type 2 interval fuzzy rule based systems in spatial analysis abstract in this paper we present a rule based system by using interval type 2 fuzzy sets it2 fs for solving problems of spatial analysis this system is encapsulated in a geographical information system gis by performing numerous tests for evaluating their reliability

a parallel implicit method for the direct numerical simulation of wall bounded compressible turbulence a new second order accurate implicit temporal numerical scheme for the direct numerical simulation of turbulent flows is presented the formulation of the implicit method and the corresponding tunable parameters are introduced the numerical simulation results are compared with the results given by explicit runge kutta schemes theoretical results and published experimental and numerical data an assessment of the accuracy and performance of the method to simulate turbulent flows is made for temporally decaying isotropic turbulence and subsonic and supersonic turbulent boundary layers whereas no significant advantage over typical explicit time integration methods are found for the incompressible flows it is shown that the implicit scheme yields significant reduction in computer cost while assuring time accurate solutions for compressible turbulence simulations

comparing the performance potentials of interval and general type 2 rule based fuzzy systems in terms of sculpting the state space this paper provides application independent perspectives on why improved performance usually occurs as one goes from an interval type 2 it2 fuzzy system to a general type 2 gt2 fuzzy system this is achieved by using the horizontal slice representation of a gt2 fuzzy set and gt2 fuzzy system and by examining first and second order rule partitions as well as novelty partitions for the horizontal slices it demonstrates that for triangle and trapezoid secondary membership functions the numbers of first and second order rule partitions are exactly the same for it2 and gt2 fuzzy systems but that a maximum amount of change always occurs in every second order rule partition of a gt2 fuzzy system this does not always occur in such partitions of an it2 fuzzy system furthermore when type reduction tr is used in a gt2 fuzzy system the total number of novelty partitions is directly proportional to the number of horizontal slices consequently there are many more such partitions in a gt2 fuzzy system that uses tr than occur in an it2 fuzzy system that also uses tr it is the author s conjecture that it is the maximum changes that occur in every second order rule partition as well as the greater number of novelty partitions when tr is used that provide a gt2 fuzzy system with the potential to outperform an it2 fuzzy system

defuzzification of the discretised generalised type 2 fuzzy set experimental evaluation the work reported in this paper addresses the challenge of the efficient and accurate defuzzification of discretised generalised type 2 fuzzy sets as created by the inference stage of a mamdani fuzzy inferencing system the exhaustive method of defuzzification for type 2 fuzzy sets is extremely slow owing to its enormous computational complexity several approximate methods have been devised in response to this defuzzification bottleneck in this paper we begin by surveying the main alternative strategies for defuzzifying a generalised type 2 fuzzy set 1 vertical slice centroid type reduction 2 the sampling method 3 the elite sampling method and 4 the a planes method we then evaluate the different methods experimentally for accuracy and efficiency for accuracy the exhaustive method is used as the standard the test results are analysed statistically by means of the wilcoxon nonparametric test and the elite sampling method shown to be the most accurate in regards to efficiency vertical slice centroid type reduction is demonstrated to be the fastest technique

interval type 2 fuzzy voter design for fault tolerant systems a voting scheme constitutes an essential component of many fault tolerant systems two types of voters are commonly used in applications of real valued systems the inexact majority and the amalgamating voters the inexact majority voter effectively isolates erroneous modules and is capable of reporting benign outputs when a significant disagreement is detected however an application specific voter threshold must be provided on the other hand amalgamating voter such as the weighted average voter reduces the influence of faulty modules by averaging the input values together unlike the majority voters amalgamating voters are not capable of producing benign outputs in the past a type 1 t1 fuzzy voting scheme was introduced allowing for both smooth amalgamation of voter inputs and effective signalization of benign outputs the presented paper proposes an extension to the fuzzy voting scheme via incorporating interval type 2 it2 fuzzy logic the it2 fuzzy logic allows for an improved handling of uncertain assumptions about the distributions of noisy and erroneous inputs which are essential for correct design of the fuzzy voting scheme the proposed voter design features robust performance when the uncertainty assumptions dynamically change over time the it2 fuzzy voter architecture was compared against the average voter inexact majority voter and the t1 fuzzy voter using a refined experimental harness the reported results demonstrate improved availability safety and reliability of the presented it2 fuzzy voting scheme

global convergence of karnik mendel algorithms karnik mendel km algorithms are the most commonly used iterative type reduction methods in interval type 2 fuzzy sets and systems as well as new techniques for computing the fuzzy weighted average fwa various extensions and improvements have been proposed however no proof has been provided for the convergence of these extensions it is necessary to provide the proof because many of the iterative algorithms may have divergence cases in the present study we provide a theoretical proof that km algorithms exhibit global convergence different initialization methods and iteration formats can always obtain the same unique optimal solution thus there are no concerns about the possibility of divergence in extensions of km algorithms our proof provides theoretical support for the applications of km algorithms especially the type reduction designs used in type 2 fuzzy systems and fwa computations because of the important roles of km algorithms in these methods

footprint of uncertainty for type 2 fuzzy sets in the paper a new and comprehensible definition is proposed for type 2 fuzzy sets t2 fss and the primary and secondary memberships function are defined respectively by using multi valued mapping a new definition and formula for the footprint of uncertainty fou is presented and based on the new definitions the relation between fou and the original definition of t2 fs is discussed finally the partition method of fou is provided to represent the primary membership grade the fou and the interval type 2 fuzzy sets it2 fs when the upper and lower membership functions umf lmf of fou are given

