Christian Lécot

Quasi-Monte Carlo Methods for Estimating Transient Measures of Discrete Time Markov Chains

We describe a new method for the transient simulation of discrete time Markov chains. It is a quasi-Monte Carlo method where different paths are simulated in parallel, but reordered at each step. We prove the convergence of the method, when the number of simulated paths increases. Using some numerical experiments, we illustrate that the error of the new algorithm is smaller than the error of standard Monte Carlo algorithms. Finally, we propose to analyze continuous time Markov chains by transforming them into a discrete time problem by using the uniformization technique.

A Quasi-Monte Carlo Scheme Using Nets for a Linear Boltzmann Equation

A quasi-Monte Carlo particle simulation for solving a linear Boltzmann equation is constructed and a convergence proof is given. The analysis is restricted to the three-dimensional equation in the space homogeneous case and the velocity domain is normalized to be I 3 = [0,1)3. A particle simulation is described. It combines a Euler scheme in time with quasi-Monte Carlo integration in velocity space. The quadratures use (0,m,s)-nets, which are sets with a very regular distribution behavior. The particles are reordered according to the components of their velocity at each time step. A deterministic error bound is obtained. Finally, numerical experiments are presented for some test problems where an exact solution is known. The accuracy of the quasi-Monte Carlo scheme is compared with a standard Monte Carlo algorithm. The results show an improvement in both magnitude of error and convergence rate of quasi-Monte Carlo over Monte Carlo approach.

A quasi-Monte Carlo method for the Boltzmann equation

A new quasi-Monte Carlo method for solving the Boltzmann equation in a simplified case is described. The analysis is restricted to a spatially homogeneous and isotropic gas; in addition, the molecular model only involves isotropic scattering. The scheme makes use of particles and combines an Euler scheme with numerical integrations. The sequence which is used for the quadratures must possess some symmetry properties which prescribe energy conservation for colliding particles. The error of the method is estimated by means of the discrepancy of the sequence which performs the quadratures. An algorithm for generating convenient sequences is proposed. In an example, where an exact solution is known, the computation of effective errors is included. INTRODUCTION Rarefied gas flows are usually simulated by Monte Carlo techniques. Besides the successful Direct Simulation Monte Carlo (DSMC) method of Bird [2], another scheme was derived by Nanbu [ 14] from the Boltzmann equation itself. A drawback of both schemes are numerical fluctuations caused by the use of pseudorandom numbers. An improved Monte Carlo scheme, which reduces fluctuations, has recently been developed at the University of Kaiserslautern [1]: it will be referred to as the KMC scheme. A fully deterministic method for solving the Boltzmann equation is proposed here. The constraints on the analysis are the following. We consider an infinite spatially homogeneous and isotropic gas (the velocity distribution is radially symmetric). The molecular model is characterized by isotropic scattering (the differential cross section a depends only on the relative speed g and not on the deflection angle). In addition, we assume that ga(g) is some nonnegative, nondecreasing, and bounded function. The hypothesis on the cross section allows physically relevant models, such as the VHS model of Bird [3], when a cutoff is used. In earlier communications [IO, I I] the simplest choice (ga(g) equals a constant) was considered. A deterministic version of the scheme of Nanbu, that we called the Low Discrepancy (LD) method, was described. An error analysis was proposed and it was shown that the LD method outperforms the original scheme of Nanbu in a test case where an exact solution is known. Received March 1, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 65M15; Secondary 65C05, llK38. @1991 American Mathematical Society 0025-5718/91

Randomized Quasi-Monte Carlo Simulation of Markov Chains with an Ordered State Space

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Quasi-Monte Carlo simulation of diffusion

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Low discrepancy sequences for solving the Boltzmann equation

We study a randomized quasi-Monte Carlo method for estimating the state distribution at each step of a Markov chain with totally ordered (discrete or continuous) state space. The number of steps in the chain can be random and unbounded. The method simulates n copies of the chain in parallel, using a (d+1)-dimensional low-discrepancy point set of cardinality n, randomized independently at each step, where d is the number of uniform random numbers required at each transition of the Markov chain. The method can be used in particular to get a lowvariance unbiased estimator of the expected total cost up to some random stopping time, when state-dependent costs are paid at each step. We provide numerical illustrations where the variance reduction with respect to standard Monte Carlo is substantial.

A quasi–Monte Carlo scheme for Smoluchowski’s coagulation equation

Abstract A particle method adapted to the simulation of diffusion problems is presented. Time is discretized into increments of length Δt . During each time step, the particles are allowed to random walk to any point by taking steps sampled from a Gaussian distribution centered at the current particle position with variance related to the time discretization Δt . Quasi-random samples are used and the particles are relabeled according to their position at each time step. Convergence is proved for the pure initial-value problem in s space dimensions. For some simple demonstration problems, the numerical results indicate that an improvement is achieved over standard random walk simulation.

Quasi-Monte Carlo Simulation of Discrete-Time Markov Chains on Multidimensional State Spaces

Abstract Some new quasi-Monte Carlo methods for the solution of the Boltzmann equation in a simplified case are described. The errors of the methods are estimated by means of the discrepancies of the sequences used for performing the Monte Carlo quadratures. Their accuracy is assessed through computation of effective errors in an example where an exact solution is known.

