Domain decomposition solution of nonlinear two-dimensional parabolic problems by random trees

Acebrón, Juan A.; Rodríguez-Rozas, Ángel; Spigler, Renato

doi:10.1016/j.jcp.2009.04.034

Cited by 24 publications

(40 citation statements)

References 26 publications

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“…means thatũ(x) is a PDD approximation obtained with tolerance a and no variance reduction; while [ũ 0 (a), ξ 0 (ũ 1 )] = IterPDD(a 0 , a 1 ), or simply IterPDD(a 0 , a 1 ) (4) means that firstũ 1 (a 1 ) = PlainPDD(a 1 ) is calculated without variance reduction, then differentiated in order to construct ξ according to (2), which in turn is used as control variate in order to reduce the variance in calculatingũ 0 with a target tolerance a 0 , which is the ultimate goal. Because the nodal values ofũ 0 can now be calculated with much less variance, statistical errors are smaller, and the time (or cost) it takes the computer to hit the tolerance a 0 is also less.…”

Section: Definitionmentioning

confidence: 99%

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A multigrid-like algorithm for probabilistic domain decomposition

Bernal

Acebrón

2016

Computers & Mathematics with Applications

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We present an iterative scheme, reminiscent of the Multigrid method, to solve large boundary value problems with Probabilistic Domain Decomposition (PDD). In it, increasingly accurate approximations to the solution are used as control variates in order to reduce the Monte Carlo error of the following iterates-resulting in an overall acceleration of PDD for a given error tolerance. The key ingredient of the proposed algorithm is the ability to approximately predict the speedup with little computational overhead and in parallel. Besides, the theoretical framework allows to explore other aspects of PDD, such as stability. One numerical example is worked out, yielding an improvement of between one and two orders of magnitude over the previous version of PDD.

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Section: Definitionmentioning

confidence: 99%

“…An alternative to deterministic methods which is specifically designed to circumvent the scalability issue is the probabilistic domain decomposition (PDD) method, which has been successfully applied to elliptic [1] and parabolic BVPs [2] [3]. PDD consists of two stages.…”

mentioning

confidence: 99%

A multigrid-like algorithm for probabilistic domain decomposition

Bernal

Acebrón

2016

Computers & Mathematics with Applications

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“…For more mathematical details, we refer to the reader to [3] and [4]. In practice, the probabilistic representation in (11) works as follows: A stochastic process initially located in (x, 0) (root of the tree) is generated and evolves in time.…”

Section: Probabilistic Representation Of Nonlinear Pdesmentioning

confidence: 99%

“…In practice, to achieve a reasonable small error in (11), (12), a large number of random trees should be generated. More details on the numerical errors related with the probabilistic representation are found in [3] and [4]. For the purpose of illustration, in Fig.…”

Section: Probabilistic Representation Of Nonlinear Pdesmentioning

confidence: 99%

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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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Domain decomposition solution of nonlinear two-dimensional parabolic problems by random trees

Cited by 24 publications

References 26 publications

A multigrid-like algorithm for probabilistic domain decomposition

A multigrid-like algorithm for probabilistic domain decomposition

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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