Parallel Hybrid Monte Carlo Algorithms for Matrix Computations

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 –1C, 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.

Parallel Monte Carlo algorithms for information retrieval

In any data mining applications, automated text and text and image retrieval of information is needed. This becomes essential with the growth of the Internet and digital libraries. Our approach is based on the latent semantic indexing (LSI) and the corresponding term-by-document matrix suggested by Berry and his co-authors. Instead of using deterministic methods to find the required number of first "k" singular triplets, we propose a stochastic approach. First, we use Monte Carlo method to sample and to build much smaller size term-by-document matrix (e.g. we build k x k matrix) from where we then find the first "k" triplets using standard deterministic methods. Second, we investigate how we can reduce the problem to finding the "k"-largest eigenvalues using parallel Monte Carlo methods. We apply these methods to the initial matrix and also to the reduced one. The algorithms are running on a cluster of workstations under MPI and results of the experiments arising in textual retrieval of Web documents as well as comparison of the stochastic methods proposed are presented. (C) 2003 IMACS. Published by Elsevier Science B.V. All rights reserved.

Monte Carlo methods for matrix computations on the grid

Many scientific and engineering applications involve inverting large matrices or solving systems of linear algebraic equations. Solving these problems with proven algorithms for direct methods can take very long to compute, as they depend on the size of the matrix. The computational complexity of the stochastic Monte Carlo methods depends only on the number of chains and the length of those chains. The computing power needed by inherently parallel Monte Carlo methods can be satisfied very efficiently by distributed computing technologies such as Grid computing. In this paper we show how a load balanced Monte Carlo method for computing the inverse of a dense matrix can be constructed, show how the method can be implemented on the Grid, and demonstrate how efficiently the method scales on multiple processors. (C) 2007 Elsevier B.V. All rights reserved.

A sparse parallel hybrid Monte Carlo algorithm for matrix computations

In this paper we introduce a new algorithm, based on the successful work of Fathi and Alexandrov, on hybrid Monte Carlo algorithms for matrix inversion and solving systems of linear algebraic equations. This algorithm consists of two parts, approximate inversion by Monte Carlo and iterative refinement using a deterministic method. Here we present a parallel hybrid Monte Carlo algorithm, which uses Monte Carlo to generate an approximate inverse and that improves the accuracy of the inverse with an iterative refinement. The new algorithm is applied efficiently to sparse non-singular matrices. When we are solving a system of linear algebraic equations, Bx = b, the inverse matrix is used to compute the solution vector x = B(-1)b. We present results that show the efficiency of the parallel hybrid Monte Carlo algorithm in the case of sparse matrices.

Computational complexity of weighted splitting schemes on parallel computers

In models of complicated physical-chemical processes operator splitting is very often applied in order to achieve sufficient accuracy as well as efficiency of the numerical solution. The recently rediscovered weighted splitting schemes have the great advantage of being parallelizable on operator level, which allows us to reduce the computational time if parallel computers are used. In this paper, the computational times needed for the weighted splitting methods are studied in comparison with the sequential (S) splitting and the Marchuk-Strang (MSt) splitting and are illustrated by numerical experiments performed by use of simplified versions of the Danish Eulerian model (DEM).

Error analysis of a Monte Carlo algorithm for computing bilinear forms of matrix powers

In this paper we present error analysis for a Monte Carlo algorithm for evaluating bilinear forms of matrix powers. An almost Optimal Monte Carlo (MAO) algorithm for solving this problem is formulated. Results for the structure of the probability error are presented and the construction of robust and interpolation Monte Carlo algorithms are discussed. Results are presented comparing the performance of the Monte Carlo algorithm with that of a corresponding deterministic algorithm. The two algorithms are tested on a well balanced matrix and then the effects of perturbing this matrix, by small and large amounts, is studied.

Collaborative virtual environment for advanced computing

Synchronous collaborative systems allow geographically distributed participants to form a virtual work environment enabling cooperation between peers and enriching the human interaction. The technology facilitating this interaction has been studied for several years and various solutions can be found at present. In this paper, we discuss our experiences with one such widely adopted technology, namely the Access Grid. We describe our experiences with using this technology, identify key problem areas and propose our solution to tackle these issues appropriately. Moreover, we propose the integration of Access Grid with an Application Sharing tool, developed by the authors. Our approach allows these integrated tools to utilise the enhanced features provided by our underlying dynamic transport layer.

Parallel implementation and one year experiments with the Danish Eulerian Model

Large scale air pollution models are powerful tools, designed to meet the increasing demand in different environmental studies. The atmosphere is the most dynamic component of the environment, where the pollutants can be moved quickly on far distnce. Therefore the air pollution modeling must be done in a large computational domain. Moreover, all relevant physical, chemical and photochemical processes must be taken into account. In such complex models operator splitting is very often applied in order to achieve sufficient accuracy as well as efficiency of the numerical solution. The Danish Eulerian Model (DEM) is one of the most advanced such models. Its space domain (4800 × 4800 km) covers Europe, most of the Mediterian and neighboring parts of Asia and the Atlantic Ocean. Efficient parallelization is crucial for the performance and practical capabilities of this huge computational model. Different splitting schemes, based on the main processes mentioned above, have been implemented and tested with respect to accuracy and performance in the new version of DEM. Some numerical results of these experiments are presented in this paper.

Nested parallelism for multi-core HPC systems using Java

Since its introduction in 1993, the Message Passing Interface (MPI) has become a de facto standard for writing High Performance Computing (HPC) applications on clusters and Massively Parallel Processors (MPPs). The recent emergence of multi-core processor systems presents a new challenge for established parallel programming paradigms, including those based on MPI. This paper presents a new Java messaging system called MPJ Express. Using this system, we exploit multiple levels of parallelism - messaging and threading - to improve application performance on multi-core processors. We refer to our approach as nested parallelism. This MPI-like Java library can support nested parallelism by using Java or Java OpenMP (JOMP) threads within an MPJ Express process. Practicality of this approach is assessed by porting to Java a massively parallel structure formation code from Cosmology called Gadget-2. We introduce nested parallelism in the Java version of the simulation code and report good speed-ups. To the best of our knowledge it is the first time this kind of hybrid parallelism is demonstrated in a high performance Java application. (C) 2009 Elsevier Inc. All rights reserved.