the sampling method of defuzzification for type 2 fuzzy sets experimental evaluation for generalised type 2 fuzzy sets the defuzzification process has historically been slow and inefficient this has hampered the development of type 2 fuzzy inferencing systems for real applications and therefore no advantage has been taken of the ability of type 2 fuzzy sets to model higher levels of uncertainty the research reported here provides a novel approach for improving the speed of defuzzification for discretised generalised type 2 fuzzy sets the traditional type reduction method requires every embedded type 2 fuzzy set to be processed the high level of redundancy in the huge number of embedded sets inspired the development of our sampling method which randomly samples the embedded sets and processes only the sample the paper presents detailed experimental results for defuzzification of constructed sets of known defuzzified value the sampling defuzzifier is compared on aggregated type 2 fuzzy sets resulting from the inferencing stage of a fis in terms of accuracy and speed with other methods including the exhaustive and techniques based on the a planes representation the results indicate that by taking only a sample of the embedded sets we are able to dramatically reduce the time taken to process a type 2 fuzzy set with very little loss in accuracy

on type 2 fuzzy relations and interval valued type 2 fuzzy sets this paper introduces new operations on the algebra of fuzzy truth values extended supremum and extended infimum which are generalizations of the extended operations of maximum and minimum between fuzzy truth values for type 2 fuzzy sets respectively using these new operations the properties of type 2 fuzzy relations are discussed especially the compositions of type 2 fuzzy relations on this basis this paper introduces interval valued type 2 fuzzy sets and interval valued type 2 fuzzy relations and discusses their properties

an adaptive implicit explicit scheme for the dns and les of compressible flows on unstructured grids an adaptive implicit explicit scheme for direct numerical simulation dns and large eddy simulation les of compressible turbulent flows on unstructured grids is developed the method uses a node based finite volume discretization with summation by parts sbp property which in conjunction with simultaneous approximation terms sat for imposing boundary conditions leads to a linearly stable semi discrete scheme the solution is marched in time using an implicit explicit runge kutta imex rk time advancement scheme a novel adaptive algorithm for splitting the system into implicit and explicit sets is developed the method is validated using several canonical laminar and turbulent flows load balance for the new scheme is achieved by a dual constraint domain decomposition algorithm the scalability and computational efficiency of the method is investigated and memory savings compared with a fully implicit method is demonstrated a notable reduction of computational costs compared to both fully implicit and fully explicit schemes is observed

defuzzification criteria and classification in this paper we contribute to the theory and development of defuzzification techniques first we define the core of a fuzzy set then we formulate a set of criteria for defuzzification in arbitrary universes ordered universes and the set of the real numbers finally we classify the most widely used defuzzification techniques into different groups and we examine the prototypes of each group with respect to the defuzzification criteria we show that the maxima methods behave well with respect to the more basic defuzzification criteria and hence are good candidates for fuzzy reasoning systems on the other hand the distribution methods and the area methods do not fulfill the basic criteria but they exhibit the property of continuity that makes them suitable for fuzzy controllers

ensuring the centroid of an interval type 2 fuzzy set in computing with words establishing a fuzzy set fs model for a word to capture its uncertainties is an important issue an interval type 2 it2 fs can be used to model a word how to establish an it2 fs from the collected data about a word has been a challenging problem it has been reported that one way is to extract the centroid of an it2 fs from the collected data and to obtain geometric parameters of its footprint of uncertainty fou such that its centroid matches the extracted one how to extract the centroid of an it2 fs from the collected data about a word has thoroughly been studied however there exists no method for obtaining fou parameters for an it2 fs such that its centroid matches the desired one to fill this gap this paper presents an approach to obtaining fou parameters for an it2 fs by establishing equations using the centroid requirement to propose this approach a sufficient and necessary condition for ensuring the centroid of an it2 fs is developed using this sufficient and necessary condition two equations about all of the fou parameters are established to obtain the fou parameters all of them except two are predetermined so that the established equations can be simplified to two single variable equations the other two fou parameters can then be determined by solving these two single variable equations using existing root finding algorithms among existing root finding algorithms the false position algorithm is recommended the overall merits of the proposed approach are its simplicity in implementation and its applicability to it2 fss with arbitrary fou shapes in addition numerical examples are provided to further illustrate how to apply the proposed approach to obtain fou parameters for an it2 fs

fuzzy sets and type 2 under algebraic product and algebraic sum abstract the concept of fuzzy sets of type 2 has been proposed by l a zadeh as an extension of ordinary fuzzy sets a fuzzy set of type 2 can be defined by a fuzzy membership function the grade or fuzzy grade of which is taken to be a fuzzy set in the unit interval 0 1 rather than a point in 0 1 this paper investigates the algebraic properties of fuzzy grades that is fuzzy sets of type 2 under the operations of algebraic product and algebraic sum which can be defined by using the concept of the extension principle and shows that fuzzy grades under these operations do not form such algebraic structures as a lattice and a semiring moreover the properties of fuzzy grades are also discussed in the case where algebraic product and algebraic sum are combined with the well known operations of join and meet for fuzzy grades and it is shown that normal convex fuzzy grades form a lattice ordered semigroup under join meet and algebraic product

localization results for the bernstein max product operator in the class of strictly positive functions strong localization results are obtained in approximation by the bernstein max product operators the results allow to approximate locally bounded strictly positive functions with very good accuracy with potential applications in e g image processing and in the approximation of fuzzy numbers which are useful concepts in statistics computer programming engineering especially communications and experimental science

