Duality theory and propagation rules for higher order nets

Higher order nets and sequences are used in quasi-Monte Carlo rules for the approximation of high dimensional integrals over the unit cube. Hence one wants to have higher order nets and sequences of high quality.In this paper we introduce a duality theory for higher order nets whose construction is not necessarily based on linear algebra over finite fields. We use this duality theory to prove propagation rules for such nets. This way we can obtain new higher order nets (sometimes with improved quality) from existing ones. We also extend our approach to the construction of higher order sequences.     

Matrix-product constructions of digital nets

We present a new construction of digital nets, and more generally of (d,k,m,s)-systems, over finite fields which is an analog of the matrix-product construction of codes. Examples show that this construction can yield digital nets with better parameters compared to competing constructions.     

Digital nets with infinite digit expansions and construction of folded digital nets for quasi-Monte Carlo integration

In this paper we study quasi-Monte Carlo integration of smooth functions using digital nets. We fold digital nets over ${\mathbb{Z}}_b$ by means of the $b$-adic tent transformation, which has recently been introduced by the authors, and employ such folded digital nets as quadrature points. We first analyze the worst-case error of quasi-Monte Carlo rules using folded digital nets in reproducing kernel Hilbert spaces. Here we need to permit digital nets with “infinite digit expansions”, which are beyond the scope of the classical definition of digital nets. We overcome this issue by considering the infinite product of cyclic groups and the characters on it. We then give an explicit means of constructing good folded digital nets as follows: we use higher order polynomial lattice point sets for digital nets and show that the component-by-component construction can find good folded higher order polynomial lattice rules that achieve the optimal convergence rate of the worst-case error in certain Sobolev spaces of smoothness of arbitrarily high order.     

Universal optimality of digital nets and lattice designs

This article considers universal optimality of digital nets and lattice designs in a regression model. Based on the equivalence theorem for matrix means and majorization theory,the necessary and sufficient conditions for lattice designs being φp-and universally optimal in trigonometric function and Chebyshev polynomial regression models are obtained. It is shown that digital nets are universally optimal for both complete and incomplete Walsh function regression models under some specified conditions,and are...     

I-binomial scrambling of digital nets and sequences

The computational complexity of the integration problem in terms of the expected error has recently been an important topic in Information-Based Complexity. In this setting, we assume some sample space of integration rules from which we randomly choose one. The most popular sample space is based on Owen's random scrambling scheme whose theoretical advantage is the fast convergence rate for certain smooth functions.This paper considers a reduction of randomness required for Owen's random scrambling by using the notion of i-binomial property. We first establish a set of necessary and sufficient conditions for digital (0,s)-sequences to have the i-binomial property. Then based on these conditions, the left and right i-binomial scramblings are defined. We show that Owen's key lemma (Lemma 4, SIAM J. Numer. Anal. 34 (1997) 1884) remains valid with the left i-binomial scrambling, and thereby conclude that all the results on the expected errors of the integration problem so far obtained with Owen's scrambling also hold with the left i-binomial scrambling.     

On the approximation of smooth functions using generalized digital nets

In this paper, we study an approximation algorithm which firstly approximates certain Walsh coefficients of the function under consideration and consequently uses a Walsh polynomial to approximate the function. A similar approach has previously been used for approximating periodic functions, using lattice rules (and Fourier polynomials), and for approximating functions in Walsh Korobov spaces, using digital nets. Here, the key ingredient is the use of generalized digital nets (which have recently been shown to achieve higher order convergence rates for the integration of smooth functions). This allows us to approximate functions with square integrable mixed partial derivatives of order α>1$\alpha >1$ in each variable. The approximation error is studied in the worst case setting in the _L2$L_2$ norm. We also discuss tractability of our proposed approximation algorithm, investigate its computational complexity, and present numerical examples.     

Construction algorithms for polynomial lattice rules for multivariate integration

We introduce a new construction algorithm for digital nets for integration in certain weighted tensor product Hilbert spaces. The first weighted Hilbert space we consider is based on Walsh functions. Dick and Pillichshammer calculated the worst-case error for integration using digital nets for this space. Here we extend this result to a special construction method for digital nets based on polynomials over finite fields. This result allows us to find polynomials which yield a small worst-case error by computer search. We prove an upper bound on the worst-case error for digital nets obtained by such a search algorithm which shows that the convergence rate is best possible and that strong tractability holds under some condition on the weights.

We extend the results for the weighted Hilbert space based on Walsh functions to weighted Sobolev spaces. In this case we use randomly digitally shifted digital nets. The construction principle is the same as before, only the worst-case error is slightly different. Again digital nets obtained from our search algorithm yield a worst-case error achieving the optimal rate of convergence and as before strong tractability holds under some condition on the weights. These results show that such a construction of digital nets yields the until now best known results of this kind and that our construction methods are comparable to the construction methods known for lattice rules.

We conclude the article with numerical results comparing the expected worst-case error for randomly digitally shifted digital nets with those for randomly shifted lattice rules.

