fuzzy rule selection by multi objective genetic local search algorithms and rule evaluation measures in data mining this paper shows how a small number of simple fuzzy if then rules can be selected for pattern classification problems with many continuous attributes our approach consists of two phases candidate rule generation by rule evaluation measures in data mining and rule selection by multi objective evolutionary algorithms in our approach first candidate fuzzy if then rules are generated from numerical data and prescreened using two rule evaluation measures i e confidence and support in data mining then a small number of fuzzy if then rules are selected from the prescreened candidate rules using multi objective evolutionary algorithms in rule selection we use three objectives maximization of the classification accuracy minimization of the number of selected rules and minimization of the total rule length thus the task of multi objective evolutionary algorithms is to find a number of non dominated rule sets with respect to these three objectives the main contribution of this paper is to propose an idea of utilizing the two rule evaluation measures as prescreening criteria of candidate rules for fuzzy rule selection an arbitrarily specified number of candidate rules can be generated from numerical data for high dimensional pattern classification problems through computer simulations we demonstrate that such a prescreening procedure improves the efficiency of our approach to fuzzy rule selection we also extend a multi objective genetic algorithm moga in our former studies to a multi objective genetic local search mogls algorithm where a local search procedure adjusts the selection i e inclusion or exclusion of each candidate rule furthermore a learning algorithm of rule weights i e certainty factors is combined with our mogls algorithm such extensions to our moga for fuzzy rule selection are another contribution of this paper

empirical study on learning in fuzzy systems by rice taste analysis abstract the aim of this paper is to examine the ability of trainable fuzzy systems as approximators of non linear mappings by computer simulations on real life data fuzzy if then rules with non fuzzy singletons in the consequent part are adjusted by a gradient descent method in fuzzy systems after examining the ability of fuzzy systems for numerical examples we apply them to the modelling of the relation among six factors in the sensory test on rice taste by computer simulations based on a random subsampling technique it is shown that the performance of fuzzy systems is comparable to that of neural networks it is also shown that pre specified conditions such as a fuzzy partition initial fuzzy if then rules and the number of iterations have a significant effect on the performance of trained fuzzy systems

numerical solutions to statistical moment equations of groundwater flow in nonstationary bounded heterogeneous media in this paper we investigate the combined effect of nonstationarity in log hydraulic conductivity and the presence of boundaries on flow in heterogeneous aquifers we derive general equations governing the statistical moments of hydraulic head for steady state flow by perturbation expansions due to their mathematical complexity we solve the moment equations by the numerical technique of finite differences the numerical approach has flexibility in handling moderately irregular geometry different boundary conditions various trends in the mean log hydraulic conductivity spatial variabilities in the magnitude and direction of mean flow and different covariance functions all of which are important factors to consider for real world applications the effect of boundaries on the first two statistical moments involving head is strong and persistent for example in the case of stationary log hydraulic conductivity the head variance is always finite in a bounded domain while the head variance may be infinite in an unbounded domain as in many other stochastic models the statistical moment equations are derived under the assumption that the variance of log hydraulic conductivity is small accounting for a spatially varying large scale trend in the log hydraulic conductivity field reduces the variance of log hydraulic conductivity although this makes the conductivity field nonstationary and significantly increases the mathematical complexity in the problem it justifies the small variance assumption for many aquifers

on the 2 tuples based genetic tuning performance for fuzzy rule based classification systems in imbalanced data sets when performing a classification task we may find some data sets with a different class distribution among their patterns this problem is known as classification with imbalanced data sets and it appears in many real application areas for this reason it has recently become a relevant topic in the area of machine learning the aim of this work is to improve the behaviour of fuzzy rule based classification systems frbcss in the framework of imbalanced data sets by means of a tuning step specifically we adapt the 2 tuples based genetic tuning approach to classification problems showing the good synergy between this method and some frbcss our empirical results show that the 2 tuples based genetic tuning increases the performance of frbcss in all types of imbalanced data furthermore when the initial rule base built by a fuzzy rule learning methodology obtains a good behaviour in terms of accuracy we achieve a higher improvement in performance for the whole model when applying the genetic 2 tuples post processing step this enhancement is also obtained in the case of cooperation with a preprocessing stage proving the necessity of rebalancing the training set before the learning phase when dealing with imbalanced data

stochastic modeling of macrodispersion in heterogeneous porous media a method for the prediction of dispersion processes of inert solutes occurring in heterogeneous porous media is presented its main features are as follows 1 the velocity field is assumed to be a space random function its moments are expressed through the physical parameters of the log transmissivity and head random fields by linearizing the flow equation the random velocity being a linear function of the head and log transmissivity which are assumed to be jointly multivariate normal mvn is thus also mvn its pdf is completely defined by its first two moments 2 the statistics of the dispersion process are obtained by the particle tracking method through monte carlo simulations which are based on gaussian conditioning the method is applied to the borden natural gradient tracer test freyberg 1986 showing favorable results

an accurate cog defuzzifier design using lamarckian co adaptation of learning and evolution this paper proposes a design technique of optimal center of gravity cog defuzzifier using the lamarckian coadaptation of learning and evolution the proposed cog defuzzifier is specified by various design parameters such as the centers widths and modifiers of mfs the design parameters are adjusted with the lamarckian co adaptation of learning and evolution where the learning performs a local search of design parameters in an individual cog defuzzifier but the evolution performs a global search of design parameters among a population of various cog defuzzifiers this co adaptation scheme allows to evolve much faster than the non learning case and gives a higher possibility of finding an optimal solution due to its wider searching capability an application to the truck backer upper control problem of the proposed co adaptive design method of cog defuzzifier is presented the approximation ability and control performance are compared with those of the conventionally simplified cog defuzzifier in terms of the fuzzy logic controller s approximation error and the average tracing distance respectively

three dimensional stochastic analysis of macrodispersion in aquifers the dispersive mixing resulting from complex flow in three dimensionally heterogeneous porous media is analyzed using stochastic continuum theory stochastic solutions of the perturbed steady flow and solute transport equations are used to construct the macroscopic dispersive flux and evaluate the resulting macrodispersivity tensor in terms of a three dimensional statistically anisotropic input covariance describing the hydraulic conductivity with a statistically isotropic input covariance the longitudinal macrodispersivity is convectively controlled but the transverse macrodispersivity is proportional to the local dispersivity and is several orders of magnitude smaller than the longitudinal term with an arbitrarily oriented anisotropic conductivity covariance all components of the macrodispersivity tensor are convectively controlled and the ratio of transverse to longitudinal dispersivity is of the order of 10 1 in this case the off diagonal components of the dispersivity tensor are significant being numerically larger than the diagonal transverse terms and the transverse dispersion process can be highly anisotropic dispersivities predicted by the stochastic theory are shown to be consistent with controlled field experiments and monte carlo simulations the theory which treats the asymptotic condition of large displacement indicates that a classical gradient transport fickian relationship is valid for large scale displacements

