On a closed-form approximation for the free energy ofd-dimensional Ising model,III

A construction of a closed-form approximation for the free energy of Ising model on ad-dimensional simple hypercubical lattice is given.     

On closed-form approximations for the free energy of 3-dimensional ising model

We show that it is possible to construct closed-form approximations to lnZ ₃, which are suitable generalizations of the Onsager formula for lnZ ₂, and whose high- and low-temperature series expansions have the same structure as the exact ones.     

An alternative view of variational Bayes and asymptotic approximations of free energy

Bayesian learning, widely used in many applied data-modeling problems, is often accomplished with approximation schemes because it requires intractable computation of the posterior distributions. In this study, we focus on two approximation methods, variational Bayes and local variational approximation. We show that the variational Bayes approach for statistical models with latent variables can be viewed as a special case of local variational approximation, where the log-sum-exp function is used to form the lower bound of the log-likelihood. The minimum variational free energy, the objective function of variational Bayes, is analyzed and related to the asymptotic theory of Bayesian learning. This analysis additionally implies a relationship between the generalization performance of the variational Bayes approach and the minimum variational free energy.     

On the approximations for multinormal integration

The system failure probability in structural reliability analysis involves the computation of multivariate normal integrals. The civil and structural engineering literature has seen the development of several approximation methods for the computation of multivariate normal integrals. However, these developments have been totally unaware of the same in the statistics literature. In this note, a 1-line program from a freely downloadable statistical software is provided for the exact computation of m-dimensional multinormal integrals. It is shown that at least two of the known methods from the civil and structural engineering literature lead to inaccurate results.     

On the use of I-divergence for generating distribution approximations

The existence of an upper bound for the error probability as a function of I-divergences between an original and an approximating distribution is proved. Such a bound is shown to be a monotonic nondecreasing function of the I-divergences, reaching the Bayes error probability when they vanish. It has been shown that if the closeness between the original and approximating distributions is assessed by the probability of error associated with a particular two-class recognition problem in which those functions are the class conditional distributions, then the best upper bound for such probability is ? regardless of the value of the I-divergences between them. Approaching the approximation problem from a rather different viewpoint, this correspondence considers the problem of a two-class discrete measurement classification where the original distributions are replaced by approximations, and its effects on the probability of error. The corresponding analysis is presented in detail.     

On multiple spline approximations for Bayesian computations

Probabilistic reasoning typically suffers from the explosive amount of information it must maintain. There are a variety of methods available for curbing this explosion. However, in doing so, it is important to avoid oversimplifying the given domain through injudicious use of assumptions such as independence. Multiple splining is an approach for compressing and approximating the probabilistic information. Instead of positing additional independence conditions, it attempts to identify patterns in the information. While the data explosion is multiplicative in nature, O(n ₁ n ₂⋯n _k), multiple splines reduces it to an additive one, O(n ₁ + n ₂ + ⋯n _k). We consider how these splines can be found and used. Since splines exploit patterns in the data, we can also use them to help in filling in missing data. As it turns out, our splining method is quite general and may be applied to other domains besides probabilistic reasoning which can benefit from data compression. This revised version was published online in June 2006 with corrections to the Cover Date.     

基于Ising模型的QR码加密算法

快速反应码(QR)二维条形码在通信时存在安全隐患,为此提出了一种加密算法。伊辛(Ising)模型具有矩阵形式,在形式上与QR码相同,且其状态变换过程支持并行计算。基于Ising模型的变换规则生成加密矩阵,然后结合QR码的编码原理计算出QR码的密文。实验结果证实该算法具有良好的随机性和密钥敏感性。该算法简单、安全、高效,可用于QR码的安全保密通信     

Analytical expressions for a finite-size 2D Ising model

Numerical methods are used to examine the thermodynamic characteristics of the twodimensional Ising model as a function of the number of spins N. Onsager’s solution is generalized to a finite-size lattice, and experimentally validated analytical expressions for the free energy and its derivatives are computed. The heat capacity at the critical point is shown to grow logarithmically with N. Due to the finite extent of the system the critical temperature can only be determined to some accuracy.     

A variational level set model with closed-form solution for bimodal image segmentation

Multimedia Tools and Applications - In this work, we present a variational level set model with closed–form solution via combining with the fuzzy clustering method for robust and efficient...     

On the existence of regular approximations

We approximate context-free, or more general, languages using finite automata. The degree of approximation is measured, roughly speaking, by counting the number of incorrect answers an automaton gives on inputs of length m$m$ and observing how these values behave for large m$m$. More restrictive variants are obtained by requiring that the automaton never accepts words outside the language or that it accepts all words in the language. A further distinction is whether a given (context-free) language has a regular approximation which is optimal under the measure of approximation degree or an approximation which is arbitrarily close to optimal. We study closure and decision properties of the approximation measure.     

