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CONVERGENCE ANALYSIS OF PARAREAL ALGORITHM BASED ON MILSTEIN SCHEME FOR STOCHASTIC DIFFERENTIAL EQUATIONS 被引量:1
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作者 Liying Zhang Jing Wang +2 位作者 Weien Zhou Landong Liu Li Zhang 《Journal of Computational Mathematics》 SCIE CSCD 2020年第3期487-501,共15页
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
关键词 Stochastic differential equations parareal algorithm CONVERGENCE Stochastic Taylor expansion Milstein scheme
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Parareal-Richardson Algorithm for Solving Nonlinear ODEs and PDEs 被引量:1
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作者 Shulin Wu Baochang Shi Chengming Huang 《Communications in Computational Physics》 SCIE 2009年第9期883-902,共20页
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
关键词 Parallel computation parareal algorithm Richardson extrapolation ACCURACY nonlinear problems
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THE REDUCED BASIS TECHNIQUE AS A COARSE SOLVER FOR PARAREAL IN TIME SIMULATIONS
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作者 Liping He 《Journal of Computational Mathematics》 SCIE CSCD 2010年第5期676-692,共17页
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
关键词 Finite element and spectral element approximations Multi-meshes and multi-degrees techniques Reduced basis technique Semi-implicit RungeoKutta scheme Offline-online procedure parareal in time algorithm.
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