a fast geometric method for defuzzification of type 2 fuzzy sets generalized type 2 fuzzy logic systems cannot currently be used for practical problems because the amount of computation required to defuzzify a generalized type 2 fuzzy set is too large this paper presents a new method for defuzzifing a type 2 fuzzy set the new much faster technique is based on geometric representations and operations the results of a real world example contained in this paper show this new approach to be over 200 000 times faster than type reduction we present a new method for assessing the accuracy of the membership function of a type 2 fuzzy set this method is used to show that the new representation used by the defuzzifier is not detrimental to the accuracy of the set we also discuss the differences between the new approach and type reduction identifying the origin of this massive improvement in execution speed

fully conservative higher order finite difference schemes for incompressible flow conservation properties of the mass momentum and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations existing finite difference schemes in regular and staggered grid systems are checked for violations of the conservation requirements and a few important discrepancies are pointed out in particular it is found that none of the existing higher order schemes for a staggered mesh system simultaneously conserve mass momentum and kinetic energy this deficiency is corrected through the derivation of a general family of fully conservative higher order accurate finite difference schemes for staggered grid systems finite difference schemes in a collocated grid system are also analyzed and a violation of kinetic energy conservation is revealed the predicted conservation properties are demonstrated numerically in simulations of inviscid white noise performed in a two dimensional periodic domain the proposed fourth order schemes in a staggered grid system are generalized for the case of a non uniform mesh and the resulting scheme is used to perform large eddy simulations of turbulent channel flow

a robust colocated implicit algorithm for direct numerical simulation of compressible turbulent flows a non dissipative robust implicit algorithm is proposed for direct numerical and large eddy simulation of compressible turbulent flows the algorithm addresses the problems caused by low mach numbers and under resolved high reynolds numbers it colocates variables in space to allow easy extension to unstructured grids and discretely conserves mass momentum and total energy the navier stokes equations are non dimensionalized using an incompressible scaling for pressure and the energy equation is used to obtain an expression for the velocity divergence a pressure correction approach is used to solve the resulting equations such that the discrete divergence is constrained by the energy equation as a result the discrete equations analytically reduce to the incompressible equations at very low mach number i e the algorithm overcomes the acoustic time scale limit without preconditioning or solution of an implicit system of equations the algorithm discretely conserves kinetic energy in the incompressible inviscid limit and is robust for inviscid compressible turbulence on the convective time scale these properties make it well suited for dns les of compressible turbulent flows results are shown for acoustic propagation the incompressible taylor problem periodic shock tube problem and isotropic turbulence

higher entropy conservation and numerical stability of compressible turbulence simulations we present a numerical formulation for the treatment of nonlinear instabilities in shock free compressible turbulence simulations the formulation is high order and contains no artificial dissipation numerical stability is enhanced through semi discrete satisfaction of global conservation properties stemming from the second law of thermodynamics and the entropy equation the numerical implementation is achieved using a conservative skew symmetric splitting of the nonlinear terms the robustness of the method is demonstrated by performing unresolved numerical simulations and large eddy simulations of compressible isotropic turbulence at a very high reynolds number results show the scheme is capable of capturing the statistical equilibrium of low mach number compressible turbulent fluctuations at infinite reynolds number comparisons with the entropy splitting technique j comput phys 162 2000 33 j comput phys 178 2002 307 staggered method j comput phys 191 2 2003 392 and skew symmetric like schemes j comput phys 161 2000 114 confirm the superiority of the current approach we also discuss a flaw in the skew symmetric splitting implemented in the literature very good results are obtained based on the proper splitting

uncertainty bounds and their use in the design of interval type 2 fuzzy logic systems we derive inner and outer bound sets for the type reduced set of an interval type 2 fuzzy logic system fls based on a new mathematical interpretation of the karnik mendel iterative procedure for computing the type reduced set the bound sets can not only provide estimates about the uncertainty contained in the output of an interval type 2 fls but can also be used to design an interval type 2 fls we demonstrate by means of a simulation experiment that the resulting system can operate without type reduction and can achieve similar performance to one that uses type reduction therefore our new design method based on the bound sets can relieve the computation burden of an interval type 2 fls during its operation which makes an interval type 2 fls useful for real time applications

an enhanced type reduction algorithm for type 2 fuzzy sets karnik and mendel proposed an algorithm to compute the centroid of an interval type 2 fuzzy set efficiently based on this algorithm liu developed a centroid type reduction strategy to carry out type reduction for type 2 fuzzy sets a type 2 fuzzy set is decomposed into a collection of interval type 2 fuzzy sets by cuts then the karnik mendel algorithm is called for each interval type 2 fuzzy set iteratively however the initialization of the switch point in each application of the karnik mendel algorithm is not a good one in this paper we present an improvement to liu s algorithm we employ the previously obtained result to construct the starting values in the current application of the karnik mendel algorithm convergence in each iteration except the first one can then speed up and type reduction for type 2 fuzzy sets can be carried out faster the efficiency of the improved algorithm is analyzed mathematically and demonstrated by experimental results

the reduction of interval type 2 lr fuzzy sets type reduction of interval type 2 it2 fuzzy sets is essential in conducting the type 2 fuzzy sets expressed with the resolution forms of it2 fuzzy sets several type reduction methods such as km ekm and centroid flow have been proposed these methods are relatively easy to implement but still computation intensive because they need to invoke an iterative switching point finding procedure this study derives a theorem and proposes a heuristic algorithm which can fast and accurately identify the minimum and maximum switching points of a piecewise smooth it2 fuzzy set it also demonstrates that it is easy to derive the close form expressions of the switching points of a piecewise smooth it2 fuzzy set if both of its upper and lower membership functions can be parameterized as lr fuzzy sets which are defined in this paper then the type reduction of piecewise smooth it2 fuzzy sets can be simplified to solve the close form expressions of their switching points in terms of lr parameters experiments on it2 fuzzy sets with various piecewise smooth membership functions including linear gaussian and hybrid shaped ones were made the results showed that the proposed type reduction method can obtain solutions which accurately approximate to the desired switching points with much lower computational overhead than the karnik mondel km and enhanced km ekm methods

