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 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.

A flexible split-step scheme for solving McKean-Vlasov Stochastic Differential Equations

“…A priori one should expect the taming to be the fastest of the three algorithms. We highlight that due to the nature of the empirical measure, after each time-step the processors need to communicate to update the calculation of the empirical measure, this leads to a well-known loss of parallelization power [9].…”

Introduction: Big data and partial differential equations

“…In [3], the authors review a particular class of domain decomposition methods, which is called probabilistic domain decomposition (PDD), pioneered by Acebron et al in [48]. The main idea is to use a stochastic representation of the PDE (via the so-called Feynman-Kac formula), then compute the solution in a few sampled points on the interfaces between the subdomains via Monte Carlo, and then use an efficient deterministic PDE solver for the solution of the PDE on each subdomain with fixed boundary conditions coming from the previously computed Monte Carlo simulation.…”

“…Also, the choice of the PDE solver on each subdomain is flexible and so can potentially be executed very efficiently. The paper [3] also serves as an introduction to the concept of PDDs and their various usage areas in data-driven applications.…”

An Unbiased Itô Type Stochastic Representation for Transport PDEs: A Toy Example

“…Stochastic techniques to solve PDEs have become increasing popular in recent times with advances in computing power and numerical techniques allowing for solutions of PDEs to be calculated to high precision. Advances in BSDEs (Backward Stochastic Differential Equations) and so-called branching diffusions also allow one to tackle nonlinear PDEs (see [BdRS17] and references therein). Stochastic representations for PDEs are useful as they give access to probabilistic Monte Carlo methods, in turn yielding strong numerical gains over deterministic based solvers, especially in high dimensional problems, see [HJW17,FTW11,BdRS17].…”

“…Advances in BSDEs (Backward Stochastic Differential Equations) and so-called branching diffusions also allow one to tackle nonlinear PDEs (see [BdRS17] and references therein). Stochastic representations for PDEs are useful as they give access to probabilistic Monte Carlo methods, in turn yielding strong numerical gains over deterministic based solvers, especially in high dimensional problems, see [HJW17,FTW11,BdRS17]. Unlike their deterministic counterparts, stochastic based PDE solvers are less prone to the curse of dimensionality.…”