on evolutionary designing and learning of flexible neuro fuzzy structures for nonlinear classification abstract in the paper the evolutionary strategy is applied for learning flexible neuro fuzzy systems in the process of evolution we determine fuzzy inference mamdani type or logical type described by an s implication specific fuzzy implication if the logical type system is found in the process of evolution or specific t norm connecting antecedents and consequences if the mamdani type system is found in the process of evolution specific t norm for aggregation of antecedents in each rule specific triangular norm describing aggregation operator shapes and parameters of fuzzy membership functions weights describing importance of antecedents of rules and weights describing importance of rules parameters of adjustable triangular norms parameters of soft triangular norms it should be noted that the crossover and mutation operators are chosen in a self adaptive way the method is tested using well known classification benchmarks

a new gradient descent approach for local learning of fuzzy neural models the majority of reported learning methods for takagi sugeno kang tsk fuzzy neural models to date mainly focus on improvement of their accuracy however one of the key design requirements in building an interpretable fuzzy model is that each obtained rule consequent must match well with the system local behavior when all the rules are aggregated to produce the overall system output this is one of the distinctive characteristics from black box models such as neural networks therefore how to find a desirable set of fuzzy partitions and hence identify the corresponding consequent models which can be directly explained in terms of system behavior presents a critical step in fuzzy neural modeling in this paper a new learning approach considering both nonlinear parameters in the rule premises and linear parameters in the rule consequents is proposed unlike the conventional two stage optimization procedure widely practiced in the field where the two sets of parameters are optimized separately the consequent parameters are transformed into a dependent set on the premise parameters thereby enabling the introduction of a new integrated gradient descent learning approach thus a new jacobian matrix is proposed and efficiently computed to achieve a more accurate approximation of the cost function by using the second order levenberg marquardt optimization method several other interpretability issues regarding the fuzzy neural model are also discussed and integrated into this new learning approach numerical examples are presented to illustrate the resultant structure of the fuzzy neural models and the effectiveness of the proposed new algorithm and compared with the results from some well known methods

radial basis function based adaptive fuzzy systems and their applications to system identification and prediction abstract in this paper we describe a neuro fuzzy system with adaptive capability to extract fuzzy if then rules from input and output sample data through learning the proposed system called radial basis function rbf based adaptive fuzzy system afs employs the gaussian functions to represent the membership functions of the premise part of fuzzy rules three architectural deviations of the rbf based afs are also presented according to different consequence types such as constant first order linear function and fuzzy variable these provide versatility of the network to handle arbitrary fuzzy inference schemes we present examples of system identification and time series prediction to illustrate how to solve these problems and to demonstrate its validity and effectiveness using the rbf based afs

a fuzzy model with online incremental svm and margin selective gradient descent learning for classification problems this paper proposes a new incremental learning approach to endow a takagi sugeno type fuzzy classification model with high generalization ability the proposed fuzzy model is learned through incremental support vector machine svm and margin selected gradient descent learning and is called fm3 in this learning approach training samples are fed incrementally one by one instead of all in one batch the fm3 evolves from an empty rule set a one pass clustering algorithm is used to determine the number of rules and initial fuzzy sets in the rule antecedent part an online incremental linear svm is proposed to tune the rule consequent parameters to endow the fm3 with high generalization ability the use of incremental instead of batch svm enables the fm3 to handle online training problems with only one new sample available at a time for antecedent parameter learning a margin selected gradient descent algorithm is proposed to prevent overtraining simulation results and comparisons with svms and fuzzy classifiers with different learning algorithms demonstrate the advantage of the fm3

scalable tsk fuzzy modeling for very large datasets using minimal enclosing ball approximation in order to overcome the difficulty in takagi sugeno kang tsk fuzzy modeling for large datasets scalable tsk stsk fuzzy model training is investigated in this study based on the core set based minimal enclosing ball meb approximation technique the specified l2 norm penalty based insensitive criterion is first proposed for tsk model training and it is found that such tsk fuzzy model training can be equivalently expressed as a center constrained meb problem with this finding an stsk fuzzy model training algorithm which is called stsk for large or very large datasets is then proposed by using the core set based meb approximation technique the proposed algorithm has two distinctive advantages over classical tsk fuzzy model training algorithms the maximum space complexity for training is not reliant on the size of the training dataset and the maximum time complexity for training is linear with the size of the training dataset as confirmed by extensive experiments on both synthetic and real world regression datasets

a self organizing ts type fuzzy network with support vector learning and its application to classification problems a self organizing takagi sugeno ts type fuzzy network with support vector learning sotfn sv is proposed in this paper the proposed sotfn sv is inspired by analysis of ts type fuzzy systems and composite kernel support vector machine svm sotfn sv is a fuzzy system constructed by the hybridization of fuzzy clustering and svm the antecedent part of sotfn sv is generated via fuzzy clustering of the input data and then svm is used to tune the consequent part parameters to give the network better generalization performance for demonstration sotfn sv is applied to several classification problems especially the skin color classification problem in the skin color classification application each color pixel is represented by hue and saturation hs color space to represent color information by histogram as accurately as possible a nonuniform partition of hs space is proposed for comparison svms and other fuzzy systems trained by svm or neural networks are applied to the same classification problems the advantages of sotfn sv are verified by comparisons with the results of these methods

