combined multilevel Monte Carlo finite volume method, allowing in particular for low [Formula: see text]-integrability of the random solution with [Formula: see text], and low deterministic convergence rates (here, the theoretical rate is [Formula: see text]). We analyze the design and error versus work of the multilevel estimators. We obtain that the maximal rate (based on optimizing possibly the pessimistic upper bounds on the discretization error) is obtained for [Formula: see text], for finite volume convergence rate of [Formula: see text]. We conclude with numerical experiments.

scholarly journals On Dynamic Parallelization of Multilevel Monte Carlo Algorithm

2020 ◽  
Vol 20 (6) ◽  
pp. 116-125
Author(s):  
Nikolay Shegunov ◽  
Oleg Iliev
Keyword(s):  
Monte Carlo ◽  
Numerical Simulations ◽  
Random Fields ◽  
Stochastic Partial Differential Equations ◽  
Computational Cost ◽  
Monte Carlo Algorithm ◽  
Multilevel Monte Carlo ◽  
Deterministic Problem ◽  
Speed Up ◽  
Main Components

AbstractMultiLevel 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.

scholarly journals Implementation and analysis of an adaptive multilevel Monte Carlo algorithm

2014 ◽  
Vol 20 (1) ◽  
pp. 1-41 ◽  
Author(s):  
Håkon Hoel ◽  
Erik von Schwerin ◽  
Anders Szepessy ◽  
Raúl Tempone
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Algorithm ◽  
Multilevel Monte Carlo

scholarly journals A continuation multilevel Monte Carlo algorithm

2014 ◽  
Vol 55 (2) ◽  
pp. 399-432 ◽  
Author(s):  
Nathan Collier ◽  
Abdul-Lateef Haji-Ali ◽  
Fabio Nobile ◽  
Erik von Schwerin ◽  
Raúl Tempone
Keyword(s):  
Monte Carlo ◽  
Monte Carlo Algorithm ◽  
Multilevel Monte Carlo

scholarly journals Multilevel Monte Carlo algorithm for quantum mechanics on a lattice

2020 ◽  
Vol 102 (11) ◽  
Author(s):  
Karl Jansen ◽  
Eike H. Müller ◽  
Robert Scheichl
Keyword(s):  
Monte Carlo ◽  
Quantum Mechanics ◽  
Monte Carlo Algorithm ◽  
Multilevel Monte Carlo
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