a method for deriving the analytical structure of a broad class of typical interval type 2 mamdani fuzzy controllers type 2 t2 fuzzy controllers are emerging as the related t2 fuzzy logic and algorithms have recently been advancing rapidly at present a t2 fuzzy controller is viewed and used as a black box function generator that produces a desired nonlinear mapping between the input and output of the controller we call the mapping analytical structure the mathematical expression of the analytical structure however is not explicitly known to the controller designer this is in sharp contrast with the analytical structure of a conventional controller which is not only always explicitly known but serves as a starting point for system analysis and design obviously the knowledge of a t2 fuzzy controller s analytical structure can have significant benefits they include 1 understanding more precisely how the controller works in the same sense as we understand how a conventional controller e g the pid controller functions 2 making t2 fuzzy control more acceptable to safety critical fields such as biomedicine 3 taking advantage of the well developed nonlinear control theory to develop better analysis and design methods for t2 control systems e g less conservative system stability criteria and 4 permitting rigorous comparative exploration on differences between the t2 and type 1 t1 fuzzy controllers and their relative merits and pitfalls e g performance and structural complexity in this paper we develop an innovative technique which is capable of deriving the analytical structure for a wide class of interval t2 mamdani fuzzy controllers the configuration of the controllers is typical and is substantially more general than the related efforts in the literature it uses any number and types of interval t2 input fuzzy sets any number and types of general or interval t2 output fuzzy sets arbitrary fuzzy rules zadeh and operator karnik mendel center of sets type reducer and the centroid defuzzifier we show in detail how the derivation method works in a general setting and provide the analytical structure of an example t2 controller as well in addition we utilize the method to prove that a subset of the t2 fuzzy controllers are the sum of two nonlinear pi or pd controllers each of which has a variable proportional gain and a variable integral gain or derivative gain plus a variable offset if and only if the input fuzzy sets are piecewise linear e g triangular and or trapezoidal the sum of the two nonlinear pi or pd controllers is a new discovery relative to the current literature

spectral methods for the navier stokes equations with one infinite and two periodic directions abstract two numerical methods were designed to solve the time dependent three dimensional incompressible navier stokes equations in boundary layers method a semi infinite domain and mixing layers or wakes method b fully infinite domain their originality lies in the use of rapidly decaying spectral basis functions to approximate the vertical dependence of the solutions combined with one method a or two method b slowly decaying extra functions for each wave vector that exactly represent the irrotational component of the solution at large distances both methods eliminate the pressure term as part of the formulation thus avoiding fractional step time integration they yield rapid convergence and are free of spurious modes in the orr sommerfeld spectra they are also efficient although the operation count is of order n 2 n is the number of modes in the infinite direction these methods have been used for extensive direct numerical simulations of transition and turbulence a new time integration scheme with low storage requirements and good stability properties is also described

geometric defuzzification revisited abstract in this paper the geometric defuzzification strategy for type 2 fuzzy sets is reappraised for both discretised and geometric fuzzy sets the techniques for type 1 interval type 2 and generalised type 2 defuzzification are presented in turn in the type 2 case the accuracy of geometric defuzzification is assessed through a series of test runs on interval type 2 fuzzy sets using exhaustive defuzzification as the benchmark method these experiments demonstrate the geometric defuzzifier to be wildly inaccurate the test sets take many shapes they are not confined to those type 2 sets with rotational symmetry that have previously been acknowledged by the technique s developers to be problematic as regards accuracy type 2 geometric defuzzification is then examined theoretically the defuzzification strategy is demonstrated to be built upon a fallacious application of the concept of centroid this explains the markedly inaccurate experimental results thus the accuracy issues of type 2 geometric defuzzification are revealed to be inevitable fundamental and significant

hierarchical collapsing method for direct defuzzification of general type 2 fuzzy sets abstract karnik mendel km and enhanced km ekm algorithms are proved to be efficient approaches when dealing with general type 2 fuzzy sets represented by their planes on the other hand collapsing method as a type reduction algorithm for interval type 2 fuzzy sets has shown to be an effectual type reduction method for interval type 2 fuzzy sets in this paper we aim at developing mathematical formulations based on collapsing procedure and plane decomposition in order to utilize geometric properties of general type 2 fuzzy sets for preparing a direct defuzzification procedure the proposed methods are fast and robust extensive simulations have shown that the proposed algorithm outperforms most of the existing approaches in this field and its results are closer to the exact optimal defuzzified values

application of fuzzy sets of type 2 to the solution of fuzzy equation systems abstract the present paper continues the authors investigation on fuzzy equation systems with tolerances we consider the situation that the outputs of an object are modelled by fuzzy sets of type 2 for special membership functions this problem can be attributed to a problem with tolerances moreover this concept enables us to define solutions for fuzzy outputs of type 1 when classical solution concepts may fail

aggregation using the linguistic weighted average and interval type 2 fuzzy sets the focus of this paper is the linguistic weighted average lwa where the weights are always words modeled as interval type 2 fuzzy sets it2 fss and the attributes may also but do not have to be words modeled as it2 fss consequently the output of the lwa is an it2 fs the lwa can be viewed as a generalization of the fuzzy weighted average fwa where the type 1 fuzzy inputs are replaced by it2 fss this paper presents the theory algorithms and an application of the lwa it is shown that finding the lwa can be decomposed into finding two fwas since the lwa can model more uncertainties it should have wide applications in distributed and hierarchical decision making