analysis of robust multigrid methods for steady viscous low mach number flows a theoretical and numerical analysis of different implicit methods in multigrid form is carried out in order to study their behaviour at various flow conditions the fourier analysis for scalar convection diffusion equations is extended to the coupled set of laminar navier stokes equations for first and second order discretized navier stokes equations the analysis of point line and block methods is in good agreement with the numerical results the influence on the convergence rate of different second order implementations in the multigrid method is also considered the combination of multigrid line methods and mixed discretization enables to tackle stiff problems in a cheap and robust way the comparison is further extended to reynolds averaged navier stokes equations with the low reynoldsk equations describing the turbulence when line and block methods are used it is advantageous not to incorporate the turbulence equations into the multigrid cycle

an improved unsteady unstructured artificial compressibility finite volume scheme for viscous incompressible flows part i theory and implementation a robust artificial compressibility scheme has been developed for modelling laminar steady state and transient incompressible flows over a wide range of reynolds and rayleigh numbers artificial compressibility is applied in a consistant manner resulting in a system of preconditioned governing equations a locally generalized preconditioner is introduced designed to be robust and offer good convergence rates free artificial compressibility parameters in the equations are automated to allow ease of use while facilitating improved or comparable convergence rates as compared with the standard artificial compressibility scheme memory efficiency is achieved through a multistage pseudo time explicit time marching solution procedure a node centred dual cell edge based finite volume discretization technique suitable for unstructured grids is used due to its computational efficiency and high resolution spatial accuracy in the interest of computational efficiency and ease of implementation stabilization is achieved via a scalar valued artificial dissipation scheme temporal accuracy is facilitated by employing a second order accurate dual time stepping method in this part of the paper the theory and implementation details are discussed in part 11 the scheme will be applied to a number of example problems to solve flows over a wide range of reynolds and rayleigh numbers

1 norm based regularization scheme for system identification of structures with discontinuous system parameters this paper presents a new class of regularization functions and the associated regularization scheme for structural system identification in particular 1 norm regularization functions are investigated to overcome the smearing effect of 2 norm regularization functions for the identification of discontinuous system parameters of structures the truncated singular value decomposition is employed to filter out noise polluted solution components and to impose the 1 norm regularization function on si the bilinear fitting method is proposed for selecting an optimal truncation number of the truncated singular value decomposition the validity of the proposed method is demonstrated through the identification of an inclusion in a square plate and damaged members in a two span truss copyright 2006 john wiley sons ltd

a species based improved electromagnetism like mechanism algorithm for tsk type interval valued neural fuzzy system optimization this paper proposes a novel tsk type interval valued neural fuzzy system with asymmetric membership functions tivnfs a which is optimized by a species based improved electromagnetism like mechanism siem algorithm the proposed tivnfs a uses interval valued asymmetric fuzzy membership functions and the tsk type consequent part to improve the approximation performance and to reduce the number of fuzzy rules in addition the corresponding type reduction procedure is integrated in the adaptive network layers to reduce the amount of computation in the system to train the tivnfs as the siem combines the advantages of electromagnetism like mechanism em algorithm and gradient descent technique to obtain faster convergence and lower computational complexity based on a similarity measurement the swarm of the siem has a dynamic population size and its population is also divided dynamically into subpopulations for optimization the gradient descent method is applied to the dominant particle in each species for a neighborhood search finally the nonlinear system identification and classification problems are shown to demonstrate the performance and effectiveness of the proposed tivnfs a with the siem

an hybrid finite volume finite element method for variable density incompressible flows this paper is devoted to the numerical simulation of variable density incompressible flows modeled by the navier stokes system we introduce an hybrid scheme which combines a finite volume approach for treating the mass conservation equation and a finite element method to deal with the momentum equation and the divergence free constraint the breakthrough relies on the definition of a suitable footbridge between the two methods through the design of compatibility condition in turn the method is very flexible and allows to deal with unstructured meshes several numerical tests are performed to show the scheme capabilities in particular the viscous rayleigh taylor instability evolution is carefully investigated

supervised hierarchical clustering in fuzzy model identification this paper presents a new supervised hierarchical clustering algorithm suhiclust for fuzzy model identification the presented algorithm solves the problem of global model accuracy together with the interpretability of local models as valid linearizations of the modeled nonlinear system the algorithm combines the advantages of supervised hierarchical algorithms which are based on heuristic tree construction algorithms together with the advantages of fuzzy product space clustering the high flexibility of the validity functions that is obtained by fuzzy clustering combined with supervised learning results in an efficient partitioning algorithm which is independent of initialization and results in a parsimonious fuzzy model furthermore the usability of suhiclust is very undemanding because it delivers in contrast with many other methods reproducible results in order to get reasonable results the user only has to set either a threshold for the maximum number of local models or a value for the maximum allowed global model error as a termination criterion for fine tuning the interpolation smoothness controls the degree of regularization the performance is illustrated on both analytical examples and benchmark problems from the literature

artificial compressibility characteristics based schemes for variable density incompressible multispecies flows part ii multigrid implementation and numerical tests the paper presents an investigation of the accuracy and efficiency of artificial compressibility characteristics based cb schemes for variable density incompressible flows the cb schemes have been implemented in conjunction with a multigrid method for accelerating numerical convergence and a fourth order explicit runge kutta method for the integration of the governing equations in time the implementation of the cb schemes is obtained in conjunction with first second and third order interpolation formulas for calculating the variables at the cell faces of the computational volume the accuracy and efficiency of the schemes are examined against analytical and experimental results for diffusion broadening in two and three dimensional microfluidic channels a problem that has motivated the development of the present methods moreover unsteady inviscid simulations have been performed for variable density mixing layer the computations revealed that accuracy and efficiency depend on the cb scheme design the best multigrid convergence rates were exhibited by the conservative cb scheme which is obtained by the fully conservative formulation of the variable density incompressible equations

