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

Published online by Cambridge University Press:  22 May 2017

FRANCISCO BERNAL ,
GONÇALO DOS REIS  and
GREIG SMITH
FRANCISCO BERNAL
Affiliation:
CMAP - Centre de Mathématiques Appliquées, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France email: [email protected]
GONÇALO DOS REIS
Affiliation:
School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD, UK email: [email protected] Centro de Matemática e Aplicações (CMA), FCT, UNL, 2829-516 Caparica, Portugal
GREIG SMITH
Affiliation:
School of Mathematics, University of Edinburgh, Edinburgh, EH9 3FD, UK email: [email protected] Maxwell Institute Graduate School in Analysis and its Applications, University of Edinburgh, Edinburgh, EH9 3FD, UK email: [email protected]
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Abstract

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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.

Keywords

Probabilistic Domain Decompositionhigh-performance parallel computingMarked branching diffusionshybrid non-linear PDE solversMonte Carlo methodsPrimary: 65C0565C30Secondary: 65N5560H3591-XX35CXX
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Special Issue 6: Big Data and Partial Differential Equations , December 2017 , pp. 949 - 972
Copyright
Copyright © Cambridge University Press 2017 

Footnotes

†

F. Bernal acknowledges funding from Centre de Mathématiques Appliquées (CMAP), École Polytechnique. G. dos Reis gratefully thanks the partial support by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project UID/MAT/00297/2013 (Centro de Matemática e Aplicações). G. Smith was supported by The Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training funded by the UK Engineering and Physical Sciences Research Council (grant [EP/L016508/01]), the Scottish Funding Council, Heriot-Watt University and the University of Edinburgh.

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