Results 1  10
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17
Improving Regression Estimation: Averaging Methods for Variance Reduction with Extensions to General Convex Measure Optimization
, 1993
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Monte Carlo complexity of global solution of integral equations
 J. COMPLEXITY
, 1998
"... The problem of global solution of Fredholm integral equations is studied. This means that one seeks to approximate the full solution function (as opposed to the local problem, where only the value of the solution in a single point or a functional of the solution is sought). The Monte Carlo complexit ..."
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Cited by 43 (7 self)
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The problem of global solution of Fredholm integral equations is studied. This means that one seeks to approximate the full solution function (as opposed to the local problem, where only the value of the solution in a single point or a functional of the solution is sought). The Monte Carlo complexity is analyzed, i. e. the complexity of stochastic solution of this problem. The framework for this analysis is provided by informationbased complexity theory. The investigations complement previous ones on stochastic complexity of local solution and on deterministic complexity of both local and global solution. The results show that even in the global case Monte Carlo algorithms can perform better than deterministic ones, although the difference is not as large as in the local case.
Monte Carlo approximation of weakly singular integral operators
"... We study the randomized approximation of weakly singular integral operators. For a suitable class of kernels having a standard type of singularity and being otherwise of finite smoothness, we develop a Monte Carlo multilevel method, give convergence estimates and prove lower bounds which show the op ..."
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Cited by 17 (13 self)
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We study the randomized approximation of weakly singular integral operators. For a suitable class of kernels having a standard type of singularity and being otherwise of finite smoothness, we develop a Monte Carlo multilevel method, give convergence estimates and prove lower bounds which show the optimality of this method and establish the complexity. As an application we obtain optimal methods for and the complexity of randomized solution of the Poisson equation in simple domains, when the solution is sought on subdomains of arbitrary dimension.
QuasiMonte Carlo Methods for Integral Equations
"... In this paper, we establish a deterministic error bound for estimating a functional of the solution of the integral transport equation via random walks that improves an earlier result of Chelson generalizing the KoksmaHlawka inequality for finite dimensional quadrature. We solve such problems b ..."
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Cited by 5 (0 self)
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In this paper, we establish a deterministic error bound for estimating a functional of the solution of the integral transport equation via random walks that improves an earlier result of Chelson generalizing the KoksmaHlawka inequality for finite dimensional quadrature. We solve such problems by simulation, using sequences that combine pseudorandom and quasirandom elements in the construction of the random walks in order to take advantage of the superior uniformity properties of quasirandom numbers and the statistical (independence) properties of pseudorandom numbers. We discuss implementation issues that arise when these hybrid sequences are used in practice. The quasiMonte Carlo techniques described in this paper have the potential to improve upon the convergence rates of both (conventional) Monte Carlo and quasiMonte Carlo simulations in many problems. Recent model problem computations confirm these improved convergence properties. 1 Introduction In the past twenty...
Approximation of Transport Equations by Matrix Equations and Sequential Sampling
 Monte Carlo Methods and Appl
, 1997
"... In this paper, we study discretizations of the integral transport equation obtained by restricting the kernel to a nite set of points in phase space. The resulting matrix equation may then be solved adaptively to obtain an approximation to the solution of the original problem. The technique may be v ..."
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Cited by 4 (4 self)
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In this paper, we study discretizations of the integral transport equation obtained by restricting the kernel to a nite set of points in phase space. The resulting matrix equation may then be solved adaptively to obtain an approximation to the solution of the original problem. The technique may be viewed as a variant of Nystom's quadrature method [1], which is normally applied by using a regular grid as the set of nodal points. We prove a theorem that establishes the error resulting from such approximations. This theorem suggests that low discrepancy nodal sets will produce smaller errors than cartesian product and other grid choices, especially when the dimension of the phase space is moderate or high. Application of Monte Carlo simulation to the matrix problem (making use of discrete random walks) has been carried out in a companion paper [2] and an improved version of this technique is applied here. Numerical results indicate the potential of this new technique for solving integral...
