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Monte Carlo versus multilevel Monte Carlo in weak error simulations of SPDE approximations

Annika Lang (Institutionen för matematiska vetenskaper, Tillämpad matematik och statistik) ; Andreas Petersson (Institutionen för matematiska vetenskaper, Tillämpad matematik och statistik)
Mathematics and Computers in Simulation (0378-4754). Vol. 143 (2018), p. 99-113.
[Artikel, refereegranskad vetenskaplig]

The simulation of the expectation of a stochastic quantity E[Y] by Monte Carlo methods is known to be computationally expensive especially if the stochastic quantity or its approximation Y-n is expensive to simulate, e.g., the solution of a stochastic partial differential equation. If the convergence of Y-n to Y in terms of the error |E[Y - Y-n]| is to be simulated, this will typically be done by a Monte Carlo method, i.e., |E[Y] - E-N [Y-n]| is computed. In this article upper and lower bounds for the additional error caused by this are determined and compared to those of |E-N [Y - Y-n]|, which are found to be smaller. Furthermore, the corresponding results for multilevel Monte Carlo estimators, for which the additional sampling error converges with the same rate as |E[Y - Y-n]|, are presented. Simulations of a stochastic heat equation driven by multiplicative Wiener noise and a geometric Brownian motion are performed which confirm the theoretical results and show the consequences of the presented theory for weak error simulations.

Nyckelord: (multilevel) Monte Carlo methods, Variance reduction techniques, Stochastic partial differential equations, Weak convergence, Upper and lower error bounds



Denna post skapades 2017-11-14.
CPL Pubid: 253137

 

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Institutioner (Chalmers)

Institutionen för matematiska vetenskaper, Tillämpad matematik och statistikInstitutionen för matematiska vetenskaper, Tillämpad matematik och statistik (GU)

Ämnesområden

Sannolikhetsteori och statistik

Chalmers infrastruktur

C3SE/SNIC (Chalmers Centre for Computational Science and Engineering)