knowledge based defuzzification abstract we suggest a new approach to the problem of defuzzification this approach is a knowledge based approach in that it uses knowledge in terms of combinability function to help to more intelligently guide the defuzzification process this approach makes use of a view of the defuzzification process as a kind of clustering problem and uses the basic idea used in the mountain clustering method

on the design of an optimal quantizer the problem of designing an optimal quantizer with a fixed number of levels for a wide class of error weighting functions and an arbitrary distribution function is discussed the existence of an optimal quantizer is proved and a two stage algorithm for its design is suggested in this algorithm at the first stage lloyd s iterative method i is applied for reducing the region where at the second stage the search for an optimal quantizer is performed using a hybrid of the dynamic programming algorithm and the lloyd max algorithm which achieves the absolute optimality of dynamic programming with much less computational effort for a continuous distribution with log concave density and an increasing convex weighting function of the absolute quantization error a reliable method is presented to compute the parameters of the optimal quantizer with a known precision using a generalization either of lloyd s method i or of the lloyd max algorithm

sufficient conditions for uniqueness of a locally optimal quantizer for a class of convex error weighting functions sufficient conditions are presented for uniqueness of a locally optimal quantizer being a stationary point of the quantization distortion measure e f x eta the expected value of an error weighting function f x eta where x is a random variable to be quantized where the probability density function p x describing x is continuous and positive on some finite or infinite interval and zero outside it and where eta is the quantization error the function f x eta is assumed convex and symmetric in eta and zero only for eta 0 it is shown that in the cases of f x eta eta 2 and f x eta eta the simple condition of concavity of in p x is sufficient for uniqueness of a locally optimal quantizer

asymptotics of the geometric mean error in the quantization for product measures on moran sets abstract assuming a separation property for moran sets we give a sufficient condition for the s 0 dimensional upper and lower quantization coefficient for of order zero to be both positive and finite when the quantization dimension exists and equals s 0 for certain product measures associated with multiscale moran sets we determine the exact value s 0 of the quantization dimension of order zero and present a subclass of such measures for which the s 0 dimensional upper and lower quantization coefficient are both positive and finite several examples are constructed to illustrate the main results

a space quantization method for numerical integration abstract we propose a new method sqm for numerical integration of c functions 0 2 defined on a convex subset c of rd with respect to a continuous distribution it relies on a space quantization of c by a n tuplex x1 xn cn fd is approximated by a weighted sum of the f xi s the integration error bound depends on the distortion en x of the voronoi tessellation of x this notion comes from information theoretists its main properties existence of a minimizing n tuple in cn asymptotics of mincn en as n are presented for a wide class of measures a simple stochastic optimization procedure is proposed to compute in any dimension d x and the characteristics of its voronoi tessellation some new results on the competitive learning vector quantization algorithm when 2 are obtained as a by product some tests simulations and provisional remarks are proposed as a conclusion

global convergence and empirical consistency of the generalized lloyd algorithm the generalized lloyd algorithm for vector quantizer design is analyzed as a descent algorithm for nonlinear programming a broad class of convex distortion functions is considered and any input distribution that has no singular continuous part is allowed a well known convergence theorem is applied to show that iterative applications of the algorithm produce a sequence of quantizers that approaches the set of fixed point quantizers the methods of the theorem are extended to sequences of algorithms yielding results on the behavior of the algorithm when an unknown distribution is approximated by a training sequence of observations it is shown that as the length of the training sequence grows large that 1 fixed point quantizers for the training sequence approach the set of fixed point quantizers for the true distribution and 2 limiting quantizers produced by the algorithm with the training sequence distribution perform no worse than limiting quantizers produced by the algorithm with the true distribution

functional quantization of gaussian processes quantization consists in studying the lr error induced by the approximation of a random vector x by a vector quantized version taking a finite number n of values for rm valued random vectors the theory and practice is quite well established and in particular the asymptotics as n of the resulting minimal quantization error for nonsingular distributions is well known it behaves like c x r m n 1 m this paper is a transposition of this problem to random vectors in an infinite dimensional hilbert space and in particular to stochastic processes xt t 0 1 viewed as l2 0 1 dt valued random vectors for gaussian vectors and the l2 error we present detailed results for stationary and optimal quantizers we further establish a precise link between the rate problem and shannon kolmogorov s entropy of x this allows us to compute the exact rate of convergence to zero of the minimal l2 quantization error under rather general conditions on the eigenvalues of the covariance operator typical rates are o logn a a 0 they are obtained for instance for the fractional brownian motion and the fractional ornstein uhlenbeck process the exponent a is closely related with the l2 regularity of the process