bayesian takagi sugeno kang fuzzy classifier in this paper the takagi sugeno kang tsk fuzzy classifier is casted into the bayesian inference framework and a new fuzzy classifier called bayesian tsk fuzzy classifier b tsk fc is proposed accordingly the proposed classifier can be constructed by learning both the antecedent and consequent parameters of the involved fuzzy rules simultaneously as a result of the introduction of bayesian inference the proposed b tsk fc can be distinguished as follows 1 unlike most existing tsk fuzzy classifiers where the antecedent and consequent parameters of fuzzy rules are learnt in a decoupled manner and the antecedent parameters are learnt only in the input space the antecedent parameters of the fuzzy rules in b tsk fc are learnt by developing a fuzzy clustering method in the input output space and the consequent parameters of fuzzy rules are learnt in accordance with the maximum margin of separation principle thereby resulting to form an intrinsic link between the input and output spaces to achieve improved classification performance and better interpretability 2 with a dirichlet prior assumption about fuzzy memberships in fuzzy clustering a markov chain monte carlo technique is employed to estimate the parameters of the proposed classifier from a sampling perspective 3 rather than being rivals fuzziness and probability in b tsk fc are collaboratively modeled to enhance the performance of tsk fuzzy classifier in terms of classification and interpretability our experimental results in synthetic datasets as well as several real world datasets confirm such merits of the proposed fuzzy classifier

determination of an optimal regularization factor in system identification with tikhonov regularization for linear elastic continua this paper presents a geometric mean scheme gms to determine an optimal regularization factor for tikhonov regularization technique in the system identification problems of linear elastic continua the characteristics of non linear inverse problems and the role of the regularization are investigated by the singular value decomposition of a sensitivity matrix of responses it is shown that the regularization results in a solution of a generalized average between the a priori estimates and the a posteriori solution based on this observation the optimal regularization factor is defined as the geometric mean between the maximum singular value and the minimum singular value of the sensitivity matrix of responses the validity of the gms is demonstrated through two numerical examples with measurement errors and modelling errors copyright 2001 john wiley sons ltd

artificial compressibility characteristics based schemes for variable density incompressible multi species flows part i derivation of different formulations and constant density limit the paper presents various formulations of characteristics based schemes in the framework of the artificial compressibility method for variable density incompressible flows in contrast to constant density incompressible flows where the characteristics based variables reconstruction leads to a single formulation in the case of variable density flows three different schemes can be obtained henceforth labeled as transport conservative and hybrid schemes the conservative scheme results in pseudo compressibility terms in the multi species density reconstruction it is shown that in the limit of constant density the transport scheme becomes the original characteristics based scheme for incompressible flows but the conservative and hybrid schemes lead to a new characteristics based variant for constant density flows the characteristics based schemes are combined with second and third order interpolation for increasing the computational accuracy locally at the cell faces of the control volume numerical experiments for constant density flows reveal that all the characteristics based schemes result in the same flow solution but they exhibit different convergence behavior the multigrid implementation and numerical studies for variable density flows are presented in part ii of this study

non conservative and conservative formulations of characteristics based numerical reconstructions for incompressible flows summary an investigation of characteristics based cb schemes for solving the incompressible navierstokes equations in conjunction with the artiflcial compressibility approach is presented both nonconservative and conservative characteristics based numerical reconstructions are derived and their accuracy and convergence properties are assessed analytically and numerically we demonstrate by means of eigenvalue analysis that there are difierences in the spectral characteristics of these formulations that result in difierent convergence properties numerical tests for two and threedimensional ows reveal that the two formulations provide similar accuracy but the non conservative formulation converges faster copyright c 2000 john wiley sons ltd

a study on reduced support vector machines recently the reduced support vector machine rsvm was proposed as an alternate of the standard svm motivated by resolving the difficulty on handling large data sets using svm with nonlinear kernels it preselects a subset of data as support vectors and solves a smaller optimization problem however several issues of its practical use have not been fully discussed yet for example we do not know if it possesses comparable generalization ability as the standard svm in addition we would like to see for how large problems rsvm outperforms svm on training time in this paper we show that the rsvm formulation is already in a form of linear svm and discuss four rsvm implementations experiments indicate that in general the test accuracy of rsvm are a little lower than that of the standard svm in addition for problems with up to tens of thousands of data if the percentage of support vectors is not high existing implementations for svm is quite competitive on the training time thus from this empirical study rsvm will be mainly useful for either larger problems or those with many support vectors experiments in this paper also serve as comparisons of 1 different implementations for linear svm and 2 standard svm using linear and quadratic cost functions

fast multigrid method for solving incompressible hydrodynamic problems with free surfaces we develop of a finite volume multigrid euler scheme for solving three dimensional fully nonlinear ship wave problems the flowfield and the a priori unknown free surface location are calculated by coupling the free surface kinematic and dynamic equations with the equations of motion for the bulk flow the evolution of the free surface boundary condition is linked to the evolution of the bulk flow via a novel iteration strategy that allows temporary leakage through the surface before the solution is converged the method of artificial compressibility is used to enforce the incompressibility constraint for the bulk flow a multigrid algorithm is used to accelerate convergence to a steady state

a new spatial regularization scheme for the identification of the geometric shape of an inclusion in a finite body this paper presents a system identification scheme to determine the geometric shape of an inclusion in a finite body the proposed algorithm is based on the minimization of the least squared errors between the measured displacement field and calculated displacement field by the finite element model the domain parameterization technique is adopted to manipulate the shape variation of an inclusion to stabilize the optimization process a new regularization function defined by the length of the boundary curve of an inclusion is added to the error function a variable regularization factor scheme is proposed for a consistent regularization effect the modified newton method with the active set method is adopted for optimization copyright 1999 john wiley sons ltd

