Results 1  10
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13
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 13 (5 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 10 (9 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.
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...
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 3 (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...
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 1 (0 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.
Table of Contents
, 2003
"... Intel products are not intended for use in medical, life saving, life sustaining, critical control or safety systems, or in nuclear facility applications. Intel may make changes to specifications and product descriptions at any time, without notice. ..."
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Intel products are not intended for use in medical, life saving, life sustaining, critical control or safety systems, or in nuclear facility applications. Intel may make changes to specifications and product descriptions at any time, without notice.
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
3.0 Documents Intel Math Kernel Library release 6.1. 07/03 4.0 Documents Intel Math Kernel Library release 7.0 Beta. 11/03 5.0 Documents Intel Math Kernel Library release 7.0 Gold. 04/04 6.0 Documents Intel Math Kernel Library release 7.0.1. 07/04 7.0 Doc
"... The information in this document is subject to change without notice and Intel Corporation assumes no responsibility or liability 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 des ..."
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The information in this document is subject to change without notice and Intel Corporation assumes no responsibility or liability 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 rights is granted by this document. The information in this document is provided in connection with Intel products and should not be construed as a commitment by Intel Corporation.
Stochastic Iterative Method for a System of Parabolic Equations
"... this paper we propose a pure stochastic computational algorithm for numerical calculation of the velocity induced by the vorticity field. The velocity is considered to be a linear functional of the vorticity being a solution to the initial value problem w(x, 0) = w 0 (x), x IR (2) for (1). Supp ..."
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this paper we propose a pure stochastic computational algorithm for numerical calculation of the velocity induced by the vorticity field. The velocity is considered to be a linear functional of the vorticity being a solution to the initial value problem w(x, 0) = w 0 (x), x IR (2) for (1). Suppose that u(x, t) is a given bounded incompressible field, deterministic to start with. Then (1) can be considered as a system of linear parabolic equations. Suppose that u i , w i , #w i , i, j = 1, 2, 3 are bounded and continuous in D = IR (0, T ] for some finite T . With these restrictions, the Cauchy problem (1), (2) is equivalent to the integraldi#erential equation (potential representation) w = # 0 # IR 3 (u #+ #u)Z w 0 Z, Z(x x # , t t # ) being the fundamental solution to the heat equation. Let w be divergence free (remind that w is supposed to be not depending on u). Then the Ostrogradsky formula implies that w satisfies the system of integral equations w i = # 0 # IR (w i u u i w) #Z # IR w 0i Z, i = 1, 2, 3, or w(x, t) = # 0 dt # dx # K(x, t; x # , t # )w(x # , t # ) + F 0 (x, t), (3) where K = {k ij i,j=1 is the matrix of weakly singular kernels. (Note, that for being locally Holder continuous with respect to x uniformly in t (3) is equivalent in C 2,1 to the Cauchy problem (1), (2) [3]). Denote by K the integral operator of this system. Then (see, e.g., [4], [7]) by induction the inequality (K F 0 ) i (x, t) (3A# ) #(1 + n/2) F  L# (D) , i = 1, 2, 3, is valid, A being a constant depending on the upper bound for u