statistical σ approximation to max product operators in this paper using the concept of statistical s convergence which is stronger than statistical convergence we obtain a statistical s approximation theorem for a general sequence of max product operators including shepard type operators although its classical limit fails we also compute the corresponding statistical s rates of the approximation

uncertain fuzzy self organization based clustering conventional unsupervised learning algorithms require knowledge of the desired number of clusters beforehand even if such knowledge is not required in advance empirical selection of the parameter values may limit the adaptive capability of the algorithm thereby restricting the clustering performance an inherent uncertainty in the number and size of clusters requires integration of fuzzy sets into a clustering algorithm in this paper we propose a type 1 t1 fuzzy art method that adaptively selects the vigilance parameter value which is critical in determining the network dynamics this results in improved clustering performance due to the added flexibility in dynamic selection of the number of clusters with the use of fuzzy sets to further manage the uncertainty associated with memberships we extend the proposed t1 fuzzy art with adaptive vigilance to an interval type 2 it2 fuzzy art method type reduction and defuzzification are then performed using the km algorithm to obtain a defuzzified vigilance parameter value we evaluate our proposed methods on several data sets to validate their effectiveness

enhanced karnik mendel algorithms the karnik mendel km algorithms are iterative procedures widely used in fuzzy logic theory they are known to converge monotonically and superexponentially fast however several usually two to six iterations are still needed before convergence occurs methods to reduce their computational cost are proposed in this paper extensive simulations show that on average the enhanced km algorithms can save about two iterations which corresponds to more than a 39 reduction in computation time an additional at least 23 computational cost can be saved if no sorting of the inputs is needed

designing of a type 2 fuzzy logic filter for improving edge preserving restoration of interlaced to progressive conversion this paper focuses on recently advanced fuzzy models and the application of type 2 fuzzy sets in video deinterlacing the final goal of the proposed deinterlacing algorithm is to exactly determine an unknown pixel value while preserving the edges and details of the image to begin we will discuss some artefacts of spatial temporal and spatio temporal domain deinterlacing methods in order to address the aforementioned issues we adopted type 2 fuzzy sets concepts to design a weight evaluating approach in the proposed method the upper and lower fuzzy membership functions of the type 2 fuzzy logic filters are derived from the type 1 or primary fuzzy membership function the weights from upper and lower membership functions are considered to be multiplied with the candidate deinterlaced pixels experimental results proved that the performance of the proposed method was superior both objectively and subjectively to other different conventional deinterlacing methods moreover the proposed method preserved the smoothness of the original image edges and produced a high quality progressive image

additive semi implicit runge kutta methods for computing high speed nonequilibrium reactive flows this paper is concerned with time stepping numerical methods for computing stiff semi discrete systems of ordinary differential equations for transient hypersonic flows with thermo chemical nonequilibrium the stiffness of the equations is mainly caused by the viscous flux terms across the boundary layers and by the source terms modeling finite rate thermo chemical processes implicit methods are needed to treat the stiff terms while more efficient explicit methods can still be used for the nonstiff terms in the equations this paper studies three different semi implicit runge kutta methods for additively split differential equations in the form ofu f u g u wherefis treated by explicit runge kutta methods andgis simultaneously treated by three implicit runge kutta methods a diagonally implicit runge kutta method and two linearized implicit runge kutta methods the coefficients of up to third order accurate additive semi implicit runge kutta methods have been derived such that the methods are both high order accurate and strongly a stable for the implicit terms the results of two numerical tests on the stability and accuracy properties of these methods are also presented in the paper

combinatorial iterative algorithms for computing the centroid of an interval type 2 fuzzy set computing the centroid of an interval type 2 fuzzy set it2 fs is an important type reduction method the aim of this paper is to develop a new method to calculate the centroid of an it2 fs when the problems of centroid computation of an it2 fs are continuous for the continuous centroid computation problems the structures of optimal solutions are strictly proven from mathematics for the first time in this paper furthermore we also prove that the structures of the optimal solutions are unique in the sense of almost everywhere equal i e if there are two optimal solutions f 1 x and f 2 x the lebesgue measure of x f 1 x neq f 2 x is equal to 0 subsequently a combinatorial iterative ci method is proposed to solve the roots of the sufficiently differentiable objective functions it is proven that the convergence of the proposed iterative method is at least sixth order based on the proposed iterative method two algorithms called ci algorithms are devised to compute the centroid of an it2 fs the efficiencies of ci algorithms are demonstrated by comparing the continuous karnik mendel algorithms and the hallye s methods with the ci algorithms through three numerical examples

methods of critical value reduction for type 2 fuzzy variables and their applications a type 2 fuzzy variable is a map from a fuzzy possibility space to the real number space it is an appropriate tool for describing type 2 fuzziness this paper first presents three kinds of critical values cvs for a regular fuzzy variable rfv and proposes three novel methods of reduction for a type 2 fuzzy variable secondly this paper applies the reduction methods to data envelopment analysis dea models with type 2 fuzzy inputs and outputs and develops a new class of generalized credibility dea models according to the properties of generalized credibility when the inputs and outputs are mutually independent type 2 triangular fuzzy variables we can turn the proposed fuzzy dea model into its equivalent parametric programming problem in which the parameters can be used to characterize the degree of uncertainty about type 2 fuzziness for any given parameters the parametric programming model becomes a linear programming one that can be solved using standard optimization solvers finally one numerical example is provided to illustrate the modeling idea and the efficiency of the proposed dea model

