Volume 22, Issue 4

The well-known random walk on spheres method (RWS) for the Laplace equation is here extended to drift-diffusion problems. First we derive a generalized spherical mean value relation which is an extension of the classical integral mean value relation for the Laplace equation. Next we give a probabilistic interpretation of the kernel. The distribution on the sphere generated by this kernel is then related to the von Mises–Fisher distribution on the sphere which can be efficiently simulated. The rigorous expressions are given for the case of constant velocity drift, but the algorithm is then extended to solve drift-diffusion problems with arbitrary varying drift velocity vector. Applications to cathodoluminescence and EBIC imaging of defects and dislocations in semiconductors are discussed.

Our aim in this note is to extend the semi-discrete technique by combine it with the split step method. We apply our new method to the Ait-Sahalia model and propose an explicit and positivity preserving numerical scheme.

We review the derivation of the Kac master equation model for random collisions of particles, its relationship to the Poisson process, and existing algorithms for simulating values from the marginal distribution of velocity for a single particle at any given time. We describe and implement a new algorithm that efficiently and more fully leverages properties of the Poisson process, show that it performs at least as well as existing methods, and give empirical evidence that it may perform better at capturing the tails of the single particle velocity distribution. Finally, we derive and implement a novel “ε-perfect sampling” algorithm for the limiting marginal distribution as time goes to infinity. In this case the importance is a proof of concept that has the potential to be expanded to more interesting (DSMC) direct simulation Monte Carlo applications.

In this paper, we consider both, the strong and weak convergence of the Euler–Maruyama approximation for one-dimensional stochastic differential equations involving the local times of the unknown process. We use a transformation in order to remove the local time Lta${L_{t}^{a}}$ from the stochastic differential equations of type Xt=X0+∫0tφ⁢(Xs)⁢𝑑Bs+∫ℝν⁢(d⁢a)⁢Lta.$X_{t}=X_{0}+\int_{0}^{t}\varphi(X_{s})\,dB_{s}+\int_{\mathbb{R}}\nu(da)L_{t}^{% a}.$ Here B is a one-dimensional Brownian motion, φ:ℝ→ℝ${\varphi:\mathbb{R}\rightarrow\mathbb{R}}$ is a bounded measurable function, and ν is a bounded measure on ℝ${\mathbb{R}}$. We provide the approximation of Euler–Maruyama for the stochastic differential equations without local time. After that, we conclude the approximation of Euler–Maruyama Xtn${X_{t}^{n}}$ of the above mentioned equation, and we provide the rate of strong convergence Error=𝔼⁢|XT-XTn|${\operatorname{Error}=\mathbb{E}\lvert X_{T}-X_{T}^{n}\rvert}$, and the rate of weak convergence Error=𝔼⁢|G⁢(XT)-G⁢(XTn)|${\operatorname{Error}=\mathbb{E}\lvert G(X_{T})-G(X_{T}^{n})\rvert}$, for any function G:ℝ→ℝ${G:\mathbb{R}\rightarrow\mathbb{R}}$ of bounded variation.

In this paper we first investigate the use of Markov Chain Monte Carlo (MCMC) methods to attack classical ciphers. MCMC has previously been used to break simple substitution, transposition and substitution-transposition ciphers. Here, we improve the accuracy of obtained results by these algorithms and we show the performance of the algorithms using quasi random numbers such as Faure, Sobol and Niederreiter sequences.

The steady state of a dilute gas enclosed within a rectangular cavity, whose upper and lower sides are in relative motion, is considered in the slip and early transition regimes. The DSMC (Direct simulation Monte Carlo) method is used to solve the Boltzmann equation for analysing a Newtonian viscous heat conducting ideal gas with the slip and jump boundary conditions (SJBC) in the vicinity of horizontal walls. The numerical results are compared with the Navier–Stokes solutions, with and without SJBC, through the velocity, temperature, and normal heat flux profiles. The parallel heat flux and shear stress are also evaluated as a function of rarefaction degree; estimated by the Knudsen number Kn${K_{n}}$. Thus, the breakdown of the classical Navier–Stokes theory is clarified in the non-equilibrium area, so-called Knudsen layer, near the top and bottom sides.

Matsumoto, Saito and Matoba recently proposed the Walsh figure of merit (WAFOM), which is a computable criterion for quasi-Monte Carlo point sets using digital nets. Several algorithms have been proposed for finding low-WAFOM point sets. In the existing algorithms, the number of points is fixed in advance, but extensible point sets are preferred in some applications. In this paper, we propose a random search algorithm for extensible low-WAFOM point sets. For this, we introduce a method that uses lookup tables to compute WAFOM faster. Numerical results show that our extensible low-WAFOM point sets are comparable with Niederreiter–Xing sequences for some low-dimensional and smooth test functions.