a dynamically bi orthogonal method for time dependent stochastic partial differential equations i derivation and algorithms we propose a dynamically bi orthogonal method dybo to solve time dependent stochastic partial differential equations spdes the objective of our method is to exploit some intrinsic sparse structure in the stochastic solution by constructing the sparsest representation of the stochastic solution via a bi orthogonal basis it is well known that the karhunen loeve expansion kle minimizes the total mean squared error and gives the sparsest representation of stochastic solutions however the computation of the kl expansion could be quite expensive since we need to form a covariance matrix and solve a large scale eigenvalue problem the main contribution of this paper is that we derive an equivalent system that governs the evolution of the spatial and stochastic basis in the kl expansion unlike other reduced model methods our method constructs the reduced basis on the fly without the need to form the covariance matrix or to compute its eigendecomposition in the first part of our paper we introduce the derivation of the dynamically bi orthogonal formulation for spdes discuss several theoretical issues such as the dynamic bi orthogonality preservation and some preliminary error analysis of the dybo method we also give some numerical implementation details of the dybo methods including the representation of stochastic basis and techniques to deal with eigenvalue crossing in the second part of our paper 11 we will present an adaptive strategy to dynamically remove or add modes perform a detailed complexity analysis and discuss various generalizations of this approach an extensive range of numerical experiments will be provided in both parts to demonstrate the effectiveness of the dybo method

comments on the benchmarks in a proposal for improving the accuracy of linguistic modeling and related articles in the above paper by cordon and herrara ieee trans fuzzy syst vol 8 p 335 44 2000 the so called accurate linguistic modeling alm method was proposed to improve the accuracy of linguistic fuzzy models a number of examples are given to demonstrate the benefits of the approach we show that 1 these examples are not suitable as benchmarks or demonstrators of nonlinear modeling techniques and 2 better results can be obtained by using both standard regression tools as well as other fuzzy modeling techniques we argue that benchmark examples that are used in articles to demonstrate the effectiveness of fuzzy modeling techniques should be selected with great care critical analysis of the results should be made and linear models should be regarded as a lower bound on the acceptable performance

identifying a fuzzy model by using the bipartite membership functions the gradient descent method and genetic algorithms have been widely used to refine fuzzy models constructed for the given data this paper approaches from another viewpoint to adjusting the fuzzy model built for the preprocessed data the membership functions defined for each premise variable are equally distributed and fixed in the transformed domain to better identify the fuzzy model either the transformation functions or consequent parts of fuzzy rules or both need to be optimized since adjusting a rule to satisfy one pattern may deteriorate the others performance and result in a lengthy tuning process we then treat each triangular membership function as two disjoint ones such that each fuzzy rule is divided into mutually independent rules this in turn benefits the refinement of consequent parts in the fuzzy rules since adjusting a rule will not affect the others not only the simulation results from the proposed model will be demonstrated but also the results from the conventional approaches will be given for comparisons we use the least squared method to calculate the desired consequent real numbers for the data located in the same region of transformed domain the conformity of the after tuned consequent parts in the fuzzy rules with the desired values further verifies the effectiveness of the presented methodology

stability and non normality of the k e equations abstract the analytical and numerical solutions of the equations of the k e turbulence model are analyzed under certain conditions on the boundary values and the interior values of k and e the analytical and numerical solutions are bounded if the steady state solution is obtained numerically by a runge kutta time stepping method then severe constraints on the time step and the non normality of the jacobian matrix make the convergence very slow the simplifications and conclusions are supported by data from a numerical solution of flow over a flat plate

a fuzzy logic based approach to qualitative modeling this paper discusses a general approach to quali tative modeling based on fuzzy logic the method of qualitative modeling is divided into two parts fuzzy modeling and linguistic approximation it proposes to use a fuzzy clustering method fuzzy c means method to identify the structure of a fuzzy model to clarify the advantages of the proposed method it also shows some examples of modeling among them a model of a dynamical process and a model of a human operator s control action

applicability of the fuzzy operators in the design of fuzzy logic controllers a study of the different roles played by the fuzzy operators in fuzzy control is developed in this paper the behavior of a very large amount of fuzzy operators is analyzed and a comparison of the accuracy of many fuzzy logic controllers designed by means of these operators is carried out in order to do that a comparison methodology is defined and two fuzzy control applications are selected the inverted pendulum problem and the fuzzy modeling of the real curve y x

ten years of genetic fuzzy systems current framework and new trends fuzzy systems have demonstrated their ability to solve different kinds of problems in various application domains currently there is an increasing interest to augment fuzzy systems with learning and adaptation capabilities two of the most successful approaches to hybridise fuzzy systems with learning and adaptation methods have been made in the realm of soft computing neural fuzzy systems and genetic fuzzy systems hybridise the approximate reasoning method of fuzzy systems with the learning capabilities of neural networks and evolutionary algorithms the objective of this paper is to provide an account of genetic fuzzy systems with special attention to genetic fuzzy rule based systems after a brief introduction to models and applications of genetic fuzzy systems the field is overviewed new trends are identified a critical evaluation of genetic fuzzy systems for fuzzy knowledge extraction is elaborated and open questions that remain to be addressed in the future are raised the paper also includes some of the key references required to quickly access implementation details of genetic fuzzy systems

stochastic analysis of head spatial variability in bounded rectangular heterogeneous aquifers groundwater flow through bounded rectangular aquifers is analyzed in a stochastic framework new analytical expressions of the head covariance and variance and the log transmissivity head cross covariance are derived for the simplified first order problem effects of different boundary conditions the aquifer size and the log transmissivity correlation scale upon head moments are studied the results are compared to analytical half plane solutions demonstrating significant differences in several cases finally the problem is solved numerically using the monte carlo simulation method the realizations of the log transmissivity field are generated in two different ways through comparisons with the analytical results the accuracy of the numerical methods is shown to be excellent

perturbation based moment equation approach for flow in heterogeneous porous media applicability range and analysis of high order terms we present detailed comparisons between high resolution monte carlo simulation mcs and low order numerical solutions of stochastic moment equations smes for the first and second statistical moments of pressure the objective is to quantify the difference between the predictions obtained from mcs and sme natural formations with high permeability variability and large spatial correlation scales are of special interest for underground resources e g oil and water consequently we focus on such formations we investigated fields with variance of log permeability y2 from 0 1 to 3 0 and correlation scales normalized by domain length of 0 05 to 0 5 in order to avoid issues related to statistical convergence and resolution level we used 9000 highly resolved realizations of permeability for mcs we derive exact discrete forms of the statistical moment equations formulations based on equations written explicitly in terms of permeability k based and log transformed permeability y based are considered the discrete forms are applicable to systems of arbitrary variance and correlation scales however equations governing a particular statistical moment depend on higher moments thus while the moment equations are exact they are not closed in particular the discrete form of the second moment of pressure includes two triplet terms that involve log permeability or permeability and pressure we combined mcs computations with full discrete sme equations to quantify the importance of the various terms that make up the moment equations we show that second moment solutions obtained using a low order y based sme formulation are significantly better than those from k based formulations especially when y2 1 as a result y based formulations are preferred the two triplet terms are complex functions of the variance level and correlation length the importance contribution of these triplet terms increases dramatically as y2 increases above one we also show that one of the triplet terms is much more important than the other when comparing k based mcs with y based sme model differences must be taken into consideration these differences model errors are due to embedded assumptions and differences in implementing the discrete forms of the equations

