Algorithm 36 SNIFF: Efficient self-tuning algorithm for numerical integration

A new adaptive algorithm for the integration of analytic functions is presented. The algorithm processes the integration interval by generating local subintervals whose length is controlled through a feedback loop. Control is performed by means of a relation derived on an analytical basis and valid for an arbitrary integration rule: two different estimates of an integral are used to compute the interval length necessary to obtain an integral estimate with accuracy within the assigned error bounds. The implied method for local generation of subintervals and an effective assumption of error partition among subintervals give rise to an adaptive algorithm provided with an accurate and very efficient integration process. The particular algorithm obtained by choosing the 6-point Gauß-Legendre integration rule is considered and extensive comparisons are made with other outstanding integration algorithms.     

Variable order composite quadrature of singular and nearly singular integrals

A class of variable order composite quadrature formulas for the numerical integration of functions with a singularity in or near to the region of integration is introduced. Exponential convergence of the method is shown for all integrands in the countably normed spaceB _β. Numerical examples are presented which demonstrate that the asymptotic exponential convergence rates obtained here are sharp and already observed for a small number of quadrature points.     

Step decision rules for multistage stochastic programming: A heuristic approach

Stochastic programming with step decision rules (SPSDR) aims to produce efficient solutions to multistage stochastic optimization problems. SPSDR, like plain multistage Stochastic Programming (SP), operates on a Monte Carlo “computing sample” of moderate size that approximates the stochastic process. Unlike SP, SPSDR does not strive to build a balanced event tree out of that sample. Rather, it defines a solution as a special type of decision rule, with the property that the decisions at each stage are piecewise constant functions on the sample of scenarios. Those pieces define a partition of the set of scenarios at each stage t, but the partition at t+1 need not be refinement of the partition at t. However, the rule is constructed so that the non-anticipativity condition is met, a necessary condition to make the rules operational. To validate the method we show how to extend a non-anticipatory decision rule to arbitrary scenarios within a very large validation sample of scenarios. We apply three methods, SPSDR, SP and Robust Optimization, to the same 12-stage problem in supply chain management, and compare them relatively to different objectives and performance criteria. It appears that SPSDR performs better than SP in that it produces a more accurate estimate (prediction) of the value achieved by its solution on the validation sample, and also that the achieved value is better.     

The convergence rate of the sandwich algorithm for approximating convex functions

The Sandwich algorithm approximates a convex function of one variable over an interval by evaluating the function and its derivative at a sequence of points. The connection of the obtained points is a piecewise linear upper approximation, and the tangents yield a piecewise linear lower approximation. Similarly, a planar convex figure can be approximated by convex polygons. Different versions of the Sandwich algorithm use different rules for selecting the next evaluation point. We consider four natural rules (interval bisection, slope bisection, maximum error rule, and chord rule) and show that the global approximation error withn evaluation points decreases by the order ofO(1/n ²), which is optimal. By special examples we show that the actual performance of the four rules can be very different from each other, and we report computational experiments which compare the performance of the rules for particular functions.     

基于遗传算法求任意函数的数值积分

针对复杂函数的数值积分问题,给出了若干个任意分割积分区间的数值积分的误差结果,并提出一种基于遗传算法的不等距节点分割的数值积分方法。该方法初始时在积分区间内任意选取一定的节点,通过遗传算法优化这些节点,在相邻节点间利用Simpson公式近似计算积分,最后得到较准确的积分结果。数值计算结果表明,该方法计算精度高,而且可以计算奇异函数及震荡函数的积分。     

Parameter learning from stochastic teachers and stochastic compulsive liars.

This paper considers a general learning problem akin to the field of learning automata (LA) in which the learning mechanism attempts to learn from a stochastic teacher or a stochastic compulsive liar. More specifically, unlike the traditional LA model in which LA attempts to learn the optimal action offered by the Environment (also here called the "Oracle"), this paper considers the problem of the learning mechanism (robot, an LA, or in general, an algorithm) attempting to learn a "parameter" within a closed interval. The problem is modeled as follows: The learning mechanism is trying to locate an unknown point on a real interval by interacting with a stochastic Environment through a series of informed guesses. For each guess, the Environment essentially informs the mechanism, possibly erroneously (i.e., with probability p), which way it should move to reach the unknown point. When the probability of a correct response is p > 0.5, the Environment is said to be informative, and thus the case of learning from a stochastic teacher. When this probability p < 0.5, the Environment is deemed deceptive, and is called a stochastic compulsive liar. This paper describes a novel learning strategy by which the unknown parameter can be learned in both environments. These results are the first reported results, which are applicable to the latter scenario. The most significant contribution of this paper is that the proposed scheme is shown to operate equally well, even when the learning mechanism is unaware of whether the Environment ("Oracle") is informative or deceptive. The learning strategy proposed herein, called CPL-AdS, partitions the search interval into d subintervals, evaluates the location of the unknown point with respect to these subintervals using fast-converging E-optimal LRI LA, and prunes the search space in each iteration by eliminating at least one partition. The CPL-AdS algorithm is shown to provably converge to the unknown point with an arbitrary degree of accuracy with probability as close to unity as desired. Comprehensive experimental results confirm the fast and accurate convergence of the search for a wide range of values for the Environment's feedback accuracy parameter p, and thus has numerous potential applications.     

