Stochastic domain decomposition for time dependent adaptive mesh generation

Bihlo, Alexander; Haynes, Ronald D.; Walsh, Emily J.

doi:10.48550/arxiv.1504.00084

Cited by 1 publication

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“…Independently, Gobet and Mairé proposed a PDD-like domain decomposition scheme for the Poisson equation in [38]. Bihlo and Haynes have applied PDD to the generation of meshes for finite elements, a PDE-based task very well suited to stochastic representations [15], [14], [16]. Moreover, in [5], the Vlasov-Poisson equations were tackled, which are of unquestionable interest in plasma physics.…”

Section: Some Applications Of Pddmentioning

confidence: 99%

Hybrid PDE solver for data-driven problems and modern branching

Bernal¹,

Reis²,

Smith³

2017

Eur. J. Appl. Math

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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 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.

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