effect of rule weights in fuzzy rule based classification systems this paper examines the effect of rule weights in fuzzy rule based classification systems each fuzzy if then rule in our classification system has antecedent linguistic values and a single consequent class we use a fuzzy reasoning method based on a single winner rule in the classification phase the winner rule for a new pattern is the fuzzy if then rule that has the maximum compatibility grade with the new pattern when we use fuzzy if then rules with certainty grades the winner is determined as the rule with the maximum product of the compatibility grade and the certainty grade in this paper the effect of rule weights is illustrated by drawing classification boundaries using fuzzy if then rules with without certainty grades it is also shown that certainty grades play an important role when a fuzzy rule based classification system is a mixture of general rules and specific rules through computer simulations we show that comprehensible fuzzy rule based systems with high classification performance can be designed without modifying the membership functions of antecedent linguistic values when we use fuzzy if then rules with certainty grades

a phase error analysis of multigrid methods for hyperbolic equations in this paper we study the effects of the coarse grid correction process on multigrid convergence for hyperbolic problems in one and two dimensions we approach this from the perspective of phase error which allows us to exploit the hyperbolic nature of the underlying pde in particular we consider three combinations of coarse grid operators and coarse grid solution approaches 1 inexact coarse grid solve with direct discretization 2 exact coarse grid solve with direct discretization and 3 exact coarse grid solve with galerkin coarse grid operator for all these approaches we show that the convergence behavior of multigrid can be precisely described by the phase error analysis of the coarse grid correction matrix and we verify our results by numerical examples in one and two dimensions

a hierarchical fuzzy clustering approach to fuzzy modeling abstract this paper introduces a new method for fuzzy modeling based on a hierarchical fuzzy clustering scheme the method consists of a sequence of steps aiming towards developing a takagi sugeno ts fuzzy model of optimal structure where the fuzzy sets in the premise part are of gaussian type starting from an initial ordinary fuzzy partition of the input space the algorithm performs a nearest neighbor search and groups the original input training data into a number of clusters the centers of these clusters are further processed using an optimal fuzzy clustering technique which is based on the weighted fuzzy c means algorithm the resulted optimal fuzzy partition defines the number of fuzzy rules and provides an initial estimation for the system parameters which in a next step are fine tuned using the well known gradient descend algorithm the proposed method is successfully applied to three test examples where the produced fuzzy models prove to be very accurate as well as compact in size

stochastic modeling of groundwater flow by unconditional and conditional probabilities the inverse problem the inverse problem is defined here as follows determine the transmissivity at varius points given the shape and boundary of the aquifer and recharge intensity and given a set of measured log transmissivity y and head h values at a few points the log transmissivity distribution is regarded as a realization of a random function of normal and stationary unconditional probability density function pdf the solution of the inverse problem is the conditional normal pdf of y conditioned on measured h and y which is expressed in terms of the unconditional joint pdf of y and h the problem is reduced to determining the unconditional head log transmissivity covariance and head variogram for a selected y covariance which depends on a few unknown parameters this is achieved by solving a first order approximation of the flow equations the method is illustrated for an exponential y covariance and the effect of head and transmissivity measurements upon the reduction of uncertainty of y is investigated systematically it is shown that measurement of h has a lesser impact than those of y but a judicious combination may lead to significant reduction of the predicted variance of y possible applications to real aquifers are outlined

designing fuzzy inference systems from data an interpretability oriented review fuzzy inference systems fis are widely used for process simulation or control they can be designed either from expert knowledge or from data for complex systems fis based on expert knowledge only may suffer from a loss of accuracy this is the main incentive for using fuzzy rules inferred from data designing a fis from data can be decomposed into two main phases automatic rule generation and system optimization rule generation leads to a basic system with a given space partitioning and the corresponding set of rules system optimization can be done at various levels variable selection can be an overall selection or it can be managed rule by rule rule base optimization aims to select the most useful rules and to optimize rule conclusions space partitioning can be improved by adding or removing fuzzy sets and by tuning membership function parameters structure optimization is of a major importance selecting variables reducing the rule base and optimizing the number of fuzzy sets over the years many methods have become available for designing fis from data their efficiency is usually characterized by a numerical performance index however for human computer cooperation another criterion is needed the rule interpretability an implicit assumption states that fuzzy rules are by nature easy to be interpreted this could be wrong when dealing with complex multivariable systems or when the generated partitioning is meaningless for experts the paper analyzes the main methods for automatic rule generation and structure optimization they are grouped into several families and compared according to the rule interpretability criterion for this purpose three conditions for a set of rules to be interpretable are defined

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 2 computational examples in paper 1 guadagnini and neuman this issue of this two part series we presented exact nonlocal equations for first and second conditional ensemble moments of hydraulic head and flux under steady state groundwater flow in bounded randomly nonuniform domains in the presence of uncertain source and boundary terms we derived recursive approximations for these equations based on expansion in powers of a small parameter y which represents the standard estimation error of log hydraulic conductivity developed a two dimensional finite element scheme for their solution under the action of deterministic forcing terms to first order in y2 showed how to localize the conditional mean flow equations so as to yield familiar looking darcian differential equations in which however all quantities are conditional on data and therefore nonunique and proposed that traditional deterministic flow models be reinterpreted in light of these localized equations we stressed that all quantities which enter into our nonlocal and localized conditional moment equations are defined on a consistent support scale thereby obviating the need for upscaling from support to grid scale or the a priori definition of a computational grid that such a grid can therefore be defined a posteriori based on the degree of smoothness one expects the conditional moments to exhibit and that conditioning points are often spaced widely enough to render these moments relatively smooth and hence resolvable on a coarser grid than is necessary for monte carlo simulations this notwithstanding we use in this paper a high resolution grid to explore superimposed mean uniform and convergent flows in unconditional and conditional log hydraulic conductivity domains and to compare the relative accuracies of corresponding nonlocal localized and monte carlo finite element solutions whereas our nonlocal solution is nominally restricted to mildly nonuniform media with y2 1 we find that it actually yields remarkably accurate results for strongly nonuniform media with y2 at least as large as 4 localized solutions deteriorate relative to nonlocal results as one approaches some points of conditioning and singularity or as y2 increases though they require much less computer time than do nonlocal solutions localized solutions have the added disadvantage that they yield no information about predictive uncertainty

on higher order corrections to the flow velocity covariance tensor second order log fluctuating conductivity variance 2 corrections to the head and velocity covariance functions are derived for a lognormal stationary hydraulic conductivity field the fourier transform method proposed by deng et al 1993 is used extensively to obtain numerical estimates of these functions for an exponential log fluctuating conductivity covariance it is shown that the velocity covariance is insensitive to second order corrections in the head field the velocity covariance on the other hand is highly sensitive to second order corrections in the velocity when the log fluctuating conductivity variance approaches unity a closed expression is derived for a second order correction to the velocity variance when there is no second order correction to the head field the longitudinal second order correction to the velocity variance is 0 4 2 different from the first order approximation in isotropic media 1 5 2 different in a highly stratified formation and no different when the ratio of vertical to horizontal integral scales approaches infinity the second order corrections to the horizontal and vertical transverse velocity variances are 2 2 different from the first order approximations for both isotropic and anisotropic systems