quantization dimension for condensation systems ii the geometric mean error abstract let f i 1 n be a family of similitudes on r 1 satisfying the strong separation condition and the self similar measure associated with f i 1 n and a probability vector t 1 t n let be the attracting measure of a condensation system associated with f i 1 n and a probability vector p 0 p 1 p n we establish a relationship between the quantization dimension of and its mass distribution on cylinder sets

a multiple grid adaptive integral method for multi region problems a multiple grid extension of the adaptive integral method aim is presented for fast analysis of scattering from piecewise homogeneous structures the proposed scheme accelerates the iterative method of moments solution of the pertinent surface integral equations by employing multiple auxiliary cartesian grids if the structure of interest is composed of k homogeneous regions it introduces k different auxiliary grids it uses the k th auxiliary grid first to determine near zones for the basis functions and then to execute aim projection propagation interpolation and near zone pre correction stages in the k th region thus the aim stages are executed a total of k times using different grids and different groups of basis functions the proposed multiple grid aim scheme requires a total of o n nz near k n k clogn k c operations per iteration where n nz near denotes the total number of near zone interactions in all regions and n k c denotes the number of nodes of the k th cartesian grid numerical results validate the method s accuracy and reduced complexity for large scale canonical structures with large numbers of regions up to 106 degrees of freedom and 103 regions moreover an investigation of hf band wave propagation in a loblolly pine forest model demonstrates the method s generality and practical applicability multiple grid aim accelerated simulations with various tree models show that higher fidelity models for the trunk material and branch geometry are needed for accurate calculation of horizontally polarized field propagation while lower fidelity models can be satisfactory for analyzing vertically polarized field propagation

singularity extraction technique for integral equation methods with higher order basis functions on plane triangles and tetrahedra a numerical solution of integral equations typically requires calculation of integrals with singular kernels the integration of singular terms can be considered either by purely numerical techniques e g duffy s method polar co ordinate transformation or by singularity extraction in the latter method the extracted singular integral is calculated in closed form and the remaining integral is calculated numerically this method has been well established for linear and constant shape functions in this paper we extend the method for polynomial shape functions of arbitrary order we present recursive formulas by which we can extract any number of terms from the singular kernel defined by the fundamental solution of the helmholtz equation or its gradient and integrate the extracted terms times a polynomial shape function in closed form over plane triangles or tetrahedra the presented formulas generalize the singularity extraction technique for surface and volume integral equation methods with high order basis functions numerical experiments show that the developed method leads to a more accurate and robust integration scheme and in many cases also a faster method than for example duffy s transformation copyright 2003 john wiley sons ltd

asymptotics of optimal quantizers for some scalar distributions we obtain semi closed forms for the optimal quantizers of some families of one dimensional probability distributions they yield the first examples of non log concave distributions for which uniqueness holds we give two types of applications of these results one is a fast computation of numerical approximations of one dimensional optimal quantizers and their use in a multidimensional framework the other is some asymptotics of the standard empirical measures associated to the optimal quantizers in terms of distribution function laplace transform and characteristic function moreover we obtain the rate of convergence in the bucklew wise theorem and finally the asymptotic size of the voronoi tessels

numerical evaluation of singular and near singular potential integrals a simple and efficient numerical procedure using a singularity cancellation scheme is presented for evaluating singular and near singular potential integrals with 1 r singularities the procedure not only has several advantages over singularity subtraction methods but also improves on some aspects of other singularity cancellation methods such as polar and duffy transformations a theoretical analysis is presented for triangles quadrilaterals tetrahedrons bricks and prisms and numerical results are presented for triangles and prisms

van der corput sequences kakutani transforms and one dimensional numerical integration abstract it is shown that many functions can be empirically integrated with the van der corput sequences with an o ln n n or o 1 n rate according to their smoothness the first rate derives from a new koksma hlawka like inequality using the diaphony then we show that as for the discrepancy estimates these properties still hold when considering an extension of these sequences generated by ergodic transforms

