Quasi-random Simulation of Linear Kinetic Equations

AbstractWe study the improvement achieved by using quasi-random sequences in place of pseudo-random numbers for solving linear spatially homogeneous kinetic equations. Particles are sampled from the initial distribution. Time is discretized and quasi-random numbers are used to move the particles in the velocity space. Quasi-random points are not blindly used in place of pseudo-random numbers: at each time step, the number order of the particles is scrambled according to their velocities. Convergence of the method is proved. Numerical results are presented for a sample problem in dimensions 1, 2 and 3. We show that by using quasi-random sequences in place of pseudo-random points, we are able to obtain reduced errors for the same number of particles

Diffusion in a nonhomogeneous medium: quasi-random walk on a lattice

International audienceWe are interested in Monte Carlo (MC) methods for solving the diffusion equation: in the case of a constant diffusion coefficient, the solution is approximated by using particles and in every time step, a constant stepsize is added to or substracted from the coordinates of each particle with equal probability. For a spatially dependent diffusion coefficient, the naive extension of the previous method using a spatially variable stepsize introduces a systematic error: particles migrate in the directions of decreasing diffusivity. A correction of stepsizes and stepping probabilities has recently been proposed and the numerical tests have given satisfactory results. In this paper, we describe a quasi-Monte Carlo (QMC) method for solving the diffusion equation in a spatially nonhomogeneous medium: we replace the random samples in the corrected MC scheme by low-discrepancy point sets. In order to make a proper use of the better uniformity of these point sets, the particles are reordered according to their successive coordinates at each time step. We illustrate the method with numerical examples: in dimensions 1 and 2, we show that the QMC approach leads to improved accuracy when compared with the original MC method using the same number of particles

A Randomized Quasi-Monte Carlo Simulation Method for Markov Chains

We introduce and study a randomized quasi-Monte Carlo method for estimating the state distribution at each step of a Markov chain, under the assumption that the chain has a 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. This technique is effective in particular to obtain a low-variance 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. The variance is reduced by factors of several thousands in some cases. We prove bounds on the convergence rate of the worst-case error and variance for special situations. In line with what is typically observed in RQMC contexts, our empirical results indicate much better convergence than what these bounds guarantee

Quasi-Monte Carlo methods for Markov chains with continuous multi-dimensional state space

International audienceWe describe a quasi-Monte Carlo method for the simulation of discrete time Markov chains with continuous multi-dimensional state space. The method simulates copies of the chain in parallel. At each step the copies are reordered according to their successive coordinates. We prove the convergence of the method when the number of copies increases. We illustrate the method with numerical examples where the simulation accuracy is improved by large factors compared with Monte Carlo simulation

Quasi-Monte Carlo methods for Markov chains with continuous multi-dimensional state space

International audienceWe describe a quasi-Monte Carlo method for the simulation of discrete time Markov chains with continuous multi-dimensional state space. The method simulates copies of the chain in parallel. At each step the copies are reordered according to their successive coordinates. We prove the convergence of the method when the number of copies increases. We illustrate the method with numerical examples where the simulation accuracy is improved by large factors compared with Monte Carlo simulation

Sequential Quasi-Monte Carlo

We derive and study SQMC (Sequential Quasi-Monte Carlo), a class of algorithms obtained by introducing QMC point sets in particle filtering. SQMC is related to, and may be seen as an extension of, the array-RQMC algorithm of L'Ecuyer et al. (2006). The complexity of SQMC is $O(N \log N)$, where $N$ is the number of simulations at each iteration, and its error rate is smaller than the Monte Carlo rate $O_P(N^{-1/2})$. The only requirement to implement SQMC is the ability to write the simulation of particle $x_t^n$ given $x_{t-1}^n$ as a deterministic function of $x_{t-1}^n$ and a fixed number of uniform variates. We show that SQMC is amenable to the same extensions as standard SMC, such as forward smoothing, backward smoothing, unbiased likelihood evaluation, and so on. In particular, SQMC may replace SMC within a PMCMC (particle Markov chain Monte Carlo) algorithm. We establish several convergence results. We provide numerical evidence that SQMC may significantly outperform SMC in practical scenarios.Comment: 55 pages, 10 figures (final version

Low discrepancy sequences for solving the Boltzmann equation

AbstractSome 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