a new multigrid approach to convection problems abstract a new multigrid method for convection problems is presented it is designed to overcome the problem of alignment the flow being aligned with the grid the technique employs semi coarsening in several directions simultaneously and gives rise to multiple coarser grids with the same total number of points per grid level but with different sizes in each co ordinate direction the amount of work per multigrid cycle is still o n as an example the method is applied to the nonlinear upwind differenced euler equations of gas dynamics in two dimensions convergence rates are estimated by two level fourier analysis for the linearised equations numerical experiments on the nonlinear equations confirm these estimates

cooperative evolutionary learning of linguistic fuzzy rules and parametric aggregation connectors for mamdani fuzzy systems there are two tasks in the design of linguistic fuzzy models for a concrete application the derivation of the linguistic rule base and the setup of the inference system and the defuzzification method traditionally the derivation of the linguistic rule base has been considered the most important task but the use of the appropriate aggregation connectors in the inference system and the defuzzification interface can improve the fuzzy system behavior in this paper we take in consideration this idea we propose an evolutionary learning method to learn a linguistic rule base and the parametric aggregation connectors of the inference and defuzzification in a single step the aim of this methodology is to make possible a high level of positive synergy between the linguistic rule base and the aggregation connectors improving the accuracy of the linguistic mamdani fuzzy systems our proposal has shown good results solving three different applications we introduce a statistical analysis of results for validating the model behavior on the applications used in the experimental study we must remark that we present an experimental study with a double intention a to compare the behavior of the new approach in comparison with those ones that first learn the rule base and then adapt the connectors and b to analyze the rule bases obtained with fixed aggregation connectors and with the adaptive ones for showing the changes on the consequent rules changes on labels that produce a better behavior of the linguistic model than the classic ones

multi objective hierarchical genetic algorithm for interpretable fuzzy rule based knowledge extraction a new scheme based on multi objective hierarchical genetic algorithm mohga is proposed to extract interpretable rule based knowledge from data the approach is derived from the use of multiple objective genetic algorithm moga where the genes of the chromosome are arranged into control genes and parameter genes these genes are in a hierarchical form so that the control genes can manipulate the parameter genes in a more effective manner the effectiveness of this chromosome formulation enables the fuzzy sets and rules to be optimally reduced some important concepts about the interpretability are introduced and the fitness function in the moga will consider both the accuracy and interpretability of the fuzzy model in order to remove the redundancy of the rule base proactively we further apply an interpretability driven simplification method to newborn individuals in our approach we first apply the fuzzy clustering to generate an initial rule based model then the multi objective hierarchical genetic algorithm and the recursive least square method are used to obtain the optimized fuzzy models the accuracy and the interpretability of fuzzy models derived by this approach are studied and presented in this paper we compare our work with other methods reported in the literature on four examples a synthetic nonlinear dynamic system a nonlinear static system the lorenz system and the mackey glass system simulation results show that the proposed approach is effective and practical in knowledge extraction

pictorial representations of fuzzy connectives part i cases of t norms t conorms and averaging operators part i of this paper shows a number of examples of existing t norms and t conorms which are known as good models of fuzzy set theoretic intersection and union and then the pictorial representations of t norms and t conorms are made with the aid of a computer several new examples of t norms and t conorms are also given averaging operators between min and max are summarized and their pictorial representations are also made

rule weight specification in fuzzy rule based classification systems this paper shows how the rule weight of each fuzzy rule can be specified in fuzzy rule based classification systems first we propose two heuristic methods for rule weight specification next the proposed methods are compared with existing ones through computer simulations on artificial numerical examples and real world pattern classification problems simulation results show that the proposed methods outperform the existing ones in the case of multiclass pattern classification problems with many classes

solution of the euler equations for two dimensional transonic flow by a multigrid method the assumption of potential o w implies that the o w is irrotational this is not strictly correct when shock waves are present an exact description of transonic inviscid o w requires the solution of the euler equations the numerical solution of the euler equations for steady transonic o ws is therefore a problem of great interest to the aeronautical community it also presents a testing challenge to applied mathematicians and numerical analysts

a proposal for the genetic lateral tuning of linguistic fuzzy systems and its interaction with rule selection linguistic fuzzy modeling allows us to deal with the modeling of systems by building a linguistic model which is clearly interpretable by human beings however since the accuracy and the interpretability of the obtained model are contradictory properties the necessity of improving the accuracy of the linguistic model arises when complex systems are modeled to solve this problem one of the research lines in recent years has led to the objective of giving more accuracy to linguistic fuzzy modeling without losing the interpretability to a high level in this paper a new postprocessing approach is proposed to perform an evolutionary lateral tuning of membership functions with the main aim of obtaining linguistic models with higher levels of accuracy while maintaining good interpretability to do so we consider a new rule representation scheme base on the linguistic 2 tuples representation model which allows the lateral variation of the involved labels furthermore the cooperation of the lateral tuning together with fuzzy rule reduction mechanisms is studied in this paper presenting results on different real applications the obtained results show the good performance of the proposed approach in high dimensional problems and its ability to cooperate with methods to remove unnecessary rules

nonlocal and localized analyses of conditional mean steady state flow in bounded randomly nonuniform domains 1 theory and computational approach we consider the effect of measuring randomly varying hydraulic conductivitiesk x on one s ability to predict numerically without resorting to either monte carlo simulation or upscaling steady state flow in bounded domains driven by random source and boundary terms our aim is to allow optimum unbiased prediction of hydraulic heads h x and fluxes q x by means of their ensemble moments h x c and q x c respectively conditioned on measurements of k x these predictors have been shown by neuman and orr 1993a to satisfy exactly an integrodifferential conditional mean flow equation in which q x c is nonlocal and non darcian here we develop complementary integrodifferential equations for second conditional moments of head and flux which serve as measures of predictive uncertainty obtain recursive closure approximations for both the first and second conditional moment equations through expansion in powers of a small parameter y which represents the standard estimation error of ln k x and show how to solve these equations to first order in y2 by finite elements on a rectangular grid in two dimensions in the special case where one treats k x as if it was locally homogeneous and mean flow as if it was locally uniform one obtains a localized darcian approximation q x c kc x h x c in which kc x is a space dependent conditional hydraulic conductivity tensor this leads to the traditional deterministic darcian steady state flow equation which however acquires a nontraditional meaning in that its parameters and state variables are data dependent and therefore inherently nonunique it further explains why parameter estimates obtained by traditional inverse methods tend to vary as one modifies the database localized equations yield no information about predictive uncertainty our stochastic derivation of these otherwise standard deterministic flow equations makes clear that uncertainty measures associated with estimates of head and flux obtained by traditional inverse methods are generally smaller often considerably so than measures of corresponding predictive uncertainty which can be assessed only by means of stochastic models such as ours we present a detailed comparison between finite element solutions of nonlocal and localized moment equations and monte carlo simulations under superimposed mean uniform and convergent flow regimes in two dimensions paper 1 presents the theory and computational approach and paper 2 guadagnini and neuman this issue describes unconditional and conditional computational results

