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Geometric Variance Reduction in Markov chains. Application to Value Function and Gradient Estimation
 Journal of Machine Learning Research
, 2006
"... We study a sequential variance reduction technique for Monte Carlo estimation of functionals in Markov Chains. The method is based on designing sequential control variates using successive approximations of the function of interest V. Regular Monte Carlo estimates have a variance of O(1=N), where N ..."
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We study a sequential variance reduction technique for Monte Carlo estimation of functionals in Markov Chains. The method is based on designing sequential control variates using successive approximations of the function of interest V. Regular Monte Carlo estimates have a variance of O(1=N), where N is the number of samples. Here, we obtain a geometric variance reduction O(N) (with < 1) up to a threshold that depends on the approximation error V AV, where A is an approximation operator linear in the values. Thus, if V belongs to the right approximation space (i.e. AV = V), the variance decreases geometrically to zero. An immediate application is value function estimation in Markov chains, which may be used for policy evaluation in policy iteration for Markov Decision Processes. Another important domain, for which variance reduction is highly needed, is gradient estimation, that is computing the sensitivity @V of the performance measure V with respect to some parameter of the transition probabilities. For example, in parametric optimization of the policy, an estimate of the policy gradient is required to perform a gradient optimization method. We show that, using two approximations, the value function and the gradient, a geometric variance reduction is also achieved, up to a threshold that depends on the approximation errors of both of those representations.
Reuse of paths in light source animation
 In Proceedings of Computer Graphics International 2004 (CGI ’04
, 2004
"... In this paper we extend the reuse of paths to the shot from a moving light source. In the classical algorithm new paths have to be cast from each new position of a light source. We show that we can reuse all paths for all positions, obtaining in this way a theoretical maximum speedup equal to the a ..."
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In this paper we extend the reuse of paths to the shot from a moving light source. In the classical algorithm new paths have to be cast from each new position of a light source. We show that we can reuse all paths for all positions, obtaining in this way a theoretical maximum speedup equal to the average length of the shooting path. 1.
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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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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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.
Applications of Monte Carlo/QuasiMonte Carlo Methods in Finance: Option Pricing
 Proceedings of a conference held at the Claremont Graduate Univ
, 1998
"... . The pricing of options is a very important problem encountered in financial markets today. The famous BlackScholes model provides explicit closed form solutions for the values of certain (European style) call and put options. But for many other options, either there are no closed form solutions, ..."
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. The pricing of options is a very important problem encountered in financial markets today. The famous BlackScholes model provides explicit closed form solutions for the values of certain (European style) call and put options. But for many other options, either there are no closed form solutions, or if such closed form solutions exist, the formulas exhibiting them are complicated and difficult to evaluate accurately by conventional methods. In this case, Monte Carlo methods may prove to be valuable. In this paper, we illustrate two separate applications of Monte Carlo and/or quasiMonte Carlo methods to the pricing of options: first, the method is used to estimate multiple integrals related to the evaluation of European style options; second, an adaptive Monte Carlo method is applied to a finite difference approximation of a partial differential equation formulation of a class of finance problems. Some of the advantages in using the Monte Carlo method for such problems are discussed....
A Parallel QuasiMonte Carlo Method for Solving Systems of Linear Equations
"... Abstract. This paper presents a parallel quasiMonte 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, th ..."
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Abstract. This paper presents a parallel quasiMonte 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 thrustworthy answers. Similar results have been reported in the quasiMonte 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. 1
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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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
c © de Gruyter Fast GPUbased reuse of paths in radiosity
"... Abstract. We present in this paper a GPUbased strategy that allows a fast reuse of paths in the context of shooting random walk applied to radiosity. Given an environment with diffuse surfaces, we aim at computing a basis of n radiosity solutions, corresponding to n lightsource positions. Thanks t ..."
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Abstract. We present in this paper a GPUbased strategy that allows a fast reuse of paths in the context of shooting random walk applied to radiosity. Given an environment with diffuse surfaces, we aim at computing a basis of n radiosity solutions, corresponding to n lightsource positions. Thanks to the reuse, paths originated at each of the positions are used to also distribute power from every other position. The visibility computations needed to make possible the reuse of paths are drastically accelerated using graphic hardware, resulting in a theoretical speedup factor of n with regard to the computation of the independent solutions. Our contribution has application to the fields of interior design, animation, and videogames.
Improving the Quality of Aggregation using data
"... (Thèse de doctorat en cotutelle) to obtain the DOCTORAT de l'UNIVERSITE DE LILLE 1 and ..."
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(Thèse de doctorat en cotutelle) to obtain the DOCTORAT de l'UNIVERSITE DE LILLE 1 and
UNC is an Equal Opportunity / Affirmative Action Institution. ON ACCELERATING MONTE CARLO TECHNIQUES FOR SOLVING LARGE SYSTEMS OF EQUATIONS
"... This paper is concerned with ways of incorporating current Monte Carlo techniques for solving large linear systems [hereinafter referredto as "plain Monte Carlo"PMC] in accelerative schemes and other numerical techniques for more rapidly solving both linear and nonlinear systems. ..."
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This paper is concerned with ways of incorporating current Monte Carlo techniques for solving large linear systems [hereinafter referredto as "plain Monte Carlo"PMC] in accelerative schemes and other numerical techniques for more rapidly solving both linear and nonlinear systems.