In this paper,we couple the parareal algorithm with projection methods of the trajectory on a specifc manifold,defined by the preservation of some conserved quantities of stochastic differential equations.First,projec...In this paper,we couple the parareal algorithm with projection methods of the trajectory on a specifc manifold,defined by the preservation of some conserved quantities of stochastic differential equations.First,projection methods are introduced as the coarse and fine propagators.Second,we apply the projection methods for systems with conserved quantities in the correction step of original parareal algorithm.Finally,three mumerical experiments are performed by different kinds of algorithms to show the property of convergence in iteration,and preservation in conserved quantities of model systems.展开更多
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
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
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,a reducedmorphological transformation model with spatially dependent composition and elastic modulus is considered.The parareal in time algorithm introduced by Lions et al.is developed for longer-time si...In this paper,a reducedmorphological transformation model with spatially dependent composition and elastic modulus is considered.The parareal in time algorithm introduced by Lions et al.is developed for longer-time simulation.The fine solver is based on a second-order scheme in reciprocal space,and the coarse solver is based on a multi-model backward Euler scheme,which is fast and less expensive.Numerical simulations concerning the composition with a randomnoise and a discontinuous curve are performed.Some microstructure characteristics at very low temperature are obtained by a variable temperature technique.展开更多
In this paper,we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation,and then develop an optimal Galerkin finite element scheme without any regularity ass...In this paper,we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation,and then develop an optimal Galerkin finite element scheme without any regularity assumption on its true solution.We show that the solution regularity of the considered problem can be affected by the maximum value of variable-order at initial time t=0.More precisely,we prove that the solution to the multi-term variable-order time-fractional diffusion equation belongs to C 2([0,T])in time provided that the maximum value has an integer limit near the initial time and the data has sufficient smoothness,otherwise the solution exhibits the same singular behavior like its constant-order counterpart.Based on these regularity results,we prove optimalorder convergence rate of the Galerkin finite element scheme.Furthermore,we develop an efficient parallel-in-time algorithm to reduce the computational costs of the evaluation of multi-term variable-order fractional derivatives.Numerical experiments are put forward to verify the theoretical findings and to demonstrate the efficiency of the proposed scheme.展开更多
基金National Natural Science Foundation of China (Nos. 11601514,11626228 and 91630312)Beijing Natural Science Foundation (No.1152002)+1 种基金National Natural Science Foundation of China (Nos.11501570 and 11571366)National Natural Science Foundation of China (Nos.11601032,11471310 and 91630312).
文摘In this paper,we couple the parareal algorithm with projection methods of the trajectory on a specifc manifold,defined by the preservation of some conserved quantities of stochastic differential equations.First,projection methods are introduced as the coarse and fine propagators.Second,we apply the projection methods for systems with conserved quantities in the correction step of original parareal algorithm.Finally,three mumerical experiments are performed by different kinds of algorithms to show the property of convergence in iteration,and preservation in conserved quantities of model systems.
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.
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.
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).
基金We wish to express our gratitude to the referees for their valuable suggestions.This research was supported by the National Natural Science Foundation of China(Grant No.11171218)the Special Foundation of Shanghai Jiaotong University for Science and Technology Innovation(Grant No.AE0710004).
文摘In this paper,a reducedmorphological transformation model with spatially dependent composition and elastic modulus is considered.The parareal in time algorithm introduced by Lions et al.is developed for longer-time simulation.The fine solver is based on a second-order scheme in reciprocal space,and the coarse solver is based on a multi-model backward Euler scheme,which is fast and less expensive.Numerical simulations concerning the composition with a randomnoise and a discontinuous curve are performed.Some microstructure characteristics at very low temperature are obtained by a variable temperature technique.
基金the National Natural Science Foundation of China(No.11971482)the Natural Science Foundation of Shandong Province(No.ZR2017MA006)+2 种基金the National Science Foundation(No.DMS-1620194)the China Postdoctoral Science Foundation(Nos.2020M681136,2021TQ0017,2021T140129)the International Postdoctoral Exchange Fellowship Program(Talent-Introduction Program)(No.YJ20210019).
文摘In this paper,we study the well-posedness and solution regularity of a multi-term variable-order time-fractional diffusion equation,and then develop an optimal Galerkin finite element scheme without any regularity assumption on its true solution.We show that the solution regularity of the considered problem can be affected by the maximum value of variable-order at initial time t=0.More precisely,we prove that the solution to the multi-term variable-order time-fractional diffusion equation belongs to C 2([0,T])in time provided that the maximum value has an integer limit near the initial time and the data has sufficient smoothness,otherwise the solution exhibits the same singular behavior like its constant-order counterpart.Based on these regularity results,we prove optimalorder convergence rate of the Galerkin finite element scheme.Furthermore,we develop an efficient parallel-in-time algorithm to reduce the computational costs of the evaluation of multi-term variable-order fractional derivatives.Numerical experiments are put forward to verify the theoretical findings and to demonstrate the efficiency of the proposed scheme.