Quantum billiards in multidimensional models with fields of forms on a product of Einstein spaces

The gravitational D-dimensional model is considered, with l scalar fields, a cosmological constant and several forms. When a cosmological block-diagonal metric, defined on a product of an 1-dimensional interval and n oriented Einstein spaces, is chosen, an electromagnetic composite brane ansatz is adopted, and certain restrictions on the branes are imposed, the conformally covariant Wheeler–DeWitt (WDW) equation for the model is studied. Under certain restrictions, asymptotic solutions to the WDWequation are found in the limit of the formation of billiard walls which reduce the problem to the socalled quantum billiard on (n + l - 1)-dimensional hyperbolic space. Several examples of billiards in the model with {pmn} non-intersecting electric branes, e.g., corresponding to hyperbolic Kac–Moody algebras, are considered. In the classical case, any of these billiards describe a never-ending oscillating behavior of scale factors while approaching to the singularity, which is either spacelike or timelike. For n = 2 the model is completely integrable in the asymptotic regime in the clasical and quantum cases.     

CLUES: A non-parametric clustering method based on local shrinking

A novel non-parametric clustering method based on non-parametric local shrinking is proposed. Each data point is transformed in such a way that it moves a specific distance toward a cluster center. The direction and the associated size of each movement are determined by the median of its K-nearest neighbors. This process is repeated until a pre-defined convergence criterion is satisfied. The optimal value of the number of neighbors is determined by optimizing some commonly used index functions that measure the strengths of clusters generated by the algorithm. The number of clusters and the final partition are determined automatically without any input parameter except the stopping rule for convergence. Experiments on simulated and real data sets suggest that the proposed algorithm achieves relatively high accuracies when compared with classical clustering algorithms.     

Constructing lattice rules based on weighted degree of exactness and worst case error

Recall that an integration rule is said to have a trigonometric degree of exactness m if it integrates exactly all trigonometric polynomials of degree ≤ m. In this paper we focus on high dimensions, say, d ? 6. We introduce three notions of weighted degree of exactness, where we use weights to characterize the anisotropicness of the integrand with respect to successive coordinate directions. Unlike in the classical unweighted setting, the minimal number of integration points needed to achieve a prescribed weighted degree of exactness no longer grows exponentially with d provided that the weights decay sufficiently fast. We present a component-by-component algorithm for the construction of a rank-1 lattice rule such that (i) it has a prescribed weighted degree of exactness, and (ii) its worst case error achieves the optimal rate of convergence in a weighted Korobov space. Then we introduce a modified, more practical, version of this algorithm which maximizes the weighted degree of exactness in each step of the construction. Both algorithms are illustrated by numerical results.     

解一类弱间断最优控制问题的一个自适应拟谱方法

针对弱间断最优控制问题,提出一种自适应拟谱方法.利用一些点序列分割时间区间为若干子区间;控制和状态函数使用分段连续多项式逼近;以数值解的收敛性为基础,证明分割时间区间的点序列可以收敛到弱间断点;依据柯西收敛原理,弱间断点位置可以由前述点序列的变化来估计,据此设计误差指示量以调整子区间和逼近多项式次数.在数值算例中,通过与两种拟谱方法比较,所提出方法在精度和效率上都有更好的表现.     

多层积分值三次样条拟插值

目的 在实际问题中,某些插值问题结点处的函数值往往是未知的,而仅仅知道一些连续等距区间上的积分值。为此提出了一种基于未知函数在连续等距区间上的积分值和多层样条拟插值技术来解决函数重构。该方法称之为多层积分值三次样条拟插值方法。方法 首先,利用积分值的线性组合来逼近结点处的函数值;然后,利用传统的三次B-样条拟插值和相应的误差函数来实现多层三次样条拟插值;最后,给出两层积分值三次样条拟插值算子的多项式再生性和误差估计。结果 选取无穷次可微函数对多层积分值三次样条拟插值方法和已有的积分值三次样条拟插值方法进行对比分析。数值实验印证了本文方法在逼近误差和数值收敛阶均稍占优。结论本文多层三次样条拟插值函数能够在整体上很好的逼近原始函数,一阶和二阶导函数。本文方法较之于已有的积分值三次样条拟插值方法具有更好的逼近误差和数值收敛阶。该方法对连续等距区间上积分值的函数重构具有普适性。     

