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Multilevel Monte Carlo methods. (English) Zbl 1316.65010

Summary: Monte Carlo methods are a very general and useful approach for the estimation of expectations arising from stochastic simulation. However, they can be computationally expensive, particularly when the cost of generating individual stochastic samples is very high, as in the case of stochastic partial differential equations. Multilevel Monte Carlo is a recently developed approach which greatly reduces the computational cost by performing most simulations with low accuracy at a correspondingly low cost, with relatively few simulations being performed at high accuracy and a high cost.
In this article, we review the ideas behind the multilevel Monte Carlo method, and various recent generalizations and extensions, and discuss a number of applications which illustrate the flexibility and generality of the approach and the challenges in developing more efficient implementations with a faster rate of convergence of the multilevel correction variance.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
65C05 Monte Carlo methods
35R60 PDEs with randomness, stochastic partial differential equations
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65M75 Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs
65N75 Probabilistic methods, particle methods, etc. for boundary value problems involving PDEs
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