stochastic analysis of nonstationary subsurface solute transport 1 unconditional moments this paper applies stochastic methods to the analysis and prediction of solute transport in heterogeneous saturated porous media partial differential equations for three unconditional ensemble moments the concentration mean concentration covariance and velocity concentration cross covariance are derived by applying perturbation techniques to the governing transport equation for a conservative solute concentration uncertainty is assumed to be the result of unmodeled small scale fluctuations in a steady state velocity field the moment expressions which describe how each moment evolves over time and space resemble the classic deterministic advection dispersion equation and can be solved using similar methods a solution procedure based on a galerkin finite element algorithm is illustrated with a hypothetical two dimensional example for this example the required steady state velocity statistics are obtained from an infinite domain spectral solution of the stochastic groundwater flow equation the perturbation solution is shown to reproduce the statistics obtained from a monte carlo simulation quite well for a natural log conductivity standard deviation of 0 5 and moderately well for a natural log conductivity standard deviation of 1 0 the computational effort required for a perturbation solution is significantly less than that required for a monte carlo solution of acceptable accuracy sensitivity analyses conducted with the perturbation approach provide qualitative confirmation of a number of results obtained by other investigators for more restrictive special cases

strategies to identify fuzzy rules directly from certainty degrees a comparison and a proposal with identification methods that learn fuzzy rules directly from certainty degrees we refer to methods that select the most promising rules from the training examples in only one pass in order to do that these methods employ a certainty measure to assess the goodness of each rule this paper aims to analyze in depth the behaviors and features of two different strategies for identifying fuzzy models from certainty degrees each of both combined with one of two well known alternatives for measuring the certainty degrees of the rules with this aim the advantages and drawbacks of each method are analyzed experimentally by considering the model error when applied to several systems besides the robustness of the results is investigated by applying the methods to noisy data as a conclusion a new method combining the best components of the previously considered methods is proposed and its results are analyzed the achieved performance in accuracy and computational cost shows the benefit of this new method

grid independent convergence of the multigrid method for first order equations the multigrid method for numerical solution of systems of first order partial differential equations is analyzed the smoothing iterations are of polynomial type and are used on all grids under certain conditions the convergence of the iterations is independent of the grid size this is explained by the propagation of smooth error modes out through the boundary and the damping of oscillatory modes an upper bound is derived on the number of multigrid v cycles needed to solve a two dimensional scalar equation numerical experiments with a simple scalar equation and the euler equations confirm the theoretical results

on choosing models for linguistic connector words for mamdani fuzzy logic systems we examine ten antecedent connector models in the framework of a singleton or nonsingleton fuzzy logic system fls to establish which models can be used in this work a usable connector model must lead to a separable firing degree that is a closed form and piecewise differentiable function of the membership function parameters and also the parameter characterizing that connector model our analysis shows that for a singleton fls where the mamdani product or mamdani minimum implication method is used all ten antecedent connector models are usable for a nonsingleton fls where the mamdani product implication method is used only one antecedent connector model is usable and for a nonsingleton fls where the mamdani minimum implication method is used none of the ten antecedent connector models is usable we also show by examples that the parameter of the antecedent connector model provides additional freedom in adjusting a fls so that the fls has the potential to achieve better performance than a fls that uses the traditional product or minimum t norm for the antecedent connections

designing and learning of adjustable quasi triangular norms with applications to neuro fuzzy systems we introduce a new class of operators called quasi triangular norms they are denoted by h and parameterized by a parameter spl nu h a sub 1 a sub 2 a sub n spl nu from the construction of function h it follows that it becomes a t norm for spl nu 0 and a dual t conorm for spl nu 1 for spl nu close to 0 function h resembles a t norm and for spl nu close to 1 it resembles a t conorm in the paper we also propose adjustable quasi implications and a new class of neuro fuzzy systems most neuro fuzzy systems proposed in the past decade employ engineering implications defined by a t norm as the minimum or product in our proposition a quasi implication i a b spl nu varies from an engineering implication t a b to corresponding s implication as spl nu goes from 0 to 1 consequently the structure of neuro fuzzy systems presented in this work is determined in the process of learning learning procedures are derived and simulation examples are presented

structure identification of fuzzy model abstract the problems of structure identification of a fuzzy model are formulated a criterion for the verification of a structure is discussed using the criterion an algorithm for identifying a structure is suggested further a successive identification algorithm of the parameters is suggested the proposed methods are applied to an example

t s fuzzy model identification with a gravitational search based hyperplane clustering algorithm in order to improve the performance of the fuzzy clustering algorithm in fuzzy space partition in the identification of the takagi sugeno t s fuzzy model a hyperplane prototype fuzzy clustering model is proposed to solve the clustering objective function which could not be handled by the gradient method as the traditional clustering method fuzzy c means does a newly developed excellent global search method which is the gravitational search algorithm gsa is employed then the gsa based hyperplane clustering algorithm gshpc is proposed and illuminated gshpc is used to partition the fuzzy space and identify premise parameters of the t s fuzzy model and orthogonal least squares is exploited to identify the consequent parameters comparative experiments are designed to verify the validity of the proposed clustering algorithm and the t s fuzzy model identification method and the results show that the new method is effective in describing a complicated nonlinear system with significantly high accuracies compared with approaches in the literature

implicit explicit methods for time dependent partial differential equations implicit explicit imex schemes have been widely used especially in conjunction with spectral methods for the time integration of spatially discretized partial differential equations pdes of diffusion convection type typically an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection term reaction diffusion problems can also be approximated in this manner in this work we systematically analyze the performance of such schemes propose improved new schemes and pay particular attention to their relative performance in the context of fast multigrid algorithms and of aliasing reduction for spectral methods for the prototype linear advection diffusion equation a stability analysis for first second third and fourth order multistep imex schemes is performed stable schemes permitting large time steps for a wide variety of problems and yielding appropriate decay of high frequency error modes are identified numerical experiments demonstrate that weak decay

analysis of the spatial structure of properties of selected aquifers data from 31 regional aquifers are analyzed to identify the horizontal spatial correlation structure of transmissivity hydraulic conductivity and storage coefficient three parameters are estimated the variance of small scale variability or nugget and the variance and integral scale range of an exponential covariance function the assumption of normality of the aforementioned geohydrologic parameters and their logarithmic transformations is also examined results are obtained which indicate that the logarithms of these properties generally pass normality tests

antecedent validity adaptation principle for fuzzy systems tuning this paper proposes an antecedent validity adaptation ava principle for fuzzy systems tuning it is suggested that the fuzzy rules may be updated with respect to the validity of antecedents this adaptation principle agrees with the human intuition and fuzzy logic reasoning the principle is first applied to table look up tl scheme to model recorded data an on line fuzzy identification algorithm is also proposed next the principle is applied to fuzzy clustering algorithm these methods are successfully applied to model nonlinear systems

