Aneta Karaivanova

Parallel resolvent Monte Carlo algorithms for linear algebra problems

In this paper, we consider Monte Carlo (MC) algorithms based on the use of the resolvent matrix for solving linear algebraic problems. Estimates for the speedup and efficiency of the algorithms are presented. Some numerical examples performed on cluster of workstations using MPI are given.

Parallel computations of eigenvalues based on a Monte Carlo approach

This paper presents a parallel Monte Carlo algorithm for evaluating the eigenvalues of matrices. This algorithm is called Resolvent Monte Carlo algorithm (RMC) and uses the resolvent matrix iterations by the Monte Carlo method. The algorithm is suitable for estimating eigenvalues of large sparse symmetric matrices. The work described in this paper has the goal to reduce the computational time when parallel machine is used, and to reach good experimental results for a speed-up and parallel efficiency for large sparse matrices. Numerical tests are performed for a number of test matrices general sparse symmetric, band symmetric matrices and dense matrices on the parallel machine Intel PARAGON. Estimations for the speed-up as well as for the parallel efficiency are obtained. It is shown that the algorithm complexity is practically independent of the size of the matrix.

Error analysis of an adaptive Monte Carlo method for numerical integration

A new adaptive technique for Monte Carlo (MC) integration is proposed and studied. An error analysis is given. It is shown that the error of the numerical integration depends on the smoothness of the integrand. A superconvergent adaptive method is presented. The method combines the idea of separation of the domain into uniformly small subdomains with the Kahn approach of importance sampling. An estimation of the probable error for functions with bounded derivatives is proved. This estimation improves the existing results. A simple adaptive Monte Carlo method is also considered. It is shown that for large dimensions d the convergence of the superconvergent adaptive MC method goes asymptotically to O(n1/2), which corresponds to the convergence of the simple adaptive method. Both adaptive methods – superconvergent and simple – are used for calculating multidimensional integrals. Numerical tests are performed on the supercomputer CRAY Y-MP C92A. It is shown that for low dimensions (up to d=5) the superconvergent adaptive method gives better results than the simple adaptive method. When the dimension increases, the simple adaptive method becomes better. One needs several seconds for evaluating 30-d integrals using the simple adaptive method, while the evaluation of the same integral using Gaussian quadrature will need more than 106 billion years if CRAY Y-MP C92A is used.

Tuning the generation of sobol sequence with owen scrambling

Sobol sequence is the most widely used low discrepancy sequence for numerical solving of multiple integrals and other quasi-Monte Carlo computations Owen first proposed scrambling of this sequence through permutation in a manner that maintained its low discrepancy Scrambling is necessary not only for error analysis but for parallel implementations Good scrambling is especially important for GRID applications However, scrambling is often difficult to implement and time consuming In this paper we propose fast generation of Sobol sequence with Owen scrambling, tuned to specific hardware Numerical and timing results, demonstrating the advantages of our approach are presented and discussed.

Parallel Monte Carlo Algorithms for Sparse SLAE Using MPI

The problem of solving sparse Systems of Linear Algebraic Equations (SLAE) by parallel Monte Carlo numerical methods is considered. The almost optimal Monte Carlo algorithms are presented. In case when a copy of the non-zero matrix elements is sent to each processor the execution time for solving SLAE by Monte Carlo on p processors is bounded by O(nNdT/p) where N is the number of chains, T is the length of the chain in the stochastic process, which are independent of matrix size n, and d is the average number of non-zero elements in the row. Finding a component of the solution vector requires O(NdT/p) time on p processors, which is independent of the matrix size n.

Iterative Monte Carlo Algorithms for Linear Algebra Problems

A common Monte Carlo approach for linear algebra problems is presented. The considered problems are inverting a matrix B, solving systems of linear algebraic equations of the form Bu=b and calculating eigenvalues of symmetric matrices. Several algorithms using the same Markov chains with different random variables are described.

Quasi-Monte Carlo integration on the grid for sensitivity studies

In this paper we present error and performance analysis of quasi-Monte Carlo algorithms for solving multidimensional integrals (up to 100 dimensions) on the grid using MPI. We take into account the fact that the Grid is a potentially heterogeneous computing environment, where the user does not know the specifics of the target architecture. Therefore parallel algorithms should be able to adapt to this heterogeneity, providing automated load-balancing. Monte Carlo algorithms can be tailored to such environments, provided parallel pseudorandom number generators are available. The use of quasi-Monte Carlo algorithms poses more difficulties. In both cases the efficient implementation of the algorithms depends on the functionality of the corresponding packages for generating pseudorandom or quasirandom numbers. We propose efficient parallel implementation of the Sobol sequence for a grid environment and we demonstrate numerical experiments on a heterogeneous grid. To achieve high parallel efficiency we use a newly developed special grid service called Job Track Service which provides efficient management of available computing resources through reservations.

Matrix Computations Using Quasirandom Sequences

The convergence of Monte Carlo method for numerical integration can often be improved by replacing pseudorandom numbers (PRNs) with more uniformly distributed numbers known as quasirandom numbers (QRNs). Standard Monte Carlo methods use pseudorandom sequences and provide a convergence rate of O(N-1/2) using N samples. Quasi-Monte Carlo methods use quasirandom sequences with the resulting convergence rate for numerical integration as good as O((logN)k) N-1).In this paper we study the possibility of using QRNs for computing matrix-vector products, solving systems of linear algebraic equations and calculating the extreme eigenvalues of matrices. Several algorithms using the same Markov chains with different random variables are described. We have shown, theoretically and through numerical tests, that the use of quasirandom sequences improves both the magnitude of the error and the convergence rate of the corresponding Monte Carlo methods. Numerical tests are performed on sparse matrices using PRNs and Sobol, Halton, and Faure QRNs.

Ultra-fast Semiconductor Carrier Transport Simulation on the Grid

We consider the problem of computer simulation of ultra-fast semiconductor carrier transport. The mathematical description of this problem includes quantum kinetic equations whose approximate solving is a computationally very intensive problem. In order to reduce the computational cost we use recently developed Monte Carlo methods as a numerical approach. We study intra-collision field effect, i.e. effective change of phonon energy, which depends on the field direction and the evolution time. In order to obtain results for different evolution times in a reasonable time-frame, we implement simulation on the computational grid. We split the task into thousands of subtasks (jobs) which are sent to different grid sites to be executed. In this paper we present new results for inhomogeneous case in the presence of electric field, and we describe our grid implementation scheme.

Parallel Importance Separation and Adaptive Monte Carlo Algorithms for Multiple Integrals

Monte Carlo Method (MCM) is the only viable method for many high-dimensional problems since its convergence is independent of the dimension. In this paper we develop an adaptive Monte Carlo method based on the ideas and results of the importance separation, a method that combines the idea of separation of the domain into uniformly small subdomains with the Kahn approach of importance sampling. We analyze the error and compare the results with crude Monte Carlo and importance sampling which is the most widely used variance reduction Monte Carlo method. We also propose efficient parallelizations of the importance separation method and the studied adaptive Monte Carlo method. Numerical tests implemented on PowerPC cluster using MPI are provided.