Hybrid PDE solver for data-driven problems and modern Branching

The numerical solution of large-scale PDEs--such as those naturally occurring in data-driven applications--unavoidably require powerful parallel computers and tailored parallel algorithms to make the best possible use of them. In fact, considerations about the parallelization and scalability of realistic problems are often critical enough to warrant acknowledgment already in the modeling phase. The purpose of this paper is to spread awareness of the Probabilistic Domain Decomposition (PDD) method, a fresh approach to the parallelization of PDEs with excellent scalability properties. The idea exploits the stochastic representation of the PDE via Monte Carlo sampling in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its range of applicability in the scope of data science. In particular, we highlight recent advances in stochastic representations for nonlinear PDEs using branching diffusions, which have significantly broadened the scope of PDD.

We envision this work as a dictionary giving large-scale PDE practitioners references on the very latest algorithms and techniques of a non-standard, yet highly parallelizable, methodology at the interface of deterministic and probabilistic numerical methods.