In this paper,we propose a parareal algorithm for stochastic differential equations(SDEs),which proceeds as a two-level temporal parallelizable integrator with the Milstein scheme as the coarse propagator and the exac...In this paper,we propose a parareal algorithm for stochastic differential equations(SDEs),which proceeds as a two-level temporal parallelizable integrator with the Milstein scheme as the coarse propagator and the exact solution as the fine propagator.The convergence order of the proposed algorithm is analyzed under some regular assumptions.Finally,numerical experiments are dedicated to illustrate the convergence and the convergence order with respect to the iteration number k,which show the efficiency of the proposed method.展开更多
The parareal algorithm,proposed firstly by Lions et al.[J.L.Lions,Y.Maday,and G.Turinici,A”parareal”in time discretization of PDE’s,C.R.Acad.Sci.Paris Ser.I Math.,332(2001),pp.661-668],is an effective algorithm to ...The parareal algorithm,proposed firstly by Lions et al.[J.L.Lions,Y.Maday,and G.Turinici,A”parareal”in time discretization of PDE’s,C.R.Acad.Sci.Paris Ser.I Math.,332(2001),pp.661-668],is an effective algorithm to solve the timedependent problems parallel in time.This algorithm has received much interest from many researchers in the past years.We present in this paper a new variant of the parareal algorithm,which is derived by combining the original parareal algorithm and the Richardson extrapolation,for the numerical solution of the nonlinear ODEs and PDEs.Several nonlinear problems are tested to show the advantage of the new algorithm.The accuracy of the obtained numerical solution is compared with that of its original version(i.e.,the parareal algorithm based on the same numerical method).展开更多
In this paper, we extend the reduced basis methods for parameter dependent problems to the parareal in time algorithm introduced by Lions et al. [12] and solve a nonlinear evolutionary parabolic partial differential e...In this paper, we extend the reduced basis methods for parameter dependent problems to the parareal in time algorithm introduced by Lions et al. [12] and solve a nonlinear evolutionary parabolic partial differential equation. The fine solver is based on the finite element method or spectral element method in space and a semi-implicit Runge-Kutta scheme in time. The coarse solver is based on a semi-implicit scheme in time and the reduced basis approximation in space. Of[line-online procedures are developed, and it is proved that the computational complexity of the on-line stage depends only on the dimension of the reduced basis space (typically small). Parareal in time algorithms based on a multi-grids finite element method and a multi-degrees finite element method are also presented. Some numerical results are reported.展开更多
基金We are very grateful to the reviewers for reading our paper carefully and providing many useful comments and suggestions.The first author is supported by NNSFC(Nos.11601514,11771444,11801556 and 11971458)The fourth author is supported by Beijing Nature Science Foundation(No.1152002)This work is also supported by NSF of Jiangsu Province of China(BK.20130779).
文摘In this paper,we propose a parareal algorithm for stochastic differential equations(SDEs),which proceeds as a two-level temporal parallelizable integrator with the Milstein scheme as the coarse propagator and the exact solution as the fine propagator.The convergence order of the proposed algorithm is analyzed under some regular assumptions.Finally,numerical experiments are dedicated to illustrate the convergence and the convergence order with respect to the iteration number k,which show the efficiency of the proposed method.
基金This work was supported by NSF of China(Nos.10671078,60773195)and by Program for NCETthe State Education Ministry of China.
文摘The parareal algorithm,proposed firstly by Lions et al.[J.L.Lions,Y.Maday,and G.Turinici,A”parareal”in time discretization of PDE’s,C.R.Acad.Sci.Paris Ser.I Math.,332(2001),pp.661-668],is an effective algorithm to solve the timedependent problems parallel in time.This algorithm has received much interest from many researchers in the past years.We present in this paper a new variant of the parareal algorithm,which is derived by combining the original parareal algorithm and the Richardson extrapolation,for the numerical solution of the nonlinear ODEs and PDEs.Several nonlinear problems are tested to show the advantage of the new algorithm.The accuracy of the obtained numerical solution is compared with that of its original version(i.e.,the parareal algorithm based on the same numerical method).
文摘In this paper, we extend the reduced basis methods for parameter dependent problems to the parareal in time algorithm introduced by Lions et al. [12] and solve a nonlinear evolutionary parabolic partial differential equation. The fine solver is based on the finite element method or spectral element method in space and a semi-implicit Runge-Kutta scheme in time. The coarse solver is based on a semi-implicit scheme in time and the reduced basis approximation in space. Of[line-online procedures are developed, and it is proved that the computational complexity of the on-line stage depends only on the dimension of the reduced basis space (typically small). Parareal in time algorithms based on a multi-grids finite element method and a multi-degrees finite element method are also presented. Some numerical results are reported.