interval type 2 mutual subsethood fuzzy neural inference system it2msfunis this paper presents an interval type 2 mutual subsethood fuzzy neural inference system it2msfunis a mutual subsethood measure between two interval type 2 fuzzy sets it2 fs has been derived and has been used in determining the similarity between the it2 fs inputs and it2 fs antecedents the consequent weights are taken to be interval sets the inputs to the system are fuzzified into it2 fss with gaussian primary membership function having fixed center and uncertain variance aggregation of type 2 mutual subsethood based activation spreads is performed using product operator the output is obtained using simplified type reduction followed by defuzzification the system learns using memetic procedure involving differential evolution for global search and gradient descent for local exploitation in solution space the mathematical modeling and empirical studies of it2msfunis bring forth its efficacy in problems pertaining to function approximation time series prediction control and classification comparisons with other type 1 and type 2 neuro fuzzy systems verify that it2msfunis compares excellently with other models with a performance better than most of them both in terms of total number of trainable parameters and result accuracy empirical studies indicate the intelligent decision making capability of the proposed model the main contribution of this paper lies in the identification of mutual subsethood to find out the correlation between it2 fss and to find out its applicability in diverse application domains the improved performance of the proposed method can be attributed to the better contrast handling capacity of mutual subsethood method and uncertainty handling capacity of it2 fss the integration of mutual subsethood with interval type 2 fuzzy logic puts forth a novel model with various merits as demonstrated amply with the help of well known problems reported in the literature

adaptive filter design using type 2 fuzzy cerebellar model articulation controller this paper aims to propose an efficient network and applies it as an adaptive filter for the signal processing problems an adaptive filter is proposed using a novel interval type 2 fuzzy cerebellar model articulation controller t2fcmac the t2fcmac realizes an interval type 2 fuzzy logic system based on the structure of the cmac due to the better ability of handling uncertainties type 2 fuzzy sets can solve some complicated problems with outstanding effectiveness than type 1 fuzzy sets in addition the lyapunov function is utilized to derive the conditions of the adaptive learning rates so that the convergence of the filtering error can be guaranteed in order to demonstrate the performance of the proposed adaptive t2fcmac filter it is tested in signal processing applications including a nonlinear channel equalization system a time varying channel equalization system and an adaptive noise cancellation system the advantages of the proposed filter over the other adaptive filters are verified through simulations

robust control of an lusm based x hbox y hbox theta motion control stage using an adaptive interval type 2 fuzzy neural network the robust control of a linear ultrasonic motor based x y thetas motion control stage to track various contours is achieved by using an adaptive interval type 2 fuzzy neural network ait2fnn control system in this study in the proposed ait2fnn control system an it2fnn which combines the merits of an interval type 2 fuzzy logic system and a neural network is developed to approximate an unknown dynamic function moreover adaptive learning algorithms are derived using the lyapunov stability theorem to train the parameters of the it2fnn online furthermore a robust compensator is proposed to confront the uncertainties including the approximation error optimal parameter vectors and higher order terms in taylor series to relax the requirement for the value of lumped uncertainty in the robust compensator an adaptive lumped uncertainty estimation law is also investigated in addition the circle and butterfly contours are planned using a nonuniform rational b spline curve interpolator the experimental results show that the contour tracking performance of the proposed ait2fnn is significantly improved compared with the adaptive type 1 fnn additionally the robustness to parameter variations external disturbances cross coupled interference and frictional force can also be obtained using the proposed ait2fnn

operations on type 2 fuzzy sets in this paper we discuss set operations on type 2 fuzzy sets including join and meet under minimum product t norm algebraic operations properties of membership grades of type 2 sets and type 2 relations and their compositions all this is needed to implement a type 2 fuzzy logic system fls

super exponential convergence of the karnik mendel algorithms for computing the centroid of an interval type 2 fuzzy set computing the centroid of an interval t2 fs is an important operation in a type 2 fuzzy logic system where it is called type reduction but it is also a potentially time consuming operation the karnik mendel km iterative algorithms are widely used for doing this in this paper we prove that these algorithms converge monotonically and super exponentially fast both properties are highly desirable for iterative algorithms and explain why in practice the km algorithms have been observed to converge very fast thereby making them very practical to use

alpha plane representation for type 2 fuzzy sets theory and applications this paper 1 reviews the alpha plane representation of a type 2 fuzzy set t2 fs which is a representation that is comparable to the alpha cut representation of a type 1 fs t1 fs and is useful for both theoretical and computational studies of and for t2 fss 2 proves that set theoretic operations for t2 fss can be computed using very simple alpha plane computations that are the set theoretic operations for interval t2 it2 fss 3 reviews how the centroid of a t2 fs can be computed using alpha plane computations that are also very simple because they can be performed using existing karnik mendel algorithms that are applied to each alpha plane 4 shows how many theoretically based geometrical properties can be obtained about the centroid even before the centroid is computed 5 provides examples that show that the mean value defuzzified value of the centroid can often be approximated by using the centroids of only 0 and 1 alpha planes of a t2 fs 6 examines a triangle quasi t2 fuzzy logic system q t2 fls whose secondary membership functions are triangles and for which all calculations use existing t1 or it2 fs mathematics and hence they may be a good next step in the hierarchy of flss from t1 to it2 to t2 and 7 compares t1 it2 and triangle q t2 flss to forecast noise corrupted measurements of a chaotic mackey glass time series

