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.

在本文中，我们将抽象演化方程的算子分裂方法与大规模优化问题的随机方法相结合。该组合产生随机分割方案，该方案在给定时间步长内不一定使用分割算子的所有部分。这与确定性分割方案形成对比，确定性分割方案总是使用每个部分至少一次，并且通常多次。因此，与此类方法相比，计算成本可以显着降低。我们在抽象设置中严格定义了随机算子分割方案，并提供了误差分析，证明该方案的时间收敛阶数至少为 1/2。我们使用随机域分解方法，通过线性和拟线性扩散问题的数值实验来说明该理论。我们得出的结论是，以某些方式选择随机化可以将阶数提高到 1。这与将向后（隐式）欧拉应用于整个问题而不进行分裂一样准确。