Parallel resolvent Monte Carlo algorithms for linear algebra problems

Dimov, Ivan; Alexandrov, Vassil; Karaivanova, Aneta

doi:10.1016/s0378-4754(00)00243-3

Cited by 34 publications

(34 citation statements)

References 22 publications

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“…Then, if required, obtaining the solution vector is found by x = A −1 b. Following the Monte Carlo method described in [7,11] we define a Markov chain and weights…”

Section: Monte Carlo and Matrix Computationmentioning

confidence: 99%

“…In general, we follow the approach described in [7,11] We can then find the solution vector through A The efficiency of inverting diagonally dominant matrices is an important part of the process enabling MC to be applied to diagonally dominant and some general matrices. Consider now the algorithm which can be used for the inversion of a general non-singular matrix A.…”

Section: The Hybrid MC Algorithmmentioning

confidence: 99%

“…Several authors have proposed different coarse grained Monte Carlo parallel algorithms for MI and SLAE [7,8,9,10,11]. In this paper, we investigate how Monte Carlo can be used for diagonally dominant and some general matrices via a general splitting and how efficient mixed (stochastic/deterministic) parallel algorithms can be derived for obtaining an accurate inversion of a given nonsingular matrix A.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we investigate how Monte Carlo can be used for diagonally dominant and some general matrices via a general splitting and how efficient mixed (stochastic/deterministic) parallel algorithms can be derived for obtaining an accurate inversion of a given nonsingular matrix A. We employ either uniform Monte Carlo (UM) or almost optimal Monte Carlo (MAO) methods [7,8,9,10,11]. The relevant experiments with dense and sparse matrices are carried out.…”

Section: Introductionmentioning

confidence: 99%

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Parallel Hybrid Monte Carlo Algorithms for Matrix Computations

Alexandrov

Atanassov

Dimov

et al. 2005

Lecture Notes in Computer Science

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Abstract. In this paper we consider hybrid (fast stochastic approximation and deterministic refinement) algorithms for Matrix Inversion (MI) and Solving Systems of Linear Equations (SLAE). Monte Carlo methods are used for the stochastic approximation, since it is known that they are very efficient in finding a quick rough approximation of the element or a row of the inverse matrix or finding a component of the solution vector. We show how the stochastic approximation of the MI can be combined with a deterministic refinement procedure to obtain MI with the required precision and further solve the SLAE using MI. We employ a splitting A = D − C of a given non-singular matrix A, where D is a diagonal dominant matrix and matrix C is a diagonal matrix. In our algorithm for solving SLAE and MI different choices of D can be considered in order to control the norm of matrix T = D −1 C, of the resulting SLAE and to minimize the number of the Markov Chains required to reach given precision. Further we run the algorithms on a mini-Grid and investigate their efficiency depending on the granularity. Corresponding experimental results are presented.

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“…Then, if required, obtaining the solution vector is found by x = A −1 b. Following the Monte Carlo method described in [7,11] we define a Markov chain and weights…”

Section: Monte Carlo and Matrix Computationmentioning

confidence: 99%

Section: The Hybrid MC Algorithmmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Parallel Hybrid Monte Carlo Algorithms for Matrix Computations

Alexandrov

Atanassov

Dimov

et al. 2005

Lecture Notes in Computer Science

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“…Further, in the next table we present a comparison of the efficiency of using Monte Carlo algorithms [1,2] for solving subproblems (e.g. linear systems arising after discretization) in the model.…”

Section: Description Of the Grid Of Sun Computersmentioning

confidence: 99%

Using Parallel Monte Carlo Methods in Large-Scale Air Pollution Modelling

Alexandrov

Zlatev²

2004

Computational Science - ICCS 2004

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Abstract. Large-scale air pollution models can successfully be used in different environmental studies. These models are described mathematically by systems of partial differential equations. Splitting procedures followed by discretization of the spatial derivatives lead to several large systems of ordinary differential equations of order up to 80 millions. These systems have to be handled numerically at up to 250 000 timesteps. Furthermore, many scenarios are often to be run in order to study the dependence of the model results on the variation of some key parameters (as, for example, the emissions). Such huge computational tasks can successfully be treated only if (i) fast and sufficiently accurate numerical methods are used and (ii) the models can efficiently be run on parallel computers. Efficient Monte Carlo methods for some subproblems will be presented and applications of the model in the solution of some environmental tasks will also be made.

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Analysis of Monte Carlo accelerated iterative methods for sparse linear systems

Benzi

Evans

Hamilton

et al. 2017

Numerical Linear Algebra App

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Summary We consider hybrid deterministic‐stochastic iterative algorithms for the solution of large, sparse linear systems. Starting from a convergent splitting of the coefficient matrix, we analyze various types of Monte Carlo acceleration schemes applied to the original preconditioned Richardson (stationary) iteration. These methods are expected to have considerable potential for resiliency to faults when implemented on massively parallel machines. We establish sufficient conditions for the convergence of the hybrid schemes, and we investigate different types of preconditioners including sparse approximate inverses. Numerical experiments on linear systems arising from the discretization of partial differential equations are presented.

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Parallel resolvent Monte Carlo algorithms for linear algebra problems

Cited by 34 publications

References 22 publications

Parallel Hybrid Monte Carlo Algorithms for Matrix Computations

Parallel Hybrid Monte Carlo Algorithms for Matrix Computations

Using Parallel Monte Carlo Methods in Large-Scale Air Pollution Modelling

Analysis of Monte Carlo accelerated iterative methods for sparse linear systems

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