On Array-RQMC for Markov Chains: Mapping Alternatives and Convergence Rates

This paper analyzes a Monte Carlo algorithm for solving Smoluchowski’s coagulation equation. A finite number of particles approximates the initial mass distribution. Time is discretized and random numbers are used to move the particles according to the coagulation dynamics. Convergence is proved when quasi-random numbers are utilized and if the particles are relabeled according to mass in every time step. The results of some numerical experiments show that the error of the new algorithm is smaller than the error of a standard Monte Carlo algorithm using pseudo-random numbers without reordering the particles. Introduction Models of coalescence (i.e., coagulation, gelation, aggregation, agglomeration, accretion, etc.) mainly stem from the work of Smoluchowski on coagulation processes in colloids [15, 16]. Smoluchowski proposed the following infinite system of differential equations for the evolution of the number N0c(i, t) of clusters of mass i for i = 1, 2, 3 . . .: (1) ∂c ∂t (i, t) = 1 2 ∑ 1≤j<i K(i− j, j)c(i− j, t)c(j, t) − ∑ j≥1 K(i, j)c(i, t)c(j, t). Here N0 is the total number of clusters at time t = 0, so that ∑ i≥1 c(i, 0) = 1, and K(i, j) is the coagulation kernel. Numerical solution of the Smoluchowski’s coagulation equation is a difficult task for deterministic methods, so several stochastic algorithms have been proposed [8, 3, 17, 7, 11, 14, 2, 4]. The Monte Carlo (MC) schemes take a system of test particles which interact and form clusters according to the dynamics described in (1). Random numbers are used to find out which clusters interact and to determine the size of the new clusters. Despite the versatility of MC methods, a drawback is their slow convergence. An approach to acceleration is to change the choice of random numbers used. Quasi–Monte Carlo (QMC) methods use quasi-random numbers instead of pseudo-random numbers and can achieve better convergence in certain cases [5]. The efficiency of a QMC method depends on the quality of the quasi-random points that are used. These points should form a low-discrepancy point set. We recall from [13] some basic notations and concepts. If s ≥ 1 is a fixed dimension, Received by the editor November 11, 2002 and, in revised form, March 14, 2003. 2000 Mathematics Subject Classification. Primary 65C05; Secondary 70-08, 82C80. Computation was supported by the Centre Grenoblois de Calcul Vectoriel du Commissariat à l’Énergie Atomique, France. c ©2004 American Mathematical Society 1953 1954 CHRISTIAN LÉCOT AND WOLFGANG WAGNER then I := [0, 1) is the s-dimensional half-open unit cube and λs denotes the sdimensional Lebesgue measure. For a point set X consisting of x0, . . . ,xN−1 ∈ I and for a Lebesgue-measurable subset Q of I we define the local discrepancy by DN (Q,X) := 1 N ∑ 0≤p<N cQ(xp)− λs(Q), where cQ is the characteristic function of Q. The discrepancy of the point set X is defined by DN (X) := sup Q |DN (Q,X)|, the supremum being taken over all subintervals of I. The star discrepancy of X is D N (X) := sup Q? |DN (Q, X)|, where Q runs through all subintervals of I with one vertex at the origin. The idea of (t,m, s)-nets is to consider point sets X for which DN (Q,X) = 0 for a large family of intervals Q. Such point sets should have a small discrepancy. For an integer b ≥ 2, an interval of the form s ∏ r=1 [ ar bdr , ar + 1 bdr ) , with integers dr ≥ 0 and 0 ≤ ar < br for 1 ≤ r ≤ s, is called an elementary interval in base b. If 0 ≤ t ≤ m are integers, a (t,m, s)-net in base b is a point set X consisting of b points in I such that DN(Q,X) = 0 for every elementary interval Q in base b with measure λs(Q) = bt−m. The sequence analog of this concept is as follows. If b ≥ 2 and t ≥ 0 are integers, a sequence x0,x1, . . . of points in I is a (t, s)-sequence in base b if, for all integers n ≥ 0 and m > t, the points xp with nb ≤ p < (n+ 1)b form a (t,m, s)-net in base b. The following result is shown in [12]. Lemma 1. Let X be a (t,m, s)-net in base b. For any elementary interval Q′ ⊂ Is−1 in base b and for any xs ∈ Ī, |Dbm(Q × [0, xs), X)| ≤ bt−m. The effectiveness of QMC methods has limitations. First, while they are valid for integration problems, they may not be directly applicable to simulations, due to the correlations between the points of a quasi-random sequence. This problem can be overcome by writing the desired result as an integral. This leads to a second limitation: the improved accuracy of QMC methods may be lost for problems in which the integrand is not smooth. It is necessary to take special measures to make optimal use of the greater uniformity associated with quasi-random sequences. This is achieved here through the additional step of reordering the particles at each time step. The aim of the paper is to construct and investigate a QMC method for Smoluchowski’s coagulation equation. In Section 1 we present a particle scheme using quasi-random numbers for the solution of the equation. In Section 2 we prove the convergence of the method, as the number of simulated particles increases. In Section 3 we carry out numerical experiments based on a comparison of the method with a standard MC scheme. A QUASI–MONTE CARLO SCHEME FOR SMOLUCHOWSKI’S EQUATION 1955 1. The algorithm We assume that the coagulation kernel K(i, j) is nonnegative and symmetric K(i, j) ≥ 0 and K(i, j) = K(j, i). Multiplying (1) by i and summing over all i, indicates that mass is conserved (2) d dt ∑

Simulation of diffusion using quasi-random walk methods

We propose and analyze a quasi-Monte Carlo (QMC) method for simulating a discrete-time Markov chain on a discrete state space of dimension s ≥ 1. Several paths of the chain are simulated in parallel and reordered at each step, using a multidimensional matching between the QMC points and the copies of the chains. This method generalizes a technique proposed previously for the case where s = 1. We provide a convergence result when the number N of simulated paths increases toward infinity. Finally, we present the results of some numerical experiments showing that our QMC algorithm converges faster as a function of N, at least in some situations, than the corresponding Monte Carlo (MC) method.