efficient least angle regression for identification of linear in the parameters models least angle regression as a promising model selection method differentiates itself from conventional stepwise and stagewise methods in that it is neither too greedy nor too slow it is closely related to l1 norm optimization which has the advantage of low prediction variance through sacrificing part of model bias property in order to enhance model generalization capability in this paper we propose an efficient least angle regression algorithm for model selection for a large class of linear in the parameters models with the purpose of accelerating the model selection process the entire algorithm works completely in a recursive manner where the correlations between model terms and residuals the evolving directions and other pertinent variables are derived explicitly and updated successively at every subset selection step the model coefficients are only computed when the algorithm finishes the direct involvement of matrix inversions is thereby relieved a detailed computational complexity analysis indicates that the proposed algorithm possesses significant computational efficiency compared with the original approach where the well known efficient cholesky decomposition is involved in solving least angle regression three artificial and real world examples are employed to demonstrate the effectiveness efficiency and numerical stability of the proposed algorithm

a maximizing discriminability based self organizing fuzzy network for classification problems a maximizing discriminability based self organizing fuzzy network mdsofn that can classify highly confusable patterns is proposed in this paper the underlying notion of the proposed mdsofn is to split the generation of fuzzy rules into linear discriminant analysis lda and gaussian mixture model gmm in lda the weights are updated by seeking directions that are efficient for discrimination in gmm parameter learning adopts the gradient descent method to reduce the cost function since lda derived fuzzy rules increase the discriminative capability among different classes the proposed mdsofn can classify highly confusable patterns the effectiveness of the proposed mdsofn is demonstrated by two classification problems a detailed comparative performance analysis for the fuzzy networks using lda principal component analysis pca and the support vector machine svm with various noise types is presented experimental results and theoretical analysis indicate that the lda derived fuzzy network performs better than the pca based fuzzy network and the svm based fuzzy network

a multilevel nonlinear method a multilevel nonlinear method mnm for the numerical solution of nonlinear partial differential equations is developed the mnm algorithm is motivated and analyzed using a simplified model which retains the essential features of the new approach it is thereby shown to combine the advantages of the two classical multigrid approaches to nonlinear problems the analysis is supported by numerical tests for nonlinear differential equations in one and two dimensions

multi species pso and fuzzy systems of takagi sugeno kang type we study a method by using the hierarchical cluster based multi species particle swarm optimization hcmspso algorithm to generate a tagaki sugeno kang tsk fuzzy system implemented in a spatial analysis problem precisely we consider an area of study divided in subzones from the data measured in each subzone a tsk fuzzy system is extracted and hence we associate an opportune root means square error rmse if the hth and kth subzones have a defined suitably similarity index s h k greater or equal than a specific threshold s t h r e s h o l d then they are merged in a new subzone and the corresponding datasets are grouped together in a single dataset thus we restart the hcmspso algorithm for generating the tsk fuzzy system of the new subzone this process is iterated until we have that s h k

gp coach genetic programming based learning of compact and accurate fuzzy rule based classification systems for high dimensional problems in this paper we propose gp coach a genetic programming based method for the learning of compact and accurate fuzzy rule based classification systems for high dimensional problems gp coach learns disjunctive normal form rules generated by means of a context free grammar coded as one rule per tree the population constitutes the rule base so it is a genetic cooperative competitive learning approach gp coach uses a token competition mechanism to maintain the diversity of the population and this obliges the rules to compete and cooperate among themselves and allows the obtaining of a compact set of fuzzy rules the results obtained have been validated by the use of non parametric statistical tests showing a good performance in terms of accuracy and interpretability

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

fuzzy rule based models with interactive rules and their granular generalization abstract processing realized in fuzzy rule based models is associated with matching individual rules with the existing input data and aggregation of the levels of matching and conclusions of the rules rules are processed as individual entities in this study we introduce an augmentation of fuzzy models by facilitating interaction among the rules leading to more flexible type membership functions of fuzzy sets forming conditions of the rules thus resulting in substantially advanced topology of the partition of the input space different ways of realizing interaction among the rules are studied in the sequel we develop a granular fuzzy model implied by the rule based model showing how granular parameters of the original rule based model enhance its quality expressed in terms of coverage of experimental data the two evaluation criteria of the constructed granular model namely coverage and specificity are studied experimental results are reported for a series of publicly available data

multigrid navier stokes calculations for three dimensional cascades a vertex based finite volume method for solving the three dimensional compressible reynolds averaged navier stokes equations is presented for calculating turbomachinery cascade flows a discretization scheme for the viscous terms is proposed the baldwin lomax algebraic turbulence model is used the scheme is verified against laminar and turbulent flows over a flat plate two and three dimensional computations were carried out for a low pressure turbine cascade at design and off design conditions results are compared with inviscid solutions and experimental data

acceleration of multigrid flow computations through dynamic adaptation of the smoothing procedure abstract the paper presents the development and investigation of an adaptive smoothing as procedure in conjunction with the full multigrid full approximation storage method the latter has been previously implemented by the authors 1 for the incompressible navier stokes equations in conjunction with the artificial compressibility method and forms the basis for investigating the current as approach the principle of adaptive smoothing is to exploit the nonuniform convergence behavior of the numerical solution during the iterations to reduce the size of the computational domain and subsequently to reduce the total computing time the implementation of the as approach is investigated in conjunction with three different adaptivity criteria for two and three dimensional incompressible flows furthermore a dynamic procedure henceforth labeled dynamic adaptivity for defining variably the as parameters during the computation is also proposed and its performance is investigated in contrast to as with constant parameters henceforth labeled static adaptivity both static and dynamic adaptivity can provide similar acceleration but the latter additionally provides more stable numerical solutions and also requires less experimentation for optimizing the performance of the method numerical experiments are presented for attached and separated flows around airfoils as well as for three dimensional flow in a curved channel for external flows the as performs better when it is applied in all grid levels of the multigrid method while for internal flows the best performance is achieved when as is applied in the finest grid only overall the results show that substantial acceleration of multigrid computations can be achieved by using adaptive smoothing

