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Multilevel Monte Carlo method with applications to stochastic partial differential equations. (English) Zbl 1270.65003

To approximate the expectation (or generally a functional) of a solution \(Y\) of a stochastic partial differential equation, taking as a Hilbert valued random variable, the multilevel Monte Carlo method is: by using a finite-dimensional subspace sequence of random variables \(Y_{\ell}\) to approximate \(Y\) first, then to take a sample of size of \(N_{\ell}\) of \(Y_{\ell}-Y_{{\ell}-1}\) to form a Monte Carlo estimate of \(E(Y_{\ell}-Y_{{\ell}-1})\), and finally to sum to get the \(\ell\)-level Monte Carlo estimate of \(E(Y_{\ell})\). Here \(N_{\ell}\) is chosen balanced to different levels to make the approximate rate better. In the present paper, the finite element approximation of the heat equation with white noise and the linearized backward Euler scheme of a first-order hyperbolic equation with white noise are treated to obtain better rates. Numerical examples are illustrated.

MSC:

65C30 Numerical solutions to stochastic differential and integral equations
65C05 Monte Carlo methods
60G60 Random fields
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
35K05 Heat equation
35L02 First-order hyperbolic equations
60H35 Computational methods for stochastic equations (aspects of stochastic analysis)
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References:

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