Juan A. Acebrón

Domain Decomposition Solution of Elliptic Boundary-Value Problems via Monte Carlo and Quasi-Monte Carlo Methods

Domain decomposition of two-dimensional domains on which boundary-value elliptic problems are formulated is accomplished by probabilistic (Monte Carlo) as well as by quasi-Monte Carlo methods, generating only a few interfacial values and interpolating on them. Continuous approximations for the trace of solution are thus obtained, to be used as boundary data for the subproblems. The numerical treatment can then proceed by standard deterministic algorithms, separately in each of the so obtained subdomains. Monte Carlo and quasi-Monte Carlo simulations may naturally exploit multiprocessor architectures, leading to parallel computing, as well as the ensuing domain decomposition does. The advantages such as scalability obtained by increasing the number of processors are shown, both theoretically and experimentally, in a number of test examples, and the possibility of using clusters of computers (grid computing) is emphasized.

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

A domain decomposition method is developed for the numerical solution of nonlinear parabolic partial differential equations in any space dimension, based on the probabilistic representation of solutions as an average of suitable multiplicative functionals. Such a direct probabilistic representation requires generating a number of random trees, whose role is that of the realizations of stochastic processes used in the linear problems. First, only few values of the sought solution inside the space-time domain are computed (by a Monte Carlo method on the trees). An interpolation is then carried out, in order to approximate interfacial values of the solution inside the domain. Thus, a fully decoupled set of sub-problems is obtained. The algorithm is suited to massively parallel implementation, enjoying arbitrary scalability and fault tolerance properties. Pruning the trees is shown to increase appreciably the efficiency of the algorithm. Numerical examples conducted in 2D, including some for the KPP equation, are given.

Efficient Parallel Solution of Nonlinear Parabolic Partial Differential Equations by a Probabilistic Domain Decomposition

Initial- and initial-boundary value problems for nonlinear one-dimensional parabolic partial differential equations are solved numerically by a probabilistic domain decomposition method. This is based on a probabilistic representation of solutions by means of branching stochastic processes. Only few values of the solution inside the space-time domain are generated by a Monte Carlo method, and an interpolation is then made so to approximate suitable interfacial values of the solution inside the domain. In this way, a fully decoupled set of sub-problems is obtained. This method allows for an efficient massively parallel implementation, is scalable and fault tolerant. Numerical examples, including some for the KPP equation and beyond are given to show the performance of the algorithm.

A new parallel solver suited for arbitrary semilinear parabolic partial differential equations based on generalized random trees

A probabilistic representation for initial value semilinear parabolic problems based on generalized random trees has been derived. Two different strategies have been proposed, both requiring generating suitable random trees combined with a Pade approximant for approximating accurately a given divergent series. Such series are obtained by summing the partial contribution to the solution coming from trees with arbitrary number of branches. The new representation greatly expands the class of problems amenable to be solved probabilistically, and was used successfully to develop a generalized probabilistic domain decomposition method. Such a method has been shown to be suited for massively parallel computers, enjoying full scalability and fault tolerance. Finally, a few numerical examples are given to illustrate the remarkable performance of the algorithm, comparing the results with those obtained with a classical method.

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

A 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 Kacs 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 difference-based 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.

A Stochastic Algorithm Based on Fast Marching for Automatic Capacitance Extraction in Non-Manhattan Geometries ∗

We present an algorithm for two- and three-dimensional capacitance analysis on multidielectric integrated circuits of arbitrary geometry. Our algorithm is stochastic in nature and as such fully parallelizable. It is intended to extract capacitance entries directly from a pixelized representation of the integrated circuit (IC), which can be produced from a scanning electron microscopy image. Preprocessing and monitoring of the capacitance calculation are kept to a minimum, thanks to the use of distance maps automatically generated with a fast marching technique. Numerical validation of the algorithm shows that the systematic error of the algorithm decreases with better resolution of the input image. Those features render the presented algorithm well suited for fast prototyping while using the most realistic IC geometry data.

Highly efficient numerical algorithm based on random trees for accelerating parallel Vlasov-Poisson simulations

An efficient numerical method based on a probabilistic representation for the Vlasov-Poisson system of equations in the Fourier space has been derived. This has been done theoretically for arbitrary dimensional problems, and particularized to unidimensional problems for numerical purposes. Such a representation has been validated theoretically in the linear regime comparing the solution obtained with the classical results of the linear Landau damping.The numerical strategy followed requires generating suitable random trees combined with a Pade approximant for approximating accurately a given divergent series. Such series are obtained by summing the partial contributions to the solution coming from trees with arbitrary number of branches. These contributions, coming in general from multi-dimensional definite integrals, are efficiently computed by a quasi-Monte Carlo method. It is shown how the accuracy of the method can be effectively increased by considering more terms of the series.The new representation was used successfully to develop a Probabilistic Domain Decomposition method suited for massively parallel computers, which improves the scalability found in classical methods. Finally, a few numerical examples based on classical phenomena such as the non-linear Landau damping, and the two streaming instability are given, illustrating the remarkable performance of the algorithm, when compared the results with those obtained using a classical method.

A fully scalable parallel algorithm for solving elliptic partial differential equations

A comparison is made between the probabilistic domain decomposition (DD) method and a certain deterministic DD method for solving linear elliptic boundary-value problems. Since in the deterministic approach the CPU time is affected by intercommunications among the processors, it turns out that the probabilistic method performs better, especially when the number of subdomains (hence, of processors) is increased. This fact is clearly illustrated by some examples. The probabilistic DD algorithm has been implemented in an MPI environment, in order to exploit distributed computer architectures. Scalability and fault-tolerance of the probabilistic DD algorithm are emphasized.

A fully scalable algorithm suited for petascale computing and beyond

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.

A new domain decomposition approach suited for grid computing

In this paper, we describe a new kind of domain decomposition strategy for solving linear elliptic boundary-value problems. It outperforms the traditional ones in complex and heterogeneous networks like those for grid computing. Such a strategy consists of a hybrid numerical scheme based on a probabilistic method along with a domain decomposition, and full decoupling can be accomplished. While the deterministic approach is strongly affected by intercommunication among the hosts, the probabilistic method is scalable as the number of subdomains, i.e., the number of processors involved, increases. This fact is clearly illustrated by an example, even operating in a grid environment.