Stabilized multilevel Monte Carlo method for stiff stochastic differential equations

A multilevel Monte Carlo (MLMC) method for mean square stable stochastic differential equations with multiple scales is proposed. For such problems, that we call stiff, the performance of MLMC methods based on classical explicit methods deteriorates because of the time step restriction to resolve th...

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Bibliographic Details
Published in:Journal of computational physics 2013-10, Vol.251, p.445-460
Main Authors: Abdulle, Assyr, Blumenthal, Adrian
Format: Article
Language:English
Subjects:
ALGORITHMS
CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS
Computation
Computational efficiency
Differential equations
Mathematical models
MONTE CARLO METHOD
Monte Carlo methods
Multilevel
Multilevel Monte Carlo
NONLINEAR PROBLEMS
PARTIAL DIFFERENTIAL EQUATIONS
S-ROCK method
SAMPLING
STABILITY
STABILIZATION
Stiffness
Stochastic differential equations
STOCHASTIC PROCESSES
Stochasticity
Strategy
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Description
Summary:A multilevel Monte Carlo (MLMC) method for mean square stable stochastic differential equations with multiple scales is proposed. For such problems, that we call stiff, the performance of MLMC methods based on classical explicit methods deteriorates because of the time step restriction to resolve the fastest scales that prevents to exploit all the levels of the MLMC approach. We show that by switching to explicit stabilized stochastic methods and balancing the stabilization procedure simultaneously with the hierarchical sampling strategy of MLMC methods, the computational cost for stiff systems is significantly reduced, while keeping the computational algorithm fully explicit and easy to implement. Numerical experiments on linear and nonlinear stochastic differential equations and on a stochastic partial differential equation illustrate the performance of the stabilized MLMC method and corroborate our theoretical findings.
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2013.05.039