The Unnormalized Dempster’s Rule of Combination: A New Justification from the Least Commitment Principle and Some Extensions

In the Transferable Belief Model, belief functions are usually combined using the unnormalized Dempster’s rule (also called the TBM conjunctive rule). This rule is used because of its intuitive appeal and because it has received formal justifications as opposed to the many other rules of combination that have been proposed in the literature. This article confirms the singularity of the TBM conjunctive rule by presenting a new formal justification based on (1) the canonical decomposition of belief functions, (2) the least commitment principle and (3) the requirement of having the vacuous belief function as neutral element of the combination. A similar result is also presented for the TBM disjunctive rule. Eventually, the existence of infinite families of rules having similar properties as those two rules is pointed out.     

A collocation method for boundary value problems of differential equations with functional arguments

A collocation procedure with polynomial and piecewise polynomial approximation is considered for second order functional differential equations with two side-conditions. The piecewise polynomials are taken in the classC ¹ and reduce to polynomials of increasing degree on each interval of a suitable assigned partition. Appropriate choices of the partition are made, according to the jump discontinuities in the derivatives caused by the functional argument, in order to optimize the rate of convergence.     

POPFNN-CRI(S): pseudo outer product based fuzzy neural network using the compositional rule of inference and singleton fuzzifier

A pseudo-outer product based fuzzy neural network using the compositional rule of inference and singleton fuzzifier [POPFNN-CRI(S)] is proposed in this paper. The correspondence of each layer in the proposed POPFNN-CRI(S) to the compositional rule of inference using standard T-norm and fuzzy relation gives it a strong theoretical foundation. The proposed POPFNN-CRI(S) training consists of two phases; namely: the fuzzy membership derivation phase using the novel fuzzy Kohonen partition (FKP) and pseudo Kohonen partition (PFKP) algorithms, and the rule identification phase using the novel one-pass POP learning algorithm. The proposed two-phase learning process effectively constructs the membership functions and identifies the fuzzy rules. Extensive experimental results based on the classification performance of the POPFNN-CRI(S) using the Anderson's Iris data are presented for discussion. Results show that the POPFNN-CRI(S) has taken only 15 training iterations and misclassify only three out of all the 150 patterns in the Anderson's Iris data.     

Numerische Lösung gewöhnlicher Differentialgleichungen mit Splinefunktionen

In this paper a general procedure to obtain spline approximations for the solutions of initial value problems for ordinary differential equations is presented. Several well-known spline approximation methods are included as special cases. It is common practice to partition the interval for which the initial value problem is defined into equidistant subintervals and to construct successively the spline approximation; thereby the spline function has to satisfy certain conditions at the knots. In the general procedure presented here additional knots are admitted in every subinterval. At these points which need not be equally spaced the spline approximation has to fulfill analogous conditions as at the original knots. Convergence and divergence theorems are proved; especially the influence of the additional knots on convergence and divergence of the method is investigated.     

A Self-Evolving Interval Type-2 Fuzzy Neural Network With Online Structure and Parameter Learning

This paper proposes a self-evolving interval type-2 fuzzy neural network (SEIT2FNN) with online structure and parameter learning. The antecedent parts in each fuzzy rule of the SEIT2FNN are interval type-2 fuzzy sets and the fuzzy rules are of the Takagi–Sugeno–Kang (TSK) type. The initial rule base in the SEIT2FNN is empty, and the online clustering method is proposed to generate fuzzy rules that flexibly partition the input space. To avoid generating highly overlapping fuzzy sets in each input variable, an efficient fuzzy set reduction method is also proposed. This method independently determines whether a corresponding fuzzy set should be generated in each input variable when a new fuzzy rule is generated. For parameter learning, the consequent part parameters are tuned by the rule-ordered Kalman filter algorithm for high-accuracy learning performance. Detailed learning equations on applying the rule-ordered Kalman filter algorithm to the SEIT2FNN consequent part learning, with rules being generated online, are derived. The antecedent part parameters are learned by gradient descent algorithms. The SEIT2FNN is applied to simulations on nonlinear plant modeling, adaptive noise cancellation, and chaotic signal prediction. Comparisons with other type-1 and type-2 fuzzy systems in these examples verify the performance of the SEIT2FNN.      

一种新的快速模糊规则提取方法

提出一种高效的规则提取算法,采用熵测量改进Chi-merge特征区间离散化方法,模糊划分输入空间闻.先为每个数据生成单条规则,再聚集相同前项的单条规则产生带概率属性的分类规则.提取的规则无需任何调整,应用模糊推理便可获得较理想的分类效果,同时支持增量规则更新.最后给出了新方法的性能测试结果.