reduction of singular surface integrals to nonsingular line integrals in integral equations for planar geometries a novel procedure is presented for the evaluation of matrix elements of the tensor green function with rao wilton glisson basis functions appearing in surface integral equations in electromagnetics the procedure at this point applicable to planar geometries or geometries composed of parallel planar sheets reduces 4 d surface integrals with singular integrands to line integrals over triangle edges with regular integrands the main advantage of the derived expressions is that they offer simplicity and easily controllable accuracy achieved at a computational cost significantly lower than for previously considered techniques in particular the conventional singularity subtraction method

reduction of volume integrals to nonsingular surface integrals for matrix elements of tensor and vector green functions of maxwell equations a procedure for the evaluation of matrix elements of tensor and vector green functions with linear basis functions appearing in volumetric integral equations in electromagnetics is presented the procedure reduces six dimensional volumetric integrals with singular integrands to four dimensional surface integrals with nonsingular integrands hence it offers simplicity and accuracy achieved at low computational cost

fast volumetric integral equation solver for acoustic wave propagation through inhomogeneous media elements are described of a volumetric integral equation based algorithm applicable to accurate large scale simulations of scattering and propagation of sound waves through inhomogeneous media the considered algorithm makes possible simulations involving realistic geometries characterized by highly subwavelength details large density contrasts and described in terms of several million unknowns the algorithm achieves its competitive performance characterized by o nlogn solution complexity and o n memory requirements where n is the number of unknowns through a fast and nonlossy fast fourier transform based matrix compression technique the adaptive integral method previously developed for solving large scale electromagnetic problems because of its ability of handling large problems with complex geometries the developed solver may constitute an efficient and high fidelity numerical simulation tool for calculating acoustic field distributions in anatomically realistic models e g in investigatin

convergence of markovian stochastic approximation with discontinuous dynamics this paper is devoted to the convergence analysis of stochastic approximation algorithms of the form theta n 1 theta n gamma n 1 h theta n x n 1 where left theta n n in mathbb n right is an mathbb r d valued sequence left gamma n n in mathbb n right is a deterministic stepsize sequence and left x n n in mathbb n right is a controlled markov chain we study the convergence under weak assumptions on smoothness in theta of the function theta mapsto h theta x it is usually assumed that this function is continuous for any x in this work we relax this condition our results are illustrated by considering stochastic approximation algorithms for adaptive quantile estimation and a penalized version of the vector quantization

asymptotic quantization error of continuous signals and the quantization dimension extensions of the limiting qnanfizafion error formula of bennet are proved these are of the form d s k n f n beta b where n is the number of output levels d s k n f is the s th moment of the metric distance between quantizer input and output beta b 0 k s beta is the signal space dimension and f is the signal distribution if a suitably well behaved k dimensional signal density f x exists b b s k int f rho x dx 1 rho rho k s k and b s k does not depend on f for k 1 s 2 this reduces to bennett s formula if f is the cantor distribution on 0 1 0 and this k equals the fractal dimension of the cantor set 12 13 random quantization optimal quantization in the presence of an output information constraint and quantization noise in high dimensional spaces are also investigated

on the existence of optimal quantizers corresp the existence of optimal n level quantizers is demonstrated for several classes of quantizers including block quantizers memoryless quantizers and uniform step size quantizers no assumptions whatsoever are made regarding the input probability distribution the distortion measure considered generalizes many of the commonly used error based distortion measures including mean squared error distortion and r th law distortion

problem based optimal scenario generation and reduction in stochastic programming scenarios are indispensable ingredients for the numerical solution of stochastic programs earlier approaches to optimal scenario generation and reduction are based on stability arguments involving distances of probability measures in this paper we review those ideas and suggest to make use of stability estimates based only on problem specific data for linear two stage stochastic programs we show that the problem based approach to optimal scenario generation can be reformulated as best approximation problem for the expected recourse function which in turn can be rewritten as a generalized semi infinite program we show that the latter is convex if either right hand sides or costs are random and can be transformed into a semi infinite program in a number of cases we also consider problem based optimal scenario reduction for two stage models and optimal scenario generation for chance constrained programs finally we discuss problem based scenario generation for the classical newsvendor problem