On the generalized successive approximations method

In this paper, denoted a continuous function from [0,1]² into [0,1] by f, we consider the iterative process:

On feasible sets for MPC and their approximations

In this paper we are interested in the computation of feasible sets for linear model predictive control techniques, based on set relations and not on the conventional orthogonal projection. Further, the problem of computing suitable inner approximations of the feasible sets is considered. Such approximations are characterized by simpler polytopic representations, and preserve essential properties as convexity, positive invariance, inclusion of the set of expected initial states.     

On the medial surface approximations of extrusions

Generating the medial surface for a general boundary representation model raises several difficulties. Problems might emerge from the complexity of the resulting equations, singularities caused by unforeseen relative boundary element positions and orientations, etc. The majority of the current algorithms are based on the topology of the boundary representation model and produce wireframes composed of straight lines regardless of the real medial surfaces. Many of the solids used in engineering can be represented by extrusions, delimited by a cross-section and an extrusion distance. This paper develops a fast and efficient method for creating the facetted approximations of the medial surfaces of extrusions generated by sweeping along the normal direction to the generating cross-section.

Stochastic reasoning, free energy, and information geometry

Belief propagation (BP) is a universal method of stochastic reasoning. It gives exact inference for stochastic models with tree interactions and works surprisingly well even if the models have loopy interactions. Its performance has been analyzed separately in many fields, such as AI, statistical physics, information theory, and information geometry. This article gives a unified framework for understanding BP and related methods and summarizes the results obtained in many fields. In particular, BP and its variants, including tree reparameterization and concave-convex procedure, are reformulated with information-geometrical terms, and their relations to the free energy function are elucidated from an information-geometrical viewpoint. We then propose a family of new algorithms. The stabilities of the algorithms are analyzed, and methods to accelerate them are investigated.     

On the geometric convergence of neural approximations

We give upper bounds rates of approximation of a set of functions from a real Hilbert space, using convex greedy iterations. The approximation method was originally proposed and analyzed by Jones (1992). Barron (1993) applied the method to the set of functions computable by single-hidden-layer feedforward neural networks. It was shown that the networks achieve an integrated squared error of order O(1/n), where n is the number of iterations, or equivalently, nodes in the network. Assuming that the functions to be approximated satisfy the so-called delta-angular condition, we show that the corresponding rate of approximation of order O((n) is achievable, where 0 =/< q < 1. Therefore, for the set of functions considered, the reported geometrical rate of approximation is an improvement of Maurey-Jones-Barron's upper bound result. In the case of orthonormal convex greedy approximations, the delta-angular condition is shown to be equivalent to the geometrically decaying expansion coefficients. In finite dimensions the delta-angular condition is proven to take place for a wide class of functions.     

Parallelization of the Ising model and its performance evaluation

We report on the parallelization of two widely used algorithms in computational physics: The Monte Carlo simulation of the Ising model and a cluster identification algorithm which is used for percolation or percolation-like problems. Both parallel algorithms were tested on a multi-transputer system using up to 128 processors. The results show that the algorithms can perform with a linear speedup. We propose a scaling law for the speedup and show that the speedup for both algorithms satisfies this scaling.     

On the convergence of Chapman-Enskog approximations for electron transport coefficients calculations

Calculations of electron transfer coefficients (electrical and thermal conductivities) for gas-plasma mixtures have been considered. These calculations have been made in Chapman-Enkog (ChE) approaches. To evaluate the convergence of these approximations, calculations ignoring electron-electron collisions have been carried out in parallel. The later one is the so called Lorenz gas for which the exact solution is known. Comparison with this gas serves to evaluate the convergence of ChE approximations. The comparison shows that the 5th approximation ensures accuracy better than 1%.     

A dedicated algorithm for calculating ground states for the triangular random bond Ising model

In the presented article we present an algorithm for the computation of ground state spin configurations for the 2d random bond Ising model on planar triangular lattice graphs. Therefore, it is explained how the respective ground state problem can be mapped to an auxiliary minimum-weight perfect matching problem, solvable in polynomial time. Consequently, the ground state properties as well as minimum-energy domain wall (MEDW) excitations for very large 2d systems, e.g. lattice graphs with up to N=384×384 spins, can be analyzed very fast.Here, we investigate the critical behavior of the corresponding T=0 ferromagnet to spin-glass transition, signaled by a breakdown of the magnetization, using finite-size scaling analyses of the magnetization and MEDW excitation energy and we contrast our numerical results with previous simulations and presumably exact results.     

On the correction of finite difference eigenvalue approximations for Sturm-Liouville problems

The use of algebraic eigenvalues to approximate the eigenvalues of Sturm-Liouville operators is known to be satisfactory only when approximations to the fundamental and the first few harmonics are required. In this paper, we show how the asymptotic error associated with related but simpler Sturm-Liouville operators can be used to correct certain classes of algebraic eigenvalues to yield uniformly valid approximations.