a two stage adaptive stochastic collocation method on nested sparse grids for multiphase flow in randomly heterogeneous porous media a new computational method is proposed for efficient uncertainty quantification of multiphase flow in porous media with stochastic permeability for pressure estimation it combines the dimension adaptive stochastic collocation method on smolyak sparse grids and the kronrod patterson hermite nested quadrature formulas for saturation estimation an additional stage is developed in which the pressure and velocity samples are first generated by the sparse grid interpolation and then substituted into the transport equation to solve for the saturation samples to address the low regularity problem of the saturation numerical examples are presented for multiphase flow with stochastic permeability fields to demonstrate accuracy and efficiency of the proposed two stage adaptive stochastic collocation method on nested sparse grids

simulation of dispersion in heterogeneous porous formations statistics first order theories convergence of computations this paper discusses the results of numerical analysis of dispersion of passive solutes in two dimensional heterogeneous porous formations statistics of flow and transport variables the accuracy and the role of approximations implicit in existing first order theories and the convergence of computational results are investigated the results suggest that quite different rates of convergence with monte carlo runs hold for different spatial moments and that over 1000 realizations are required to stabilize second moments even for relatively mild heterogeneity sigma y 2 1 6 this has implications for the extent of the spatial domain for single realization numerical studies of the same type a comparison of the variance of plumes with the results of linear theories 0 05 sigma y 2 1 6 shows an unexpectedly broad validity field for the theoretical solution obtained from a suitable linearization of flow and transport reformulation of the same problem linearizing in tum the flow or the transport equations shows opposite deviations from the linear theory the interesting consequence is that the errors induced by linearizations in the flow or the transport equations have different signs and their effects on the moments of dispersing plumes are compensating thereby yielding consistent formulations unexpected features of the statistics of probability distributions of longitudinal and transverse velocities and travel times are also computed and discussed

a proposal for improving the accuracy of linguistic modeling we propose accurate linguistic modeling a methodology to design linguistic models that are accurate to a high degree and may be suitably interpreted this approach is based on two main assumptions related to the interpolative reasoning developed by fuzzy rule based systems a small change in the structure of the linguistic model based on allowing the linguistic rule to have two consequents associated and a different way to obtain the knowledge base based on generating a preliminary fuzzy rule set composed of a large number of rules and then selecting the subset of them best cooperating moreover we introduce two variants of an automatic design method for these kinds of linguistic models based on two well known inductive fuzzy rule generation processes and a genetic process for selecting rules the accuracy of the proposed methods is compared with other linguistic modeling techniques with different characteristics when solving of three different applications

a fast genetic method for inducting descriptive fuzzy models under certain inference mechanisms fuzzy rule bases can be regarded as extended additive models this relationship can be applied to extend some statistical techniques to learn fuzzy models from data the interest in this parallelism is twofold theoretical and practical first extended additive models can be estimated by means of the matching pursuit algorithm which has been related to support vector machines boosting and radial basis neural networks learning this connection can be exploited to better understand the learning of fuzzy models in particular the technique we propose here can be regarded as the counterpart to boosting fuzzy classifiers in the field of fuzzy modeling second since matching pursuit is very efficient in time we can expect to obtain faster algorithms to learn fuzzy rules from data we show that the combination of a genetic algorithm and the backfitting process learns faster than ad hoc methods in certain datasets

on the order of prolongations and restrictions in multigrid procedures abstract it is well known in the world of multigrid that the order of the prolongation and the order of the restriction in a multigrid method should satisfy certain conditions a rule of thumb is that the sum of the orders of the prolongation and of the restriction should at least be equal to the order of the differential equation solved in this note we show the correctness of this rule we notice that we have to distinguish between low frequency and high frequency orders for the transfer operators for the restriction the low frequency order is related with its accuracy whereas for the interpolation operator both orders are related with the accuracy of the result of the interpolation if an interpolation rule leaves all polynomials of degree k 1 invariant then both the low and the high frequency order are equal to k it is the high frequency order that plays a role in the above mentioned rule of thumb

nonlocal and localized analyses of conditional mean transient flow in bounded randomly heterogeneous porous media 1 we consider the numerical prediction of transient flow in bounded randomly heterogeneous porous media driven by random sources initial heads and boundary conditions without resorting to monte carlo simulation after applying the laplace transform to the governing stochastic flow equations we derive exact nonlocal integro differential equations for the mean and variance covariance of transformed head and flux conditioned on measured values of log conductivity y ln k approximating these conditional moment equations recursively to second order in the standard deviation y of y we solve them by finite elements for superimposed mean uniform and convergent flows in a two dimensional domain an alternative conditional mean solution is obtained through localization of the exact moment expressions the nonlocal and localized solutions are obtained using a highly efficient parallel algorithm and inverted numerically back into the time domain a comparison with monte carlo simulations demonstrates that the moment solutions are remarkably accurate for strongly heterogeneous media with y2 as large as 4 the nonlocal solution is only slightly more accurate than the much simpler localized solution but the latter does not yield information about predictive uncertainty the accuracy of each solution improves markedly with conditioning a preliminary comparison of computational efficiency suggests that both the nonlocal and localized solutions for mean head and its variance require significantly less computer time than is required for monte carlo statistics to stabilize when the same direct matrix solver is used for all three we do not presently know how using iterative solvers would have affected this conclusion this is true whether the laplace inversion and monte carlo simulations are conducted sequentially or in parallel on multiple processors and regardless of problem size the underlying exact and recursive moment equations as well as the proposed computational algorithm are valid in both two and three dimensions only the numerical implementation of our algorithm is two dimensional

can approximate reasoning be consistent abstract a fuzzy expert system with one block of rules is said to be consistent if when the input data exactly matches the antecedent of a certain rule then the final conclusion of the fuzzy expert system exactly matches the consequence of that rule we investigate the two types of approximate reasoning 1 first infer then aggregate fita and 2 first aggregate then infer fati we argue that fita can be consistent but fati is in general not consistent

convergence analysis of iterative methods by pseudodifference operators abstract the convergence of iterative methods to solve linear partial differential equations numerically is analyzed by the theory of pseudodifference operators approximate inverses are determined from the symbol of the iterative operator so that implicit and preconditioned methods are covered the behavior of the symbol for higher wave numbers describes the convergence rate for the corresponding error modes the results are applied to krylov subspace methods stationary iterations of gauss seidel type the multigrid algorithm and time stepping methods for flow problems

first order analysis of uncertainty in numerical models of groundwater flow part 1 mathematical development in part 1 of this paper first and second order analysis of uncertainty is applied to numerical models of groundwater flow the models are cast in state space form with boundary conditions and inputs that are functionally dependent but statistically independent of time using a compact matrix calculus notation first and second order taylor series expansions of the model equations are derived and used to estimate the mean and variance covariance properties of piezometric head predictions given corresponding statistics for aquifer parameters material properties initial conditions boundary conditions and inputs the mathematical results demonstrate that the prediction uncertainty is a function of the magnitude of the parameter uncertainty and sensitivity of the predictions to the parameters furthermore the first order estimate of the piezometric head is identical to the dete ministic result part 2 of this paper to be presented later will illustrate these and other results through numerous applications of the methodology