Parallel resolvent Monte Carlo algorithms for linear algebra problems
, 2005
"... 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. ..."
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Cited by 4 (1 self)
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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.
Implementation of Monte Carlo Algorithms for Eigenvalue Problem Using MPI
"... Abstract. The problem of evaluating the dominant eigenvalue of real matrices using Monte Carlo numerical methods is considered. Three almost optimal Monte Carlo algorithms are presented: { Direct Monte Carlo algorithm (DMC) for calculating the largest eigenvalue of a matrix A. The algorithm uses it ..."
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Cited by 1 (0 self)
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Abstract. The problem of evaluating the dominant eigenvalue of real matrices using Monte Carlo numerical methods is considered. Three almost optimal Monte Carlo algorithms are presented: { Direct Monte Carlo algorithm (DMC) for calculating the largest eigenvalue of a matrix A. The algorithm uses iterations with the given matrix. { Resolvent Monte Carlo algorithm (RMC) for calculating the smallest or the largest eigenvalue. The algorithm uses Monte Carlo iterations with the resolvent matrix and includes parameter controlling the rate of convergence; { Inverse Monte Carlo algorithm (IMC) for calculating the smallest eigenvalue. The algorithm uses iterations with inverse matrix. Numerical tests are performed for a number of large sparse test matrices using MPI on a cluster of workstations. 1
ARTICLE IN PRESS Available online at www.sciencedirect.com Mathematics and Computers in Simulation xxx (2009) xxx–xxx
"... Monte Carlo (MC) linear solvers can be considered stochastic realizations of deterministic stationary iterative processes. That is, they estimate the result of a stationary iterative technique for solving linear systems. There are typically two sources of errors: (i) those from the underlying determ ..."
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Monte Carlo (MC) linear solvers can be considered stochastic realizations of deterministic stationary iterative processes. That is, they estimate the result of a stationary iterative technique for solving linear systems. There are typically two sources of errors: (i) those from the underlying deterministic iterative process and (ii) those from the MC process that performs the estimation. Much progress has been made in reducing the stochastic errors of the MC process. However, MC linear solvers suffer from the drawback that, due to efficiency considerations, they are usually stochastic realizations of the Jacobi method (a diagonal splitting), which has poor convergence properties. This has limited the application of MC linear solvers. The main goal of this paper is to show that efficient MC implementations of nondiagonal splittings too are feasible, by constructing efficient implementations for one such splitting. As a secondary objective, we also derive conditions under which this scheme can perform better than MC Jacobi, and demonstrate this experimentally. The significance of this work lies in proposing an approach that can lead to efficient MC implementations of a wider variety of deterministic iterative processes. © 2009 IMACS. Published by Elsevier B.V. All rights reserved.
Variance reduction by means of deterministic computation: Collision estimate
, 1996
"... We study the collision estimate of Monte Carlo methods for the solution of integral equations. A new variance technique is proposed and analyzed. It consists in the separation of the main part by constructing a neighboring equation based on deterministic numerical methods. ..."
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We study the collision estimate of Monte Carlo methods for the solution of integral equations. A new variance technique is proposed and analyzed. It consists in the separation of the main part by constructing a neighboring equation based on deterministic numerical methods.
SALE AND/OR USE OF INTEL PRODUCTS INCLUDING LIABILITY OR WARRANTIES RELATING TO FITNESS FOR
, 2005
"... for any errors or inaccuracies that may appear in this document or any software that may be provided in association with this document. This document and the software described in it are furnished under license and may only be used or copied in accordance with the terms of the license. No license, e ..."
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for any errors or inaccuracies that may appear in this document or any software that may be provided in association with this document. This document and the software described in it are furnished under license and may only be used or copied in accordance with the terms of the license. No license, express or implied, by estoppel or otherwise, to any intellectual property