Hybrid PDE solver for data-driven problems and modern branching

Francisco Bernal, Gonçalo Dos Reis, Greig Smith

Research output: Contribution to journalArticle

2 Citations (Scopus)

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 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.

Original languageEnglish
Pages (from-to)949-972
Number of pages24
JournalEuropean Journal of Applied Mathematics
Volume28
Issue number6
DOIs
Publication statusPublished - Dec 2017

Fingerprint

Data-driven
Stochastic Representation
Scalability
Branching
Domain Decomposition
Decomposition
Domain decomposition methods
Parallelization
Glossaries
Parallel algorithms
Numerical methods
Nonlinear PDE
Probabilistic Methods
Fully Nonlinear
Domain Decomposition Method
Parallel Computers
Parallel Algorithms
High Performance
Numerical Methods
Numerical Solution

Keywords

  • 2010 AMS Subject Classification: Primary: 65C05 65C30 Secondary: 65N55 60H35 91-XX 35CXX

Cite this

@article{8ad3932842d64aa68eeebd588088dd3b,
title = "Hybrid PDE solver for data-driven problems and modern branching",
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 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 = "2010 AMS Subject Classification: Primary: 65C05 65C30 Secondary: 65N55 60H35 91-XX 35CXX",
author = "Francisco Bernal and {Dos Reis}, Gon{\cc}alo and Greig Smith",
note = "Sem PDF. Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) (UID/MAT/00297/2013) Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training - UK Engineering and Physical Sciences Research Council (EP/L016508/01)",
year = "2017",
month = "12",
doi = "10.1017/S0956792517000109",
language = "English",
volume = "28",
pages = "949--972",
journal = "European Journal of Applied Mathematics",
issn = "0956-7925",
publisher = "Cambridge University Press",
number = "6",

}

Hybrid PDE solver for data-driven problems and modern branching. / Bernal, Francisco; Dos Reis, Gonçalo; Smith, Greig.

In: European Journal of Applied Mathematics, Vol. 28, No. 6, 12.2017, p. 949-972.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Hybrid PDE solver for data-driven problems and modern branching

AU - Bernal, Francisco

AU - Dos Reis, Gonçalo

AU - Smith, Greig

N1 - Sem PDF. Fundacao para a Ciencia e a Tecnologia (Portuguese Foundation for Science and Technology) (UID/MAT/00297/2013) Maxwell Institute Graduate School in Analysis and its Applications, a Centre for Doctoral Training - UK Engineering and Physical Sciences Research Council (EP/L016508/01)

PY - 2017/12

Y1 - 2017/12

N2 - 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.

AB - 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.

KW - 2010 AMS Subject Classification: Primary: 65C05 65C30 Secondary: 65N55 60H35 91-XX 35CXX

UR - http://www.scopus.com/inward/record.url?scp=85021118641&partnerID=8YFLogxK

U2 - 10.1017/S0956792517000109

DO - 10.1017/S0956792517000109

M3 - Article

VL - 28

SP - 949

EP - 972

JO - European Journal of Applied Mathematics

JF - European Journal of Applied Mathematics

SN - 0956-7925

IS - 6

ER -