     

Construction algorithms for higher order polynomial lattice rules

Higher order polynomial lattice point sets are special types of digital higher order nets which are known to achieve almost optimal convergence rates when used in a quasi-Monte Carlo algorithm to approximate high-dimensional integrals over the unit cube. The existence of higher order polynomial lattice point sets of “good” quality has recently been established, but their construction was not addressed.We use a component-by-component approach to construct higher order polynomial lattice rules achieving optimal convergence rates for functions of arbitrarily high smoothness and at the same time–under certain conditions on the weights–(strong) polynomial tractability. Combining this approach with a sieve-type algorithm yields higher order polynomial lattice rules adjusting themselves to the smoothness of the integrand up to a certain given degree. Higher order Korobov polynomial lattice rules achieve analogous results.     

Quadrature rules based on partial fraction expansions

Quadrature rules are typically derived by requiring that all polynomials of a certain degree be integrated exactly. The nonstandard issue discussed here is the requirement that, in addition to the polynomials, the rule also integrates a set of prescribed rational functions exactly. Recurrence formulas for computing such quadrature rules are derived. In addition, Fejér's first rule, which is based on polynomial interpolation at Chebyshev nodes, is extended to integrate also rational functions with pre-assigned poles exactly. Numerical results, showing a favorable comparison with similar rules that have been proposed in the literature, are presented. An error analysis of a representative test problem is given. This revised version was published online in June 2006 with corrections to the Cover Date.     

A comprehensive approach of planning and scheduling based on petri nets

The main results available on the use of black-and-white Petri nets for modelling, planning and scheduling manufacturing systems are presented. In the first part of the paper, the basics of Petri nets necessary to understand the subsequent presentation are introduced. Particular attention is paid to event graphs, a particular type of Petri nets used for modelling and evaluating ratio-driven systems. The second part of the paper is devoted to ratio-driven systems, their modelling and their scheduling. Job-shops, assembly systems, and KANBAN systems are used to illustrate this section. Finally, the general case is investigated of manufacturing systems subject to changing demands. An approach based on conflict-free Petri nets with input and output transitions is proposed for planning and scheduling this type of system.     

TSD based framework for mining the induction rules

Sex determination mainly encompasses two aspects: genotypic sex determination (GSD) and temperature-dependent sex determination (TSD). Genotypic sex determination performs its task by observing the presence of sex chromosomes. In many reptiles sex determination is greatly influenced by the environmental conditions such as temperature of the nest, weight and size of eggs. A nature inspired algorithm which mimics the mechanism of temperature dependent sex determination (TSD) has been introduced for mining the classification rules from datasets. A comparison of proposed TSD algorithm with other well known rule induction algorithms like PRISM, C4.5, 1-R, CN2, and NN has been evaluated on some bench mark datasets.     

The Construction of cubature rules for multivariate highly oscillatory integrals

We present an efficient approach to evaluate multivariate highly oscillatory integrals on piecewise analytic integration domains. Cubature rules are developed that only require the evaluation of the integrand and its derivatives in a limited set of points. A general method is presented to identify these points and to compute the weights of the corresponding rule.

The accuracy of the constructed rules increases with increasing frequency of the integrand. For a fixed frequency, the accuracy can be improved by incorporating more derivatives of the integrand. The results are illustrated numerically for Fourier integrals on a circle and on the unit ball, and for more general oscillators on a rectangular domain.

     

A universal compiler system based on production rules

This paper describes a compiler system which makes use of production rules for the translation. The source language syntax is defined in terms of a phrase structure grammar. Semantic rules are provided by an extension of the production rules, and special symbols are introduced for this purpose. Recognition of symbol strings is facilitated by a special syntactic filter routine. An example of a simple macro compiler is given to illustrate the basic concepts of the system.     

Vehicle dispatching system based on Taguchi-tuned fuzzy rules

This paper presents the development of a dispatching system for a fleet of automated guided vehicles in a flexible manufacturing environment which is based on a hybrid Fuzzy–Taguchi approach. A fuzzy decision-making system emulates the human behavior necessary for multi-objective directed decision making in a dynamically evolving environment. A statistical approach based on the Taguchi method tunes the fuzzy rules to achieve near optimal performance. Simulation results demonstrate the effectiveness of this marriage of computational tools in dealing with the well-known NP-complete scheduling problem.     

Symmetric quadrature rules for tetrahedra based on a cubic close-packed lattice arrangement

A family of quadrature rules for integration over tetrahedral volumes is developed. The underlying structure of the rules is based on the cubic close-packed (CCP) lattice arrangement using 1, 4, 10, 20, 35, and 56 quadrature points. The rules are characterized by rapid convergence, positive weights, and symmetry. Each rule is an optimal approximation in the sense that lower-order terms have zero contribution to the truncation error and the leading-order error term is minimized. Quadrature formulas up to order 9 are presented with relevant numerical examples.     

Randomized Smolyak algorithms based on digital sequences for multivariate integration

Gunther Leobacher ^{In this paper, we consider Smolyak algorithms based on quasi-MonteCarlo rules for high-dimensional numerical integration. Thequasi-Monte Carlo rules employed here use digital (t, , ß,, d)-sequences as quadrature points. We consider the worst-caseerror for multivariate integration in certain Sobolev spacesand show that our quadrature rules achieve the optimal rateof convergence. By randomizing the underlying digital sequences,we can also obtain a randomized Smolyak algorithm. The boundon the worst-case error holds also for the randomized algorithmin a statistical sense. Further, we also show that the randomizedalgorithm is unbiased and that the integration error can beapproximated as well.}