A generalization of short-period Tausworthe generators and its application to Markov chain quasi-Monte Carlo

A one-dimensional sequence $u_0, u_1, u_2, \ldots \in [0, 1)$ is said to be completely uniformly distributed (CUD) if overlapping $s$-blocks $(u_i, u_{i+1}, \ldots , u_{i+s-1})$, $i = 0, 1, 2, \ldots$, are uniformly distributed for every dimension $s \geq 1$. This concept naturally arises in Markov chain quasi-Monte Carlo (QMC). However, the definition of CUD sequences is not constructive, and thus there remains the problem of how to implement the Markov chain QMC algorithm in practice. Harase (2021) focused on the $t$-value, which is a measure of uniformity widely used in the study of QMC, and implemented short-period Tausworthe generators (i.e., linear feedback shift register generators) over the two-element field $\mathbb{F}_2$ that approximate CUD sequences by running for the entire period. In this paper, we generalize a search algorithm over $\mathbb{F}_2$ to that over arbitrary finite fields $\mathbb{F}_b$ with $b$ elements and conduct a search for Tausworthe generators over $\mathbb{F}_b$ with $t$-values zero (i.e., optimal) for dimension $s = 3$ and small for $s \geq 4$, especially in the case where $b = 3, 4$, and $5$. We provide a parameter table of Tausworthe generators over $\mathbb{F}_4$, and report a comparison between our new generators over $\mathbb{F}_4$ and existing generators over $\mathbb{F}_2$ in numerical examples using Markov chain QMC

Simulation and theory of abnormal grain growth--anisotropic grain boundary energies and mobilities

Abnormal grain growth has been studied by means of a computer-based Monte Carlo model. This model has previously been shown to reproduce many of the essential features of normal grain growth. The simulations presented in this work are based on a modified model in which two distinct types of grains are present. These two grain types might correspond to two components of different crystallographic orientation, for example. This results in three classes of grain boundaries: 1. (a) between unlike types,2. (b)between grains of the first type and3. (c) between grains of the second type, to which different grain boundary energies or different mobilities can be assigned. Most simulations started with a single grain of the first type embedded in a matrix of grains of the second type. Anisotropie grain boundary energies were modeled by assigning a higher energy to boundaries between like type than to boundaries between grains of unlike type. For this case, abnormal grain growth only occurred for an energy ratio greater than 2 and then wetting of the matrix by the abnormal grain occurred. Anisotropie grain boundary mobilities were modeled by assigning a lower mobility to boundaries between grains of like type than to boundaries between unlike type. For this case the extent of abnormal grain growth varied with the ratio of mobilities and it is tentatively concluded that there is a limiting ratio of size of the abnormal grain relative to the matrix. A simple treatment of anisotropic grain boundary mobility was developed by modifying Hillert's grain growth model [Acta metall. 13, 227 (1965)]. This theoretical treatment also produced a limiting ratio of relative size that is a simple function of the mobility ratio.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/28006/1/0000442.pd