on advanced computing with words using the generalized extension principle for type 1 fuzzy sets in this paper we propose and demonstrate an effective methodology for implementing the generalized extension principle to solve advanced computing with words acww problems such problems involve implicit assignments of linguistic truth probability and possibility to begin we establish the vocabularies of the words involved in the problems and then collect data from subjects about the words after which fuzzy set models for the words are obtained by using the interval approach ia or the enhanced interval approach eia next the solutions of the acww problems which involve the fuzzy set models of the words are formulated using the generalized extension principle because the solutions to those problems involve complicated functional optimization problems that cannot be solved analytically we then develop a numerical method for their solution finally the resulting fuzzy set solutions are decoded into natural language words using jaccard s similarity measure we explain how acww problems can solve some potential prototype engineering problems and connect the methodology of this paper with perceptual computing

on clarifying some definitions and notations used for type 2 fuzzy sets as well as some recommended changes this paper is about some important changes in type 2 fuzzy set t2 fs definitions and notations that have occurred during the past 16 years it summarizes the evolution of how the primary membership jx has been used in both the mathematical descriptions of a t2 fs and its footprint of uncertainty fou discusses notational problems associated with the secondary membership function explains why and when one should distinguish between the fou and the domain of uncertainty dou explains why no errors have been introduced into t2 fs computations because of notation and it provides recommendations notational changes that can be used by all authors

novel weighted averages versus normalized sums in computing with words in this paper some properties of novel weighted averages that are related to the concepts of possibility theory are examined it is shown that novel weighted averages have certain interpretations in terms of addition of interactive interval or fuzzy constraints to do this alternative forms of novel weighted averages are provided in particular an alternative form of novel weighted averages represented by the extension principle is determined it is shown that when fuzzy set models of words are obtained by collecting data from subjects in a computing with words setting interactive addition of fuzzy sets is not a well defined method and the optimization problems related to it may have no solutions although interactive addition is recommended in the literature for solving multicriteria decision making problems and for dealing with uncertain probabilities on the other hand novel weighted averages perform a specific normalization that guarantees that they always exist

centroid of a type 2 fuzzy set abstract in this paper we introduce the centroid and generalized centroid of a type 2 fuzzy set both of which are essential for implementing a type 2 fuzzy logic system and explain how to compute them for practical use we show how to compute the centroid of interval and gaussian type 2 fuzzy sets an exact computation procedure is provided for an interval type 2 set whereas an approximation is provided for both interval and gaussian type 2 sets examples are given that compare the exact computational results with the approximate results

general type 2 fuzzy logic systems made simple a tutorial the purpose of this tutorial paper is to make general type 2 fuzzy logic systems gt2 flss more accessible to fuzzy logic researchers and practitioners and to expedite their research designs and use to accomplish this the paper 1 explains four different mathematical representations for general type 2 fuzzy sets gt2 fss 2 demonstrates that for the optimal design of a gt2 fls one should use the vertical slice representation of its gt2 fss because it is the only one of the four mathematical representations that is parsimonious 3 shows how to obtain set theoretic and other operations for gt2 fss using type 1 t1 fs mathematics cuts play a central role 4 reviews mamdani and tsk interval type 2 it2 flss so that their mathematical operations can be easily used in a gt2 fls 5 provides all of the formulas that describe both mamdani and tsk gt2 flss 6 explains why center of sets type reduction should be favored for a gt2 fls over centroid type reduction 7 provides three simplified gt2 flss two are for mamdani gt2 flss and one is for a tsk gt2 fls all of which bypass type reduction and are generalizations from their it2 fls counterparts to gt2 flss 8 explains why gradient based optimization should not be used to optimally design a gt2 fls 9 explains how derivative free optimization algorithms can be used to optimally design a gt2 fls and 10 provides a three step approach for optimally designing flss in a progressive manner from t1 to it2 to gt2 each of which uses a quantum particle swarm optimization algorithm by virtue of which the performance for the it2 fls cannot be worse than that of the t1 fls and the performance for the gt2 fls cannot be worse than that of the it2 fls

the max product generalized sampling operators convergence and quantitative estimates abstract in this paper we study the max product version of the generalized sampling operators based upon a general kernel function in particular we prove pointwise and uniform convergence for the above operators together with a certain quantitative jackson type estimate based on the first order modulus of continuity of the function being approximated the proof of the proposed results are based on the definition of the so called generalized absolute moments by the proposed approach the achieved approximation results can be applied for several type of kernels not necessarily duration limited such as the sinc function the fejer kernel and many others examples of kernels with compact support for which the above theory holds can be given for example by the well known central b splines

application of implicit explicit high order runge kutta methods to discontinuous galerkin schemes despite the popularity of high order explicit runge kutta erk methods for integrating semi discrete systems of equations erk methods suffer from severe stability based time step restrictions for very stiff problems we implement a discontinuous galerkin finite element method dgfem along with recently introduced high order implicit explicit runge kutta imex rk schemes to overcome geometry induced stiffness in fluid flow problems the imex algorithms solve the non stiff portions of the domain using explicit methods and isolate and solve the more expensive stiff portions using an l stable stiffly accurate explicit singly diagonally implicit runge kutta method esdirk furthermore we apply adaptive time step controllers based on the embedded temporal error predictors we demonstrate in a number of numerical test problems that imex methods in conjunction with efficient preconditioning become more efficient than explicit methods for systems exhibiting high levels of grid induced stiffness