local solution acceleration method for the euler and navier stokes equations the solution of the compressible euler and navier stokes equations via an upwind finite volume scheme is obtained for the inviscid fluxes the monotone upstream centered scheme for conservation laws muscl has been incorporated into a riemann solver the muscl scheme is used for the unfactored implicit equations that are solved by a newton form and relaxation is performed via gauss seidel relaxation technique the solution on the fine grid is obtained by iterating first on a sequence of coarser grids and then interpolating the solution up to the next refined grid since the distribution of the numerical error is nonuniform the local solution of the equations can be obtained in regions where the numerical errors are large the construction of the partial meshes in which the iterations will be continued is determined by an adaptive procedure taking into account some convergence criteria reduction of the computational work units for two dimensional problems is obtained via the local adaptive mesh solution which is expected to be more effective in three dimensional complex flow computations

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

identification and simplification of t s fuzzy neural networks based on incremental structure learning and similarity analysis abstract this paper proposes a novel identification method and a simplification scheme for t s fuzzy neural networks which consists of two steps the first step refers to the structure design based on an incremental clustering approach whose basic ideas are that the structure identification of fuzzy neural networks is guided by the attenuation of output approximation error in each cluster and processed by a recursive refined clustering iteration with the input space clustering and sub clustering as the main steps once the structure of a t s fuzzy neural network is identified by the incremental clustering approach its parameters are further learned and refined by the levenberg marquardt optimization algorithm the second step refers to the structure simplification including removing redundant fuzzy rules and merging highly similar fuzzy rules furthermore the performances of several similarity calculating methods are analyzed and discussed which provides a basis for the selection of the appropriate similarity analysis and effective calculating method for the merging of fuzzy rules and system simplification that is the given performance analysis provides the methodology basis and design guide for structure simplification based on similarity analysis and merger of fuzzy rules several experiments are implemented to illustrate the feasibility and effectiveness of the proposed approach

design of a high performance unsteady naiver stokes solver using a flexible cycle additive correction multigrid technique a multigrid technique based on a flexible cycle additive correction multigrid fcac mg scheme is utilized to design a high performance solver for the unsteady incompressible navier stokes n s equations the unsteady incompressible n s solver discussed here incorporates the fractional step method and finite volume discretization over a staggered cartesian grid the current research indicates that the fcac mg acceleration technique is highly efficient reliable and robust which makes it feasible for cpu intensive computations such as large eddy simulation les and direct numerical simulation dns the high efficiency and the robustness of the solver are achieved through developing the fcac mg acceleration technique as well as flow physics oriented solving strategies the flow solver based on the fcac mg technique is applied to both temporal and spatial turbulence simulations using both les and dns the residuals of the large scale algebraic equation system are guaranteed to be continuously driven down to the level of the computer machine round off error on each time marching step which warrants strong conservations of mass and momentum satisfied over all the control volumes in addition the flow solver developed in the current research possesses the potential capability for handling complex geometry flows since the performance of the solver does not rely on the use of special type of operations on the large scale algebraic equation system unlike the cyclic reduction algorithm but instead it is driven by a flow physics oriented solving strategy les for flows in a square duct a square annular duct and a confined square coaxial jet are performed to present the advantages of the fcac mg technique and the flow physics oriented solving strategy the analyses of the simulation results for these three flow configurations provide strong evidence that the flow solver based on fcac mg method is capable of capturing the major characteristics of turbulence physics and correctly predicting the relevant turbulent flow phenomena

an efficient ls svm based method for fuzzy system construction this paper proposes an efficient learning mechanism to build fuzzy rule based systems through the construction of sparse least squares support vector machines ls svms in addition to the significantly reduced computational complexity in model training the resultant ls svm based fuzzy system is sparser while offers satisfactory generalization capability over unseen data it is well known that the ls svms have their computational advantage over conventional svms in the model training process however the model sparseness is lost which is the main drawback of ls svms this is an open problem for the ls svms to tackle the nonsparseness issue a new regression alternative to the lagrangian solution for the ls svm is first presented a novel efficient learning mechanism is then proposed in this paper to extract a sparse set of support vectors for generating fuzzy if then rules this novel mechanism works in a stepwise subset selection manner including a forward expansion phase and a backward exclusion phase in each selection step the implementation of the algorithm is computationally very efficient due to the introduction of a few key techniques to avoid the matrix inverse operations to accelerate the training process the computational efficiency is also confirmed by detailed computational complexity analysis as a result the proposed approach is not only able to achieve the sparseness of the resultant ls svm based fuzzy systems but significantly reduces the amount of computational effort in model training as well three experimental examples are presented to demonstrate the effectiveness and efficiency of the proposed learning mechanism and the sparseness of the obtained ls svm based fuzzy systems in comparison with other svm based learning techniques