an efficient high order perturbation approach for flow in random porous media via karhunen loeve and polynomial expansions in this study we attempt to obtain higher order solutions of the means and co variances of hydraulic head for saturated flow in randomly heterogeneous porous media on the basis of the combination of karhunen loeve decomposition polynomial expansion and perturbation methods we first decompose the log hydraulic conductivity y ln ks as an infinite series on the basis of a set of orthogonal gaussian standard random variables i the coefficients of the series are related to eigenvalues and eigenfunctions of the covariance function of log hydraulic conductivity we then write head as an infinite series whose terms h n represent head of nth order in terms of y the standard deviation of y and derive a set of recursive equations for h n we then decompose h n with polynomial expansions in terms of the products of n gaussian random variables the coefficients in these series are determined by substituting decompositions of y and h n into those recursive equations we solve the mean head up to fourth order in y and the head variances up to third order in y2 we conduct monte carlo mc simulation and compare mc results against approximations of different orders from the moment equation approach based on karhunen loeve decomposition klme we also explore the validity of the klme approach for different degrees of medium variability and various correlation scales it is evident that the klme approach with higher order corrections is superior to the conventional first order approximations and is computationally more efficient than the monte carlo simulation

regularity and sparse approximation of the recursive first moment equations for the lognormal darcy problem abstract we study the darcy boundary value problem with lognormal permeability field we adopt a perturbation approach expanding the solution in taylor series around the nominal value of the coefficient and approximating the expected value of the stochastic solution of the pde by the expected value of its taylor polynomial the recursive deterministic equation satisfied by the expected value of the taylor polynomial first moment equation is formally derived well posedness and regularity results for the recursion are proved to hold in sobolev space valued holder spaces with mixed regularity the recursive first moment equation is then discretized by means of a sparse approximation technique and the convergence rates are derived

a new approach to fuzzy rule generation fuzzy extension matrix this paper proposes a new approach to fuzzy rule generation from a set of examples with fuzzy representation the new approach called fuzzy extension matrix incorporates the fuzzy entropy to search for paths and generalizes the concept of crisp extension matrix by discussing paths of the fuzzy extension matrix a new heuristic algorithm for generating fuzzy rules is introduced compared with the crisp extension matrix the proposed method has the capability of handling fuzzy representation and tolerating noisy data or missing data a case study shows that the proposed heuristic algorithm partially inherits the advantages from the crisp case such as simplicity of rules and high learning accuracy the proposed approach offers a new practical way to automatically acquire imprecise knowledge

development of a systematic methodology of fuzzy logic modeling this paper proposes a systematic methodology of fuzzy logic modeling for complex system modeling it has a unified parameterized reasoning formulation an improved fuzzy clustering algorithm and an efficient strategy of selecting significant system inputs and their membership functions the reasoning mechanism introduces 4 parameters whose variation provides a continuous range of inference operation as a result we are no longer restricted to standard extremes in any step of reasoning the fuzzy model itself can then adjust the reasoning process by optimizing the inference parameters based on input output data the fuzzy rules are generated through fuzzy c means fcm clustering major bottlenecks are addressed and analytical solutions are suggested we also address the classification process to extend the derived fuzzy partition to the entire output space in order to select suitable input variables among a finite number of candidates unlike traditional approaches we suggest a new strategy through which dominant input parameters are assigned in one step and no iteration process is required furthermore a clustering technique called fuzzy fine clustering is introduced to assign the input membership functions in order to evaluate the proposed methodology two examples a nonlinear function and a gas furnace dynamic procedure are investigated in detail the significant improvement of the model is concluded compared to other fuzzy modeling approaches

a monte carlo assessment of eulerian flow and transport perturbation models monte carlo studies of flow and transport in two dimensional synthetic conductivity fields are employed to evaluate first order flow and eulerian transport theories hydraulic conductivity is assumed to obey fractional brownian motion fbm statistics with infinite integral scale or to have an exponential covariance structure with finite integral scale the flow problem is solved via a block centered finite difference scheme and a random walk approach is employed to solve the transport equation for a conservative tracer the model is tested for mass conservation and convergence of computed statistics and found to yield accurate results it is then used to address several issues in the context of flow and transport the validity of the first order relation between the fluctuating velocity covariance and the fluctuating log conductivity is examined the simulations show that for exponential covariance this approximation is justified in the mean flow direction for log conductivity variance f2 of the order of unity however as f2 increases the relation for the transverse velocity component deviates from the fully nonlinear monte carlo results eulerian transport models neglect triplet correlation functions that appear in the nonlocal macroscopic flux the relative importance of the triplet correlation term for conservative chemicals is examined this term appears to be small relative to the convolution flux term in mildly heterogeneous media as f2 increases or the integral scale grows the triplet correlation becomes significant in purely convective transport the triplet correlation term is significant if the heterogeneity is evolving the exact nonlocal macroscale flux for the purely convective case significantly differs from that of the convective dispersive transport this is in agreement with recent theoretical analysis and numerical studies and it suggests that neglecting local scale dispersion may lead to large errors localization errors in the flux term are evaluated using monte carlo simulations the nonlocal in time model significantly differs from the fully nonlocal model for small variance and integral scale there is a slight difference between the fully localized flux and the fully nonlocal convolution flux this is also in agreement with recent theories that suggest that moments through the second for the two models are nearly identical for conservative tracers the fully localized model does not perform well in the purely convective cases

eulerian lagrangian approach for modeling of flow and transport in heterogeneous geological formations this paper presents a new eulerian lagrangian method for modeling flow and transport of passive solutes in heterogeneous porous formations the physically plausible random velocity fields are generated by a geostatistical based model the ability of the method to correctly reproduce the velocity field statistics and to satisfy mass balance is tested and demonstrated for the case of two dimensional flow the method is found particularly useful in dealing with large fields where numerical fixed grid methods such as finite difference or finite elements become very demanding from a computational point of view the transport problem is solved in a lagrangian framework by the particle tracking approach which uses a suitable grid refinement to correctly handle the velocity fluctuation at any a priori defined resolution the new method is employed to compute the probability distribution functions of the concentration and travel times and to investigate the limitations of existing methods for predicting concentrations

stochastic modeling of groundwater flow by unconditional and conditional probabilities 1 conditional simulation and the direct problem mathematical modeling of groundwater flow in heterogeneous porous formations of large extent is investigated the formation properties hydraulic conductivity transmissivity as well as flow variables head specific discharge solute concentration are regarded as random variables subjected to uncertainty the main aim of the study is to analyze the influence of conditional probability of input variables upon the statistical structure of the dependent variables the unconditional probability density functions are supposed to be stationary multivariate normal while conditioning accounts for the measured values at a few points of the formation two problems of groundwater flow are discussed in part 1 conditional simulation and the direct problem for steady flow analytical results are obtained by using perturbation approximations average uniform head gradient flows as well as recharge and flow to wells are discussed in unconditional modeling the input variable the conductivity or transmissivity log is regarded as stationary and is represented by its constant mean and variogram the ensemble of formations on which statistical calculations are carried out represents all aquifers with same probability density distributions in conditional modeling the measured values at a few points are kept fixed and uncertainty prevails only at other points consequently statistical computations are performed for the subensemble of aquifers which preserve the measured values and as a result both input and output variables are nonstationary the main effect of conditioning is to reduce the variance i e the uncertainty of these variables this effect is particularly important in the case of flow toward wells application of the modeling method to field problems is outlined

