Volume 25, Issue 3

We propose the use of randomized (scrambled) quasirandom sequences for the purpose of providing practical error estimates for quasi-Monte Carlo (QMC) applications. One popular quasirandom sequence among practitioners is the Halton sequence. However, Halton subsequences have correlation problems in their highest dimensions, and so using this sequence for high-dimensional integrals dramatically affects the accuracy of QMC. Consequently, QMC studies have previously proposed several scrambling methods; however, to varying degrees, scrambled versions of Halton sequences still suffer from the correlation problem as manifested in two-dimensional projections. This paper proposes a modified Halton sequence (MHalton), created using a linear digital scrambling method, which finds the optimal multiplier for the Halton sequence in the linear scrambling space. In order to generate better uniformity of distributed sequences, we have chosen strong MHalton multipliers up to 360 dimensions. The proposed multipliers have been tested and proved to be stronger than several sets of multipliers used in other known scrambling methods. To compare the quality of our proposed scrambled MHalton sequences with others, we have performed several extensive computational tests that use L2{L_{2}}-discrepancy and high-dimensional integration tests. Moreover, we have tested MHalton sequences on Mortgage-backed security (MBS), which is one of the most widely used applications in finance. We have tested our proposed MHalton sequence numerically and empirically, and they show optimal results in QMC applications. These confirm the efficiency and safety of our proposed MHalton over scrambling sequences previously used in QMC applications.

Monte Carlo simulations have been utilized to make a comparative study between diffusion approximation (DA) and the Gillespie algorithm and its dependence on population in the information diffusion model. Diffusion approximation is one of the widely used approximation methods which have been applied in queuing systems, biological systems and other fields. The Gillespie algorithm, on the other hand, is used for simulating stochastic systems. In this article, the validity of diffusion approximation has been studied in relation to the Gillespie algorithm for varying population sizes. It is found that diffusion approximation results in large fluctuations which render forecasting unreliable particularly for a small population. The relative fluctuations in relation to diffusion approximation, as well as to the Gillespie algorithm have been analyzed. To carry out the study, a nonlinear stochastic model of innovation diffusion in a finite population has been considered. The nonlinearity of the problem necessitates use of approximation methods to understand the dynamics of the system. A stochastic differential equation (SDE) has been used to model the innovation diffusion process, and corresponding sample paths have been generated using Monte Carlo simulation methods.

We consider stochastic processes Y⁢(t){Y(t)} which can be represented as Y⁢(t)=(X⁢(t))s{Y(t)=(X(t))^{s}}, s∈ℕ{s\in\mathbb{N}}, where X⁢(t){X(t)} is a stationary strictly sub-Gaussian process, and build a wavelet-based model that simulates Y⁢(t){Y(t)} with given accuracy and reliability in Lp⁢([0,T]){L_{p}([0,T])}. A model for simulation with given accuracy and reliability in Lp⁢([0,T]){L_{p}([0,T])} is also built for processes Z⁢(t){Z(t)} which can be represented as Z⁢(t)=X1⁢(t)⁢X2⁢(t){Z(t)=X_{1}(t)X_{2}(t)}, where X1⁢(t){X_{1}(t)} and X2⁢(t){X_{2}(t)} are independent stationary strictly sub-Gaussian processes.

We introduce a parallel scheme for simulated annealing, a widely used Markov chain Monte Carlo (MCMC) method for optimization. Our method is constructed and analyzed under the classical framework of MCMC. The benchmark function for optimization is used for validation and verification of the parallel scheme. The experimental results, along with the proof based on statistical theory, provide us with insights into the mechanics of the parallelization of simulated annealing for high parallel efficiency or scalability for large parallel computers.

The paper shows a new weak approximation method for stochastic differential equations as a generalization and an extension of Heath–Platen’s scheme for multidimensional diffusion processes. We reformulate the Heath–Platen estimator from the viewpoint of asymptotic expansion. The proposed scheme is implemented by a Monte Carlo method and its variance is much reduced by the asymptotic expansion which works as a kind of control variate. Numerical examples for the local stochastic volatility model are shown to confirm the efficiency of the method.

A Monte Carlo method used for the estimation of convergent von Neumann series solutions of a Fredholm equation of second kind is considered. The sum z(d)⁢(x){z^{(d)}(x)} of d initial terms of the von Neumann series estimating the solution z⁢(x){z(x)} of the equation is represented as a d -dimensional integral over the unit cube Hd{H_{d}}. This note presents three examples calculating z(d)⁢(x){z^{(d)}(x)} for different kernels with norms ∥K∥<1{\lVert K\rVert<1}. We found that z(d)⁢(x){z^{(d)}(x)} calculated using a quasi-Monte Carlo (QMC) method converges significantly faster than the corresponding Monte Carlo (MC) estimates in the entire range of ∥K∥{\lVert K\rVert} values. We also found that the average dimension d^{\hat{d}} of the integrand in all our examples is small, less than 3. We suggest that the average dimensions d^{\hat{d}} of our d -dimensional integrands are bounded as d→∞{d\to\infty}.

A quick generation method of k -wise independent uniformly distributed m -bit random variables with small randomness is proposed with applications to the Monte Carlo method.

A meshless stochastic algorithm for solving anisotropic transient diffusion problems based on an extension of the classical Random Walk on Spheres method is developed. Direct generalization of the Random Walk on Spheres method to anisotropic diffusion equations is not possible, therefore, we have derived approximations of the probability densities for the first passage time and the exit point on a small sphere. The method can be conveniently applied to solve diffusion problems with spatially varying diffusion coefficients and is simply implemented for complicated three-dimensional domains. Particle tracking algorithm is highly efficient for calculation of fluxes to boundaries. We present some simulation results in the case of cathodoluminescence and electron beam induced current in the vicinity of a dislocation in a semiconductor material.