support vector learning for fuzzy rule based classification systems to design a fuzzy rule based classification system fuzzy classifier with good generalization ability in a high dimensional feature space has been an active research topic for a long time as a powerful machine learning approach for pattern recognition problems the support vector machine svm is known to have good generalization ability more importantly an svm can work very well on a high or even infinite dimensional feature space this paper investigates the connection between fuzzy classifiers and kernel machines establishes a link between fuzzy rules and kernels and proposes a learning algorithm for fuzzy classifiers we first show that a fuzzy classifier implicitly defines a translation invariant kernel under the assumption that all membership functions associated with the same input variable are generated from location transformation of a reference function fuzzy inference on the if part of a fuzzy rule can be viewed as evaluating the kernel function the kernel function is then proven to be a mercer kernel if the reference functions meet a certain spectral requirement the corresponding fuzzy classifier is named positive definite fuzzy classifier pdfc a pdfc can be built from the given training samples based on a support vector learning approach with the if part fuzzy rules given by the support vectors since the learning process minimizes an upper bound on the expected risk expected prediction error instead of the empirical risk training error the resulting pdfc usually has good generalization moreover because of the sparsity properties of the svms the number of fuzzy rules is irrelevant to the dimension of input space in this sense we avoid the curse of dimensionality finally pdfcs with different reference functions are constructed using the support vector learning approach the performance of the pdfcs is illustrated by extensive experimental results comparisons with other methods are also provided

an interval type 2 neural fuzzy classifier learned through soft margin minimization and its human posture classification application this paper proposes an interval type 2 neural fuzzy classifier learned through soft margin minimization it2nfc smm and applies it to human body posture classification the it2nfc smm consists of interval type 2 zero order takagi sugeno t s fuzzy rules established through online structure learning the antecedent part of the it2nfc smm uses interval type 2 fuzzy sets to decrease the number of rules and manage noisy data for parameter learning the consequent parameters are learned through a linear support vector machine svm for soft margin minimization to improve the generalization ability the proposed svm based learning addresses the problem that the orders of the fuzzy rules in computing the outputs of an interval type 2 fuzzy system depend on the consequent values that are unknown in advance to address this problem the it2nfc smm uses weighted bound set boundaries to simplify the type reduction operation and a novel crisp to interval linear svm learning algorithm based on the soft margin minimization the antecedent parameters are tuned using the gradient descent algorithm the it2nfc smm is applied to a vision based human body posture classification system the system uses two cameras and novel classification features extracted from a silhouette of the human body to classify the four postures of standing bending sitting and lying the classification performance of the it2nfc smm is verified through results in clean and noisy classification examples and through the posture classification problem as well as through comparisons with various type 1 and type 2 fuzzy classifiers the overall result shows that the it2nfc smm achieves higher classification rates with a smaller or similar model size than the classifiers used for comparison especially for noisy classification problems

an interval type 2 neural fuzzy system for online system identification and feature elimination we propose an integrated mechanism for discarding derogatory features and extraction of fuzzy rules based on an interval type 2 neural fuzzy system nfs in fact it is a more general scheme that can discard bad features irrelevant antecedent clauses and even irrelevant rules high dimensional input variable and a large number of rules not only enhance the computational complexity of nfss but also reduce their interpretability therefore a mechanism for simultaneous extraction of fuzzy rules and reducing the impact of or eliminating the inferior features is necessary the proposed approach namely an interval type 2 neural fuzzy system for online system identification and feature elimination it2nfs sife uses type 2 fuzzy sets to model uncertainties associated with information and data in designing the knowledge base the consequent part of the it2nfs sife is of takagi sugeno kang type with interval weights the it2nfs sife possesses a self evolving property that can automatically generate fuzzy rules the poor features can be discarded through the concept of a membership modulator the antecedent and modulator weights are learned using a gradient descent algorithm the consequent part weights are tuned via the rule ordered kalman filter algorithm to enhance learning effectiveness simulation results show that it2nfs sife not only simplifies the system architecture by eliminating derogatory irrelevant antecedent clauses rules and features but also maintains excellent performance

fully adaptive multigrid methods adaptivity is a key concept for the effective numerical solution of differential equations the multilevel solution of elliptic partial differential equations can be combined with adaptive mesh refinement and an adaptive choice of the discretization order additionally adaptivity may be built into the relaxation and the multilevel cycling strategy the goal of these fully adaptive methods is to spend work only where it is most effective in the solution process this approach includes concepts like local relaxation and not only leads to particularly fast convergence but also to additional robustness and generality the efficient implementation of fully adaptive multilevel methods in a finite element framework will be discussed

a hybrid multigrid method for the unsteady incompressible navier stokes equations an approach is presented for the high reynolds incompressible flow based on multigrid method and sequential regular method in this paper the velocity pressure increment and sequential regular equations are derived from the navier stokes equation the pressure increment optimum sweep algorithm is discussed the numerical results of the close square flow show that our algorithm improves the convergence rate for high reynolds unsteady flow

topology selection for particle swarm optimization particle swarm optimization pso uses a social topology for particles to share information among neighbors during optimization a large number of existing literatures have shown that the topology affects the performance of pso and an optimal topology is problem dependent but currently there is a lack of study on this issue in this paper we first analyze a class of deterministic regular topologies with regard to what affect the optimality of algorithmic parameters e g the number of particles and the topological degree so as to provide a guide to topology selections for pso both theoretical analysis and numerical experiments are performed and reported in detail the theoretical analysis reveals that the optimality of algorithmic parameters is dependent on the computational budget available in particular the optimal number of particles increases unstrictly as the computational budget increases while for any fixed number of particles the optimal degree decreases unstrictly as computational budget increases the only condition is that the computational budget cannot exceed a constant measuring the hardness of the benchmark function set with a total of 198 regular topologies and 9 different numbers of particles tested on 90 benchmark functions using a recently reported data profiling technique numerical experiments verify the theoretical derivations based on these results two formulas are developed to help choose optimal topology parameters for increased ease and applicability of pso to real world problems