a simple but powerful heuristic method for generating fuzzy rules from numerical data abstract in this paper we propose a simple but powerful heuristic method for automatically generating fuzzy if then rules from numerical data fuzzy if then rules with nonfuzzy singletons i e real numbers in the consequent parts are generated by the proposed heuristic method the main advantage of the proposed heuristic method is its simplicity i e it involves neither time consuming iterative learning procedures nor complicated rule generation mechanisms we also suggest a linguistic representation method for deriving linguistic rules from fuzzy if then rules with consequent real numbers the proposed linguistic approximation method consists of two linguistic rule tables which can realize exactly the same nonlinear mapping as an original system based on fuzzy if then rules with consequent real numbers using computer simulations on rice taste data we demonstrate the high performance of the proposed heuristic method and illustrate the proposed linguistic representation method

slide a simple adaptive defuzzification method we introduce a parameterized family of defuznfica tion operators called the semi linear defuzzification slide method this method is based upon a simple transformation of the fuzzy output set of the controller we suggest an algorithm for the learning of the parameter from a data set in an attempt to simplify the parameter learning we suggest a modified version of the slide method which results in a simple learning algorithm the development of the learning algorithm is based upon the use of the kalman filter n fuzzy logic control systems l 2 the defuzzification i step involves the selection of one value as the output of the controller more specifically starting with a fuzzy subset possibility distribution f over the output space x of the controller the defuzzification step uses this fuzzy subset to select a representative element x the two most often used methods of defuzzification found in the literature are the center of area coa and mean of maxima mom 3 6 methods we recall that the mom method takes as its defuzzified value the mean of the elements that attain the maximum membership grade in f the coa method takes as its defuzzified value d ci ui xi where ui f xi cj f xj in 7 and 8 we formulated a general defuzzification method via basic defuzzification distribution badd transformations the main idea of this badd approach is to transform the possibility distribution f into a probability distribution p and then the defuzzified value is the expected value of the probability distribution the process of transforming f into p was seen as a two step process the first step was to transform f into a new possibility distribution e and then to normalize e to obtain p the step of obtaining e from f involved the use of a badd transformation in which e xi f z where 6 e o co it was shown in 7 that the generalized badd method implies the coa and mom methods as special cases in particular the coa method is obtained for a 1 and the mom is obtained for 6 00 when s 0 we get a defuzzified value that is the unweighted mean of the output space d xi xi the badd transformation has the advantage of being based on a single parameter which allows us the potential of adaptively learning the best method of defuzzification for a given controller while the family of all defuzzified values that can be obtained by the generalized badd defuzzification method was parameterized an undesirable property of the method was the nonlinear

stochastic theory of field scale fickian dispersion in anisotropic porous media a three dimensional theory is described for field scale fickian dispersion in anisotropic porous media due to the spatial variability of hydraulic conductivities the study relies partly on earlier work by the authors the attributes of which are briefly reviewed it leads to results which differ in important ways from earlier theoretical conclusions about dispersion in anisotropic media we express the dispersion tensor d as the sum of a local component d and a field scale component the local component is assumed to be independent of velocity which is most appropriate if it represents molecular diffusion and its principal terms are taken to act parallel and normal to the mean velocity vector the field scale component is written as where is a dispersivity tensor and we show that at large peclet numbers p the dispersivity tensor reduces to a single principal component parallel to the mean velocity regardless of how is oriented this result valid for arbitrary covariance functions of log hydraulic conductivity differs from that of l w gelhar and c l axness 1983 according to whom the asymptotic dispersivity tensor may possess more than one nonzero eigen value they calculate the direction of the largest principal dispersivity to be offset from the mean velocity toward the direction of least spatial correlation or away from the stratification in typical layered media we show that this principal dispersivity is offset in the opposite direction at small and intermediate peclet numbers but rotates toward the mean velocity as p increases the largest eigen value is constant and dominated by field scale velocity fluctuations at large p values the other two eigen values diminish asymptotically in proportion to p 1 and are controlled by d as well as by field scale differential convection the range of small peclet numbers has not been previously investigated under anisotropic conditions yet is of much importance for transport in low permeability rocks or soils we show that at low p values all three principal dispersivities are proportional to p and thus is proportional to 2 a phenomenon reminiscent of taylor diffusion when the mean velocity is inclined to the axes of anisotropy the eigen values of are neither parallel nor normal to however since d is dominated by d at small peclet numbers the principal dispersion coefficients are asymptotically as p 0 parallel and normal to the mean velocity just like when p is large their maximum deviation from these directions occurs at intermediate p values

fourier analysis of multigrid methods for general systems of pdes most iteration methods for solving boundary value problems can be viewed as approximations of a time dependent differential equation in this paper we show that the multigrid method has the effect of increasing the timestep for the smooth part of the solution leading back to an increase of the convergence rate for the nonsmooth part the convergence is an effect of damping fourier analysis is used to find the relation between the convergence rate for multigrid methods and singlegrid methods the analysis is performed for general partial differential equations and an arbitrary number of grids the difference in the behavior of the iterations between firstand second order equations is discussed the theoretical results are confirmed in simple numerical experiments

a robust design criterion for interpretable fuzzy models with uncertain data we believe that nonlinear fuzzy filtering techniques may be turned out to give better robustness performance than the existing linear methods of estimation h sup 2 and h sup spl infin filtering techniques because of the fact that not only linear parameters consequents but also the nonlinear parameters membership functions attempt to identify the uncertain behavior of the unknown system however the fuzzy identification methods must be robust to data uncertainties and modeling errors to ensure that the fuzzy approximation of unknown system s behavior is optimal in some sense this study presents a deterministic approach to the robust design of fuzzy models in the presence of unknown but finite uncertainties in the identification data we consider online identification of an interpretable fuzzy model based on the robust solution of a regularized least squares fuzzy parameters estimation problem the aim is to resolve the difficulties associated with the robust fuzzy identification method due to lack of a priori knowledge about upper bounds on the data uncertainties the study derives an optimal level of regularization that should be provided to ensure the robustness of fuzzy identification strategy by achieving an upper bound on the value of energy gain from data uncertainties and modeling errors to the estimation errors a time domain feedback analysis of the proposed identification approach is carried out with emphasis on stability robustness and steady state issues the simulation studies are provided to show the superiority of the proposed fuzzy estimation over the classical estimation methods

fuzzy transforms method in prediction data analysis we present a prediction method based on fuzzy transforms for forecasting problems and we compare it with the well known wang mendel s one another comparison is made on the local linear wavelet neural network method via forecasting time series we apply these concepts to a mackey glass chaotic time series and to the monthly nao climatic index time series