Sequential numerical methods for integrating initial value problems (IVPs) can be prohibitively expensive when high numerical accuracy is required over the entire interval of integration. One remedy is to integrate in a parallel fashion, "predicting" the solution serially using a cheap (coarse) solver and "correcting" these values using an expensive (fine) solver that runs in parallel on a number of temporal subintervals. In this work, we propose a time-parallel algorithm (GParareal) that solves IVPs by modelling the correction term, i.e. the difference between fine and coarse solutions, using a Gaussian process emulator. This approach compares favourably with the classic parareal algorithm and we demonstrate, on a number of IVPs, that GParareal can converge in fewer iterations than parareal, leading to an increase in parallel speed-up. GParareal also manages to locate solutions to certain IVPs where parareal fails and has the additional advantage of being able to use archives of legacy solutions, e.g. solutions from prior runs of the IVP for different initial conditions, to further accelerate convergence of the method -- something that existing time-parallel methods do not do.
翻译:整合初始值问题( IVPs) 的序列数方法, 在整个整合期需要高数字精确度时, 可能会非常昂贵。 一种补救措施是同时整合, 使用廉价( 粗) 求解器和“ 校正” 这些值, 使用一个价格昂贵( 纯) 求解器, 与一些时间的次interval并行运行。 在这项工作中, 我们提议一个时间单算法( GParareal), 通过模拟校正术语解决IVPs, 即精细和粗粗的解决方案之间的差别, 使用高斯进程模拟器。 这种方法优于典型的模拟算法, 我们用一些IVPs表示, GParalalalalalalal, 其迭代代之以比仿真少, 导致同步加速速度增长。 GPalalalalalalalalal还设法将解决方案定位给某些具有超正数的IVPs,, 其额外优势是能够使用遗留解决方案的档案, 例如, 与以前运行的 IVP 的解决方案相比, 不同的初始方法不会加速现有方法的聚合。