an improvement of fractional step methods for the incompressible navier stokes equations abstract a numerical method for computing three dimensional unsteady incompressible flows is presented the method is a predictor corrector technique combined with a fractional step method each time step is advanced in three sub steps the novel feature of the present scheme is that the poisson equation for the pressure is solved only at the final sub step resulting in substantial savings in computing time it is shown that the method allows a larger cfl number and reduces the computing cost without loss of accuracy by satisfying the continuity equation only at the last sub step numerical solutions for the decaying vortices and flow over a backward facing step are obtained and compared with analytical and other numerical results

high order fluxes for conservative skew symmetric like schemes in structured meshes developing high order non dissipative schemes is an important research task for both steady and unsteady flow computations we take as a starting point the built in de aliasing property of the discretized skew symmetric form for the non linear terms of the navier stokes equations recalled in kravchenko and moin 1 two families of high order locally conservative schemes matching this discretized skew symmetric form are considered and rewritten in terms of telescopic fluxes for both finite difference and finite volume approximations in the context of compressible flows the jameson s scheme 2 is shown to be the second order member of larger families of skew symmetric like centered schemes the fourth order finite volume and finite difference and the sixth order finite difference schemes which belong to this family are provided the proposed schemes are extended to shock capturing schemes either by modifying the jameson s artificial viscosity or by hybriding the centered flux with weno 3 fluxes an adapted interpolation is proposed to extend the use of the proposed schemes to non regular grids several tests are provided showing that the conjectured order is properly recovered even with irregular meshes and that the shock capturing properties allow us to improve the second order results for standard test cases the improvement due to fourth order is then confirmed for the estimation of the growth of two ts waves and three crow instability dimensional unstable modes for both confined and free shear flows the last application concerns the steady computation using the spalart allmaras model of a separated boundary layer it confirms that the use of a high order scheme improves the results even in this type of steady applications

stable and accurate schemes for the compressible navier stokes equations minimal stencil width discretizations of combined mixed and non mixed second order derivatives are analyzed with respect to accuracy and stability we show that these discretizations lead to stability for cauchy problems with a careful boundary treatment we also show that the stability holds for initial boundary value problems the analysis is verified by numerical simulations of burgers and navier stokes equations in two and three space dimensions

hybrid explicit implicit unconditionally stable scheme for unsteady compressible flows a variable explicit implicit scheme was proposed by collins et al ccg scheme for one dimensional hyperbolic systems by suitable blending of explicit and implicit schemes this approach aims to ensure a maximum norm diminishing mnd property at all courant numbers although proven for linear equations however the ccg scheme fails to maintain this property unconditionally for non linear equations and requires a specific time step restriction the remedy for this limitation of the ccg scheme is shown and a new general method for the design of unconditionally mnd hybrid schemes for nonlinear hyperbolic equations is proposed an extension to the compressible navier stokes equations is also discussed the method is tested and verified by calculation of the burgers equation on a highly stiff nonuniform grid it is also applied to simulate the reflection of a normal shock wave from the end wall of a two dimensional channel which is attended with unsteady viscous interaction processes including shock bifurcation

answering an open problem on t norms for type 2 fuzzy sets abstract this paper proves that a binary operation on 0 1 ensuring that the binary operation is a t norm or is a t conorm is a t norm where and are special convolution operations defined by f g x sup f y g z y z x f g x sup f y g z y z x for any f g map 0 1 0 1 where and are a continuous t norm and a continuous t conorm on 0 1 answering negatively an open problem posed in 8 besides some characteristics of t norm and t conorm are obtained in terms of the binary operations and

type 1 owa operators for aggregating uncertain information with uncertain weights induced by type 2 linguistic quantifiers the owa operator proposed by yager has been widely used to aggregate experts opinions or preferences in human decision making yager s traditional owa operator focuses exclusively on the aggregation of crisp numbers however experts usually tend to express their opinions or preferences in a very natural way via linguistic terms these linguistic terms can be modelled or expressed by type 1 fuzzy sets in this paper we define a new type of owa operator the type 1 owa operator that works as an uncertain owa operator to aggregate type 1 fuzzy sets with type 1 fuzzy weights which can be used to aggregate the linguistic opinions or preferences in human decision making with linguistic weights the procedure for performing type 1 owa operations is analysed in order to identify the linguistic weights associated to the type 1 owa operator type 2 linguistic quantifiers are proposed the problem of how to derive linguistic weights used in type 1 owa aggregation given such type of quantifier is solved examples are provided to illustrate the proposed concepts

geometric type 1 and type 2 fuzzy logic systems this paper presents a novel approach to the representation of type 1 and type 2 fuzzy sets utilising computational geometry to achieve this our approach borrows ideas from the field of computational geometry and applies these techniques in the novel setting of fuzzy logic we provide new algorithms for various operations on type 1 and type 2 fuzzy sets and for defuzzification results of experiments indicate that this approach reduces the execution speed of these operations

the collapsing method of defuzzification for discretised interval type 2 fuzzy sets this paper proposes a new approach for defuzzification of interval type 2 fuzzy sets the collapsing method converts an interval type 2 fuzzy set into a type 1 representative embedded set res whose defuzzified values closely approximates that of the type 2 set as a type 1 set the res can then be defuzzified straightforwardly the novel representative embedded set approximation resa to which the method is inextricably linked is expounded stated and proved within this paper it is presented in two forms simple resa this approximation deals with the most simple interval fou in which a vertical slice is discretised into 2 points interval resa this approximation concerns the case in which a vertical slice is discretised into 2 or more points the collapsing method simple resa version was tested for accuracy and speed with excellent results on both criteria the collapsing method proved more accurate than the karnik mendel iterative procedure kmip for an asymmetric test set for both a symmetric and an asymmetric test set the collapsing method outperformed the kmip in relation to speed