论文标题
部分可观测时空混沌系统的无模型预测
A randomized operator splitting scheme inspired by stochastic optimization methods
论文作者
论文摘要
在本文中，我们将抽象演化方程的操作员分裂方法与用于大规模优化问题的随机方法的分裂方法相结合。该组合会导致随机分裂方案，在给定的时间步中，该方案并不一定使用拆分操作员的所有部分。这与确定性分裂方案形成鲜明对比的是，这些方案总是至少使用一次，而且通常几次使用每个部分。结果，与此类方法相比，计算成本可以显着降低。我们严格地在抽象设置中定义了一个随机操作员分裂方案，并提供了错误分析，我们证明该方案的时间收敛顺序至少为1/2。我们使用随机结构域分解方法通过数值实验来说明理论。我们得出的结论是，以某些方式选择随机化可以将顺序提高到1。这与应用一样准确，例如向后（隐式）欧拉（Euler）到整个问题，而不会分裂。
In this paper, we combine the operator splitting methodology for abstract evolution equations with that of stochastic methods for large-scale optimization problems. The combination results in a randomized splitting scheme, which in a given time step does not necessarily use all the parts of the split operator. This is in contrast to deterministic splitting schemes which always use every part at least once, and often several times. As a result, the computational cost can be significantly decreased in comparison to such methods. We rigorously define a randomized operator splitting scheme in an abstract setting and provide an error analysis where we prove that the temporal convergence order of the scheme is at least 1/2. We illustrate the theory by numerical experiments on both linear and quasilinear diffusion problems, using a randomized domain decomposition approach. We conclude that choosing the randomization in certain ways may improve the order to 1. This is as accurate as applying e.g. backward (implicit) Euler to the full problem, without splitting.