sgerd a steady state genetic algorithm for extracting fuzzy classification rules from data this paper considers the automatic design of fuzzy rule based classification systems from labeled data the performance of classifiers and the interpretability of generated rules are of major importance in these systems in past research some genetic based algorithms have been used for the rule learning process these genetic fuzzy systems have utilized different approaches to encode rules in this paper we have proposed a novel steady state genetic algorithm to extract a compact set of good fuzzy rules from numerical data sgerd the selection mechanism of this algorithm is nonrandom and only the best individuals can survive our approach is very simple and fast and can be applied to high dimensional problems with numerical attributes to select the rules having high generalization capabilities our algorithm makes use of some rule and data dependent parameters we have also proposed an enhancing function that modifies the rule evaluation measures in order to assess the candidate rules more effectively before their selection experiments on some well known data sets are performed to show the performance of sgerd

design of an fir filter for the displacement reconstruction using measured acceleration in low frequency dominant structures this paper presents a new class of displacement reconstruction scheme using only acceleration measured from a structure for a given set of acceleration data the reconstruction problem is formulated as a boundary value problem in which the acceleration is approximated by the second order central finite difference of displacement the displacement is reconstructed by minimizing the least squared errors between measured and approximated acceleration within a finite time interval an overlapping time window is introduced to improve the accuracy of the reconstructed displacement the displacement reconstruction problem becomes ill posed because the boundary conditions at both ends of each time window are not known a priori furthermore random noise in measured acceleration causes physically inadmissible errors in the reconstructed displacement a tikhonov regularization scheme is adopted to alleviate the ill posedness it is shown that the proposed method is equivalent to an fir filter designed in the time domain the fundamental characteristics of the proposed method are presented in the frequency domain using the transfer function and the accuracy function the validity of the proposed method is demonstrated by a numerical example a laboratory experiment and a field test copyright 2009 john wiley sons ltd

functional equivalence between radial basis function networks and fuzzy inference systems it is shown that under some minor restrictions the functional behavior of radial basis function networks rbfns and that of fuzzy inference systems are actually equivalent this functional equivalence makes it possible to apply what has been discovered learning rule representational power etc for one of the models to the other and vice versa it is of interest to observe that two models stemming from different origins turn out to be functionally equivalent

hierarchical cluster based multispecies particle swarm optimization for fuzzy system optimization this paper proposes a hierarchical cluster based multispecies particle swarm optimization hcmspso algorithm for fuzzy system optimization the objective of this paper is to learn takagi sugeno kang tsk type fuzzy rules with high accuracy in the hcmspso designed fuzzy system fs each rule defines its own fuzzy sets which implies that the number of fuzzy sets for each input variable is equal to the number of fuzzy rules a swarm in hcmspso is clustered into multiple species at an upper hierarchical level and each species is further clustered into multiple subspecies at a lower hierarchical level for an fs consisting of r rules r species swarms are formed in the upper level where one species optimizes a single fuzzy rule initially there are no species in hcmspso an online cluster based algorithm is proposed to generate new species fuzzy rules automatically in the lower layer subspecies within the same species are formed adaptively in each iteration during the particle update several simulations are conducted to verify hcmspso performance comparisons with other neural learning genetic and pso algorithms demonstrate the superiority of hcmspso performance

a multigrid semi implicit line method for viscous incompressible and low mach number flows on high aspect ratio grids abstract discretization of viscous incompressible and viscous low mach number flows often leads to a system of equations which is very difficult to solve there are two reasons first the use of high aspect ratio grids results in a very numerically anisotropic behaviour of the diffusive and acoustic terms and second in low mach number flow the ratio of the convective and acoustic eigenvalues of the inviscid system becomes very high we implemented an ausm based discretization method using an explicit third order discretization for the convective part and a line implicit central discretization for the acoustic part and for the diffusive part the lines are chosen in the direction of the grid points with shortest connection the preconditioned semi implicit line method is used in multistage form because of the explicit third order discretization of the convective part multigrid is used as an acceleration technique it is shown that the convergence is very good independent of grid aspect ratio and mach number

on the multilevel adaptive iterative method the multilevel adaptive iterative method is a technique for solving the sparse matrix equations that typically arise from partial differential systems its core consists of a relaxation scheme and an active set strategy the active set is used to monitor where the iteration efficiently reduces the error it is incrementally updated by exploiting the current solution and the matrix structure and most arithmetic operations are restricted to it the algorithm can be extended to a multilevel structure by additionally tracing the dependencies between unknowns on different levels it improves the robustness and efficiency of classical multilevel methods in particular it is an almost ideal supplement to adaptive refinement techniques

efficient and non reflecting far field boundary conditions for incompressible flow calculations abstract traditionally the artificial compressibility ac method of chorin is used for simulation of low speed and incompressible flows in this method the difficulty of continuity and momentum equations decoupling is removed by adding an artificial time derivative of pressure to the continuity equation for the first time a fully two dimensional upwind scheme was presented for ac equations by using characteristic structure of equations by the authors in this paper a new remote boundary calculation method is presented by using the idea of characteristics for these equations instead of simple boundary conditions which usually are employed for incompressible flows the flow quantities at the far field boundaries are evaluated by compatibility relations of characteristic equations this is implemented by assuming a row of ghost cells outside of the far field boundary and using the flux calculation method based on characteristics similar to the inside of computational domain the idea in conjunction with multidimensional characteristic based scheme was tested for incompressible flow around circular cylinder in comparison with conventional far field boundary condition and showed good improvements in the terms of accuracy and convergence speed

