Hybrid PDE solver for data-driven problems and modern branching

• Published in: European journal of applied mathematics Vol. 28; no. 6; pp. 949 - 972
• Main Authors: BERNAL, FRANCISCO, DOS REIS, GONÇALO, SMITH, GREIG
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.12.2017
• Subjects: Algorithms; Applied mathematics; Computer simulation; Domain decomposition methods; Mathematical models; Numerical methods; Parallel computers; Parallel processing; Probabilistic methods; Solvers; Probabilistic Domain Decomposition; high-performance parallel computing; Monte Carlo methods; 65C30; 35CXX; Primary: 65C05; Marked branching diffusions; hybrid non-linear PDE solvers; Secondary: 65N55; 60H35; 91-XX
• ISSN: 0956-7925; 1469-4425; 1469-4425
• DOI: 10.1017/S0956792517000109

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 non-linear 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 non-linear case and open research questions.