Volume 26, Issue 4

Monte Carlo algorithms are developed to simulate the electron transport in semiconductors. In particular, the drift velocity in GaN semiconductors is calculated, and a comparison with experimental measurements is discussed. Explicit expressions for the scattering probabilities and distributions of the scattering angle of electrons on polar optical and intervalley phonons, and acoustic deformation potential as well are given. A good agreement of the simulation results and the experimental measurements reveals that the M-L valley is located at 0.7 eV higher than the Γ-valley. This value agrees with other experimental studies, while it is lower compared to ab initio calculations.

In this article, we consider the general problem of checking the correctness of matrix multiplication. Given three n×nn\times n matrices 𝐴, 𝐵 and 𝐶, the goal is to verify that A×B=CA\times B=C without carrying out the computationally costly operations of matrix multiplication and comparing the product A×BA\times B with 𝐶, term by term. This is especially important when some or all of these matrices are very large, and when the computing environment is prone to soft errors. Here we extend Freivalds’ algorithm to a Gaussian Variant of Freivalds’ Algorithm (GVFA) by projecting the product A×BA\times B as well as 𝐶 onto a Gaussian random vector and then comparing the resulting vectors. The computational complexity of GVFA is consistent with that of Freivalds’ algorithm, which is O⁢(n2)O(n^{2}). However, unlike Freivalds’ algorithm, whose probability of a false positive is 2-k2^{-k}, where 𝑘 is the number of iterations, our theoretical analysis shows that, when A×B≠CA\times B\neq C, GVFA produces a false positive on set of inputs of measure zero with exact arithmetic. When we introduce round-off error and floating-point arithmetic into our analysis, we can show that the larger this error, the higher the probability that GVFA avoids false positives. Moreover, by iterating GVFA 𝑘 times, the probability of a false positive decreases as pkp^{k}, where 𝑝 is a very small value depending on the nature of the fault on the result matrix and the arithmetic system’s floating-point precision. Unlike deterministic algorithms, there do not exist any fault patterns that are completely undetectable with GVFA. Thus GVFA can be used to provide efficient fault tolerance in numerical linear algebra, and it can be efficiently implemented on modern computing architectures. In particular, GVFA can be very efficiently implemented on architectures with hardware support for fused multiply-add operations.

In this work, we propose a new method for calculating the mathematical expectation of nonlinear functionals from random processes. The method is based on using Wiener chaos expansion and approximate formulas, exact for functional polynomials of given degree. Examples illustrating approximation accuracy are considered.

We describe an implementation of the de-biased estimator using mixed sequences; these are sequences obtained from pseudorandom and low-discrepancy sequences. We use this implementation to numerically solve some stochastic differential equations from computational finance. The mixed sequences, when combined with Brownian bridge or principal component analysis constructions, offer convergence rates significantly better than the Monte Carlo implementation.

In our paper [A. Nasroallah and K. Elkimakh, HMM with emission process resulting from a special combination of independent Markovian emissions, Monte Carlo Methods Appl. 23 2017, 4, 287–306] we have studied, in a first scenario, the three fundamental hidden Markov problems assuming that, given the hidden process, the observed one selects emissions from a combination of independent Markov chains evolving at the same time. Here, we propose to conduct the same study with a second scenario assuming that given the hidden process, the emission process selects emissions from a combination of independent Markov chain evolving according to their own clock. Three basic numerical examples are studied to show the proper functioning of the iterative algorithm adapted to the proposed model.

Methods from integral geometry and geometric probability allow us to estimate geometric size measures indirectly. In this article, a Monte Carlo algorithm for simultaneous estimation of hyper-volumes and hyper-surface areas of a class of compact sets in Euclidean space is developed. The algorithm is based on Santalo’s formula and the Hadwiger formula from integral geometry, and employs a comparison principle to assign geometric probabilities. An essential component of the method is to be able to generate uniform sets of random lines on the sphere. We utilize an empirically established method to generate these random chords, and we describe a geometric randomness model associated with it. We verify our results by computing measures for hyper-ellipsoids and certain non-convex sets.

In this paper, to construct a confidence interval (general and shortest) for quantiles of normal distribution in one population, we present a pivotal quantity that has non-central t distribution. In the case of two independent normal populations, we propose a confidence interval for the ratio of quantiles based on the generalized pivotal quantity, and we introduce a simple method for extracting its percentiles, based on which a shorter confidence interval can be created. Also, we provide general and shorter confidence intervals using the method of variance estimate recovery. The performance of five proposed methods will be examined by using simulation and examples.

A Random Walk on Ellipsoids (RWE) algorithm is developed for solving a general class of elliptic equations involving second- and zero-order derivatives. Starting with elliptic equations with constant coefficients, we derive an integral equation which relates the solution in the center of an ellipsoid with the integral of the solution over an ellipsoid defined by the structure of the coefficients of the original differential equation. This integral relation is extended to parabolic equations where a first passage time distribution and survival probability are given in explicit forms. We suggest an efficient simulation method which implements the RWE algorithm by introducing a notion of a separation sphere. We prove that the logarithmic behavior of the mean number of steps for the RWS method remains true for the RWE algorithm. Finally we show how the developed RWE algorithm can be applied to solve elliptic and parabolic equations with variable coefficients. A series of supporting computer simulations are given.