Parareal is a well-studied algorithm for numerically integrating systems of time-dependent differential equations by parallelising the temporal domain. Given approximate initial values at each temporal sub-interval, the algorithm locates a solution in a fixed number of iterations using a predictor-corrector, stopping once a tolerance is met. This iterative process combines solutions located by inexpensive (coarse resolution) and expensive (fine resolution) numerical integrators. In this paper, we introduce a stochastic parareal algorithm with the aim of accelerating the convergence of the deterministic parareal algorithm. Instead of providing the predictor-corrector with a deterministically located set of initial values, the stochastic algorithm samples initial values from dynamically varying probability distributions in each temporal sub-interval. All samples are then propagated by the numerical method in parallel. The initial values yielding the most continuous (smoothest) trajectory across consecutive sub-intervals are chosen as the new, more accurate, set of initial values. These values are fed into the predictor-corrector, converging in fewer iterations than the deterministic algorithm with a given probability. The performance of the stochastic algorithm, implemented using various probability distributions, is illustrated on systems of ordinary differential equations. When the number of sampled initial values is large enough, we show that stochastic parareal converges almost certainly in fewer iterations than the deterministic algorithm while maintaining solution accuracy. Additionally, it is shown that the expected value of the convergence rate decreases with increasing numbers of samples.
翻译:paralreal 是一种通过平行时间域来对基于时间的差别方程式进行数字整合的算法。 根据每个时间子间替法的大约初始值, 算法将解决方案定位在使用预测器- 校正器的固定迭代数中, 一旦满足了容忍度, 就会停止。 这个迭代进程将廉价( 粗分辨率) 和昂贵( 分辨率) 数字整合器的解决方案结合起来。 在本文中, 我们引入了一种随机准方程算法, 目的是加速确定性准异方程式的趋同。 这些数值被输入到预测者- 校正值中, 最初值的位置是确定性标定数的一组, 随机算算法从动态变化的概率分布开始, 然后通过数字方法平行传播。 产生最连续次间替( 分辨率) 轨迹的初始值是新的、 更准确的、 初始值组合。 这些数值被输入到预测者- 预测者- 初始值的精确性值, 预测算法的初始值在几乎递归正数中, 显示性运算法的概率的概率值是显示的数值的概率值的数值, 。 显示的概率的概率值是不同的数值, 。 度的概率值的数值在不同的计算中, 显示的概率值的数值在不同的计算中, 的数值在不同的计算中, 的概率值的概率值在不同的计算中, 的概率值在不同的计算中, 的概率值在不同的计算中, 显示的数值在不同的计算中, 的数值是不同的计算中, 。