Quasi-Monte Carlo Methods for some Linear Algebra Problems. Convergence and Complexity

We present quasi-Monte Carlo analogs of Monte Carlo methods for some linear algebra problems: solving systems of linear equations, computing extreme eigenvalues, and matrix inversion. Reformulating the problems as solving integral equations with a special kernels and domains permits us to analyze the quasi-Monte Carlo methods with bounds from numerical integration. Standard Monte Carlo methods for integration 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)). 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 considered Monte Carlo methods. We also analyze the complexity of considered quasi-Monte Carlo algorithms and compare them to the complexity of the analogous Monte Carlo and deterministic algorithms.* This work is supported by the National Science Fund of Bulgaria under Grant No. D002-146/16.12.2008

Statistical Numerical Methods for Eigenvalue Problem. Parallel Implementation

MSC subject classification: 65C05, 65U05.The problem of evaluating the smallest eigenvalue of real symmetric matrices using statistical numerical methods is considered

Towards Effective Bulgarian Competence Centre in High Performance Computing – Service Portfolio and Competences

Three years ago a new European initiative started: establishing a network of Centres of Competence in HPC in the European countries through the project EuroCC (2020-2022) in its second phase EuroCC2 (2023-2025). The goal is to accelerate the improvement of national and thus European capabilities in the area of HPC+ technologies, where HPC+ means High Performance Computing (HPC) and HPC application in HPDA and AI. In order to form an effective National Competence Centre in HPC+ technologies, we performed extensive competence mapping and now devise an extensive service portfolio, open to users from academia, public administration and industry. In this paper we present our analysis of the situation in Bulgaria as well as our approaches to make NCC and effective focus point for those that can benefit from use of HPC+ technology in their research or business

The 14th National Information Day: Open Science, Open Data, Open Access, Bulgarian Open Science Cloud

The paper gives an overview on the current landscape and the activities on national and institutional level regarding Open Science, Open Access to scientific information, Open Data, Bulgarian Open Science Cloud

Solving BVPs using quasirandom walks on the

The "random walk on the boundary" Monte Carlo method has been successfully used for solving boundary-value problems. This method has significant advantages when compared to random walks on spheres, balls or a grid, when solving exterior problems, or when solving a problem at an arbitrary number of points using a single random walk. In this paper we study the properties of the method when we use quasirandom sequences instead of pseudorandom numbers to construct the walks on the boundary. Theoretical estimates of the convergence rate are given and numerical experiments are presented in an attempt to confirm the convergence results. The numerical results show that for "walk on the boundary" quasirandom sequences provide a slight improvement over ordinary Monte Carlo

A Parallel Quasi-Monte Carlo Method for Solving Systems of Linear Equations

Abstract. This paper presents a parallel quasi-Monte Carlo method for solving general sparse systems of linear algebraic equations. In our parallel implementation we use disjoint contiguous blocks of quasirandom numbers extracted from a given quasirandom sequence for each processor. In this case, the increased speed does not come at the cost of less thrust-worthy answers. Similar results have been reported in the quasi-Monte Carlo literature for parallel versions of computing extremal eigenvalues [8] and integrals [9]. But the problem considered here is more complicated- our algorithm not only uses an s−dimensional quasirandom sequence, but also its k−dimensional projections (k =1,2,...,s−1) onto the coordinate axes. We also present numerical results. In these test examples of matrix equations, the martrices are sparse, randomly generated with condition numbers less than 100, so that each corresponding Neumann series is rapidly convergent. Thus we use quasirandom sequences with dimension less than 10.

c ○ VSP 2004 Parallel Quasirandom Walks on the Boundary

used for solving boundary-value problems. This method has significant advantages when compared with random walks on spheres, balls or on discrete grids when an exterior Dirichlet or Neumann problem is solved, or when we are interested in computing the solution to a problem at an arbitrary number of points using a single random walk. In this paper we will investigate ways: • to increase the convergence rate of this method by using quasirandom sequences instead of pseudorandom numbers for the construction of the boundary walks, • to find an efficient parallel implementation of this method on a cluster using MPI. In our parallel implementation we use disjoint contiguous blocks of quasirandom numbers extracted from a given quasirandom sequence for each processor. In this case, the increased convergence rate does not come at the cost of less trustworthy answers. We also present some numerical examples confirming both the increased rate of convergence and the good parallel efficiency of the method.

HP-SEE User Forum 2012

This book is a collection of carefully reviewed papers presented during the HP-SEE User Forum, the meeting of the High-Performance Computing Infrastructure for South East Europe’s (HP-SEE) Research Communities, held in October 17-19, 2012, in Belgrade, Serbia. HP-SEE aims at supporting and integrating regional HPC infrastructures; implementing solutions for HPC in the region; and making HPC resources available to research communities in SEE, region, which are working in a number of scientific fields with specific needs for massively parallel execution on powerful computing resources. HP-SEE brings together research communities and HPC operators from 14 different countries and enables them to share HPC facilities, software, tools, data and research results, thus fostering collaboration and strengthening the regional and national human network; the project specifically supports research groups in the areas of computational physics, computational chemistry and the life sciences. The contributions presented in this book are organized in four main sections: computational physics; computational chemistry; the life sciences; and scientific computing and HPC operations.