A Multigrid-like Algorithm for Probabilistic Domain Decomposition
This work addresses the computational bottleneck of PDD for solving large-scale boundary value problems, offering a significant speedup.
The paper introduces an iterative algorithm that accelerates Probabilistic Domain Decomposition (PDD) for solving large boundary value problems, achieving one to two orders of magnitude improvement in computational efficiency.
We present an iterative scheme, reminiscent of the Multigrid method, to solve large boundary value problems with Probabilistic Domain Decomposition (PDD). In it, increasingly accurate approximations to the solution are used as control variates in order to reduce the Monte Carlo error of the following iterates--resulting in an overall acceleration of PDD for a given error tolerance. The key ingredient of the proposed algorithm is the ability to approximately predict the speedup with little computational overhead and in parallel. Besides, the theoretical framework allows to explore other aspects of PDD, such as stability. One numerical example is worked out, yielding an improvement of between one and two orders of magnitude over the previous version of PDD.