Multi-level Monte Carlo Finite Element method for parabolic stochastic partial differential equations
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Date
2011-05Type
- Report
ETH Bibliography
yes
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Abstract
We analyze the convergence and complexity of multi-level Monte Carlo (MLMC) discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show, under regularity assumptions on the solution that are minimal under certain criteria, that the judicious combination of piecewise linear, continuous multi-level Finite Element discretizations in space and Euler--Maruyama discretizations in time yields mean square convergence of order one in space and of order $1/2$ in time to the expected value of the mild solution. The complexity of the multi-level estimator is shown to scale log-linearly with respect to the corresponding work to generate a single solution path on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mesh. Examples are provided for Lévy driven SPDEs as well as equations for randomly forced surface diffusions. Show more
Publication status
publishedExternal links
Journal / series
SAM Research ReportVolume
Publisher
Seminar for Applied Mathematics, ETH ZurichSubject
Multi-level Monte Carlo; Stochastic partial differential equations; Stochastic finite element methods; Stochastic parabolic equation; Multi-level approximationsOrganisational unit
03435 - Schwab, Christoph / Schwab, Christoph
Funding
247277 - Automated Urban Parking and Driving (EC)
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ETH Bibliography
yes
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