Probabilistically induced domain decomposition methods for elliptic boundary-value problems

Acebrón, Juan A.; Busico, Maria Pia; Lanucara, Piero; Spigler, Renato

doi:10.1016/j.jcp.2005.04.014

Cited by 18 publications

(24 citation statements)

References 29 publications

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“…As an alternative the method can be combined with the FDTD method through a domain decomposition method. This idea, called probabilistic domain decomposition method [2], was already proposed in literature for different problems, but can also be applied, adapting conveniently, for the telegraph equations.…”

Section: Resultsmentioning

confidence: 99%

“…(1) Generate a two-point discrete random variable ξ 1 taking the values 0, and [2] procedure update left(X, sign, τ, solution)…”

Section: Probabilistic Formulation Of Telegraph Equations With Boundamentioning

confidence: 99%

“…When this formula is used for numerical purpose, to obtain accurate solutions requires to compute precisely the first exit time of a diffusion process (or a Brownian motion in particular for a Laplace operator). Numerical experiments show that the error made estimating the first exit time may be dominant over the other sources of numerical errors and is therefore of paramount importance in providing an accurate value of such a quantity [2]. Rather, the corresponding probabilistic method for the telegraph equations does not exhibit this issue, mainly because the nature of the generated random process is completely different.…”

Section: Algorithm 1 Telegraph Equations With Boundary Conditionsmentioning

confidence: 99%

“…An alternative consists in combining the Monte Carlo method with other classical techniques, such as the domain decomposition method, computing merely the solution in a few points along some chosen interfaces inside the domain. This method is called probabilistic domain decomposition method (PDD), and was successfully used to solve a variety of problems modeled by elliptic, and linear and semilinear parabolic partial differential equations [2,3].…”

Section: Introductionmentioning

confidence: 99%

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A Monte Carlo method for solving the one-dimensional telegraph equations with boundary conditions

Acebrón

Ribeiro

2016

Journal of Computational Physics

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Please cite this article in press as: J.A. Acebrón, M.A. Ribeiro, A Monte Carlo method for solving the one-dimensional telegraph equations with boundary conditions, J. Comput. Phys. (2015), http://dx.doi.org/10.1016/j.jcp. 2015.10.027 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. A Monte Carlo AbstractA Monte Carlo algorithm is derived to solve the one-dimensional telegraph equations in a bounded domain subject to resistive and non-resistive boundary conditions. The proposed numerical scheme is more efficient than the classical Kac's theory because it does not require the discretization of time. The algorithm has been validated by comparing the results obtained with theory and the Finite-difference time domain (FDTD) method for a typical two-wire transmission line terminated at both ends with general boundary conditions. We have also tested transmission line heterogeneities to account for wave propagation in multiple media. The algorithm is inherently parallel, since it is based on Monte Carlo simulations, and does not suffer from the numerical dispersion and dissipation issues that arise in finite differencebased numerical schemes on a lossy medium. This allowed us to develop an efficient numerical method, capable of outperforming the classical FDTD method for large scale problems and high frequency signals.

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Section: Resultsmentioning

confidence: 99%

“…(1) Generate a two-point discrete random variable ξ 1 taking the values 0, and [2] procedure update left(X, sign, τ, solution)…”

Section: Probabilistic Formulation Of Telegraph Equations With Boundamentioning

confidence: 99%

Section: Algorithm 1 Telegraph Equations With Boundary Conditionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Monte Carlo method for solving the one-dimensional telegraph equations with boundary conditions

Acebrón

Ribeiro

2016

Journal of Computational Physics

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“…In fact, in [1,2], an algorithm for numerically solving linear two-dimensional boundary-value problems for ellip tic PDEs, exploiting the probabilistic representation of solutions, were investigated for the first time. This consists in a hybrid algorithm, which combines the classical domain decomposition method [15], and the probabilistic method, and was called "probabilistic domain decomposition method" (PDD for short).…”

Section: Introductionmentioning

confidence: 99%

A fully scalable algorithm suited for petascale computing and beyond

Acebrón¹,

Rodríguez-Rozas²,

Spigler

2010

Comput Sci Res Dev

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Nowadays supercomputers have already entered in the petascale computing era and peak rate performance is dramatically increasing year after year. However, most of current algorithms are not capable of exploiting fully such a technology due to the well-known parallel programming related issues, such as synchronization, communication and fault tolerance. The aim of this paper is to present a probabilistic domain decomposition algorithm based on generating suitable random trees for solving nonlinear parabolic partial differential equations. These are of paramount importance since many important scientific and engineering problems are modeled by such type of differential equations. We stress that such algorithm is perfectly suited for both current and future high performance supercomputers, showing a remarkable performance and arbitrary scalability.While classical algorithms based on a deterministic domain decomposition exhibits strong limitations when increasing the size of the problem and the number of processors involved, probabilistic methods rather allow us to exploit efficiently massively parallel architectures, being the problem fully decoupled. Large-scale simulations runned on a high performance supercomputer confirm such properties.

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The PDD Method for Solving Linear, Nonlinear, and Fractional PDEs Problems

Rodríguez-Rozas

Acebrón

Spigler

2021

SEMA SIMAI Springer Series

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We review the Probabilistic Domain Decomposition (PDD) method for the numerical solution of linear and nonlinear Partial Differential Equation (PDE) problems. This Domain Decomposition (DD) method is based on a suitable probabilistic representation of the solution given in the form of an expectation which, in turns, involves the solution of a Stochastic Differential Equation (SDE). While the structure of the SDE depends only upon the corresponding PDE, the expectation also depends upon the boundary data of the problem. The method consists of three stages: (i) only few values of the sought solution are solved by Monte Carlo or Quasi-Monte Carlo at some interfaces; (ii) a continuous approximation of the solution over these interfaces is obtained via interpolation; and (iii) prescribing the previous (partial) solutions as additional Dirichlet boundary conditions, a fully decoupled set of sub-problems is finally solved in parallel.For linear parabolic problems, this is based on the celebrated Feynman-Kac formula, while for semilinear parabolic equations requires a suitable generalization based on branching diffusion processes. In case of semilinear transport equations and the Vlasov-Poisson system, a generalization of the probabilistic representation was also obtained in terms of the Method of Characteristics (characteristic curves).Finally, we present the latest progress towards the extension of the PDD method for nonlocal fractional operators.The algorithm notably improves the scalability of classical algorithms and is suited to massively parallel implementation, enjoying arbitrary scalability and fault tolerance properties. Numerical examples conducted in 1D and 2D, including some for the KPP equation and Plasma Physics, are given.

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Probabilistically induced domain decomposition methods for elliptic boundary-value problems

Cited by 18 publications

References 29 publications

A Monte Carlo method for solving the one-dimensional telegraph equations with boundary conditions

A Monte Carlo method for solving the one-dimensional telegraph equations with boundary conditions

A fully scalable algorithm suited for petascale computing and beyond

The PDD Method for Solving Linear, Nonlinear, and Fractional PDEs Problems

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