In this article,we address two issues related to the perturbation method introduced by Zhang and Lu(J Comput Phys 194:773-794,2004),and applied to solving linear stochastic parabolic PDE.Those issues are the construct...In this article,we address two issues related to the perturbation method introduced by Zhang and Lu(J Comput Phys 194:773-794,2004),and applied to solving linear stochastic parabolic PDE.Those issues are the construction of the perturbation series,and its convergence.展开更多
In this article, we propose a space-time Multi-Index Monte Carlo （MIMC） estimator for a one-dimensional parabolic stochastic partial differential equation （SPDE） of Zakai type. We compare the complexity with the M...In this article, we propose a space-time Multi-Index Monte Carlo （MIMC） estimator for a one-dimensional parabolic stochastic partial differential equation （SPDE） of Zakai type. We compare the complexity with the Multilevel Monte Carlo （MLMC） method of Giles and Reisinger （2012）, and find, by means of Fourier analysis, that the MIMC method： （i） has suboptimal complexity of 0（ε^-21 ｜ogε｜） for a root mean square error （RMSE） z if the same spatial discretisation as in the MLMC method is used; （ii） has a better complexity of 0（ε^-21 ｜ogε｜） if a carefully adapted discretisation is used; （iii） has to be adapted for non-smooth functionals. Numerical tests confirm these findings empirically.展开更多
文摘In this article,we address two issues related to the perturbation method introduced by Zhang and Lu(J Comput Phys 194:773-794,2004),and applied to solving linear stochastic parabolic PDE.Those issues are the construction of the perturbation series,and its convergence.
文摘In this article, we propose a space-time Multi-Index Monte Carlo （MIMC） estimator for a one-dimensional parabolic stochastic partial differential equation （SPDE） of Zakai type. We compare the complexity with the Multilevel Monte Carlo （MLMC） method of Giles and Reisinger （2012）, and find, by means of Fourier analysis, that the MIMC method： （i） has suboptimal complexity of 0（ε^-21 ｜ogε｜） for a root mean square error （RMSE） z if the same spatial discretisation as in the MLMC method is used; （ii） has a better complexity of 0（ε^-21 ｜ogε｜） if a carefully adapted discretisation is used; （iii） has to be adapted for non-smooth functionals. Numerical tests confirm these findings empirically.
