题　目：A time-parallel time-integration method for ordinary and partial differential equations
    报告人：Professor Stefan Vandewalle 　　　　K. U. Leuven大学计算机科学系
    地　点：理学院会议室（中二楼2222室）
    时　间：2006年12月6日（星期三）　　　　下午3: 00－4:00
    欢迎广大师生届时参加！
    Abstract: During the last twenty years several algorithms have been suggested for solving time dependent problems parallel in time. In such algorithms parts of the solution later in time are approximated simultaneously to parts of the solution earlier in time. A very recent method was presented in 2001 by Lions, Maday and Turinici, who called their algorithm the parareal algorithm. The name was chosen for the iterative algorithm to indicate that it is well suited for parallel real time computations of evolution problems whose solution can not be obtained in real time using one processor only. The method is not meant as a method to be used on a one processor computer. One iteration of the method costs already as much as the sequential solution of the entire problem, when used on one processor only. If however several processors are used, then the algorithm can lead to an approximate solution in less time than the time needed to compute the solution sequentially. The parareal algorithm has received a lot of attention over the past few years and extensive experiments have been done for fluid and structure problems. In this talk, we will show that the parareal algorithm can be reformulated as a two-level space-time multigrid method with a strong semi-coarsening in the time-dimension. The method can also be seen as a multiple shooting method with a coarse grid Jacobian approximation. These equivalences have opened up new paths for the convergence analysis of the algorithm, which is the topic of the second part of this talk. First, we will show a sharp linear, and a new superlinear convergence result for the parareal algorithm applied to ordinary differential equations. We then use Fourier analysis to derive convergence results for the parareal algorithm applied to partial differential equations. We show that the algorithm converges superlinearly on bounded time intervals, both for parabolic and hyperbolic problems. On long time intervals the algorithm converges linearly for parabolic PDEs. For hyperbolic problems however there is no such convergence estimate on long time intervals. 

    投稿人:陈曦
    单位:理学院
    电话:82668559