# Hybrid PDE solver for data-driven problems and modern branching

## TL;DR

This paper introduces the Probabilistic Domain Decomposition (PDD) method, a scalable parallel approach for solving large-scale PDEs in data-driven applications, leveraging stochastic representations and Monte Carlo techniques.

## Contribution

It presents the PDD method, combining probabilistic and deterministic PDE solvers, and discusses recent advances in stochastic representations for nonlinear PDEs using branching diffusions.

## Key findings

- PDD exhibits excellent scalability for large-scale PDEs.
- Recent stochastic representations enable solving nonlinear PDEs with branching diffusions.
- The paper provides a comprehensive overview of advanced parallel algorithms for PDEs.

## Abstract

The numerical solution of large-scale PDEs, such as those 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 acknowledgement in the modelling 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 and its approximation via Monte Carlo in combination with deterministic high-performance PDE solvers. We describe the ingredients of PDD and its 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. We close this work with an invitation to the fully nonlinear case and open research questions.