an online self constructing neural fuzzy inference network and its applications a self constructing neural fuzzy inference network sonfin with online learning ability is proposed in this paper the sonfin is inherently a modified takagi sugeno kang tsk type fuzzy rule based model possessing neural network learning ability there are no rules initially in the sonfin they are created and adapted as online learning proceeds via simultaneous structure and parameter identification in the structure identification of the precondition part the input space is partitioned in a flexible way according to an aligned clustering based algorithm as to the structure identification of the consequent part only a singleton value selected by a clustering method is assigned to each rule initially afterwards some additional significant terms selected via a projection based correlation measure for each rule will be added to the consequent part incrementally as learning proceeds the combined precondition and consequent structure identification scheme can set up an economic and dynamically growing network a main feature of the sonfin in the parameter identification the consequent parameters are tuned optimally by either least mean squares or recursive least squares algorithms and the precondition parameters are tuned by a backpropagation algorithm to enhance the knowledge representation ability of the sonfin a linear transformation for each input variable can be incorporated into the network so that much fewer rules are needed or higher accuracy can be achieved

reconstruction of dynamic displacement and velocity from measured accelerations using the variational statement of an inverse problem this paper presents two types of finite impulse response fir filters to reconstruct dynamic displacement induced by structural vibration from measured acceleration the governing equation for the reconstruction is derived by taking the variation of a minimization problem which defines an inverse problem on displacement a regularization function for overcoming the ill posedness of the inverse problem is included in the minimization problem the governing equation of the inverse problem becomes the same type of differential equation as that of a beam on an elastic foundation the conventional fir cfir filter directly approximates the transfer function of the governing equation while the fem based fir ffir filter is formulated by the discretization of the minimization problem with the finite element method for the finite element discretization the hermitian shape function is utilized the proposed ffir filter is capable of reconstructing displacement and velocity simultaneously the fundamental characteristics of the proposed filters are investigated in the frequency domain using the transfer and accuracy functions it is shown that the proposed fir filters suppress low frequency noise components in measured accelerations effectively and reconstruct physically meaningful displacement accurately the validity of the proposed filters is demonstrated through a numerical simulation study a field experiment and an evaluation of flutter derivatives using measurements taken from a wind tunnel test

support vector machines for histogram based image classification traditional classification approaches generalize poorly on image classification tasks because of the high dimensionality of the feature space this paper shows that support vector machines svm can generalize well on difficult image classification problems where the only features are high dimensional histograms heavy tailed rbf kernels of the form k x y e sup spl rho spl sigma sub i sup xia yia b with a spl les 1 and b spl les 2 are evaluated on the classification of images extracted from the corel stock photo collection and shown to far outperform traditional polynomial or gaussian radial basis function rbf kernels moreover we observed that a simple remapping of the input x sub i spl rarr x sub i sup a improves the performance of linear svm to such an extend that it makes them for this problem a valid alternative to rbf kernels

multigrid solution of the navier stokes equations at low speeds with large temperature variations multigrid methods for the navier stokes equations at low speeds and large temperature variations are investigated the compressible equations with time derivative preconditioning and preconditioned flux difference splitting of the inviscid terms are used three implicit smoothers have been incorporated into a common multigrid procedure both full coarsening and semi coarsening with directional fine grid defect correction have been studied the resulting methods have been tested on four 2d laminar problems over a range of reynolds numbers on both uniform and highly stretched grids two of the three methods show efficient and robust performance over the entire range of conditions in addition none of the methods has any difficulty with the large temperature variations

support vector learning mechanism for fuzzy rule based modeling a new approach this paper describes a fuzzy modeling framework based on support vector machine a rule based framework that explicitly characterizes the representation in fuzzy inference procedure the support vector learning mechanism provides an architecture to extract support vectors for generating fuzzy if then rules from the training data set and a method to describe the fuzzy system in terms of kernel functions thus it has the inherent advantage that the model does not have to determine the number of rules in advance and the overall fuzzy inference system can be represented as series expansion of fuzzy basis functions the performance of the proposed approach is compared to other fuzzy rule based modeling methods using four data sets

structural system identification in time domain using measured acceleration this paper presents a system identification scheme in time domain to estimate stiffness and damping parameters of a structure using measured acceleration an error function is defined as the time integral of the least squared errors between measured accelerations and calculated accelerations by a numerical model of a structure to alleviate the ill posedness of si problems a regularization technique is employed and a new regularization function for the time domain si is proposed the regularization factor is determined by the geometric mean scheme the validity of the proposed method is demonstrated by a numerical simulation study on a two span truss bridge and by an experimental laboratory study on a three story shear building model

improved convergence to the steady state of the euler equations by enhanced wave propagation the convergence to the numerical solution of the stationary euler equations in two and three dimensions is studied the basic iterative algorithms are runge kutta time stepping gmres and a modified gmres method convergence acceleration is achieved by two preconditioning techniques residual smoothing and multigrid iteration the preconditioners are such that they increase the propagation of smooth error modes out of the computational domain runge kutta time stepping and the modified gmres method guarantee this wave propagation the results from a number of numerical experiments are reported

support vector based fuzzy neural network for pattern classification fuzzy neural networks fnns for pattern classification usually use the backpropagation or c cluster type learning algorithms to learn the parameters of the fuzzy rules and membership functions from the training data however such kinds of learning algorithms usually cannot minimize the empirical risk training error and expected risk testing error simultaneously and thus cannot reach a good classification performance in the testing phase to tackle this drawback a support vector based fuzzy neural network svfnn is proposed for pattern classification in this paper the svfnn combines the superior classification power of support vector machine svm in high dimensional data spaces and the efficient human like reasoning of fnn in handling uncertainty information a learning algorithm consisting of three learning phases is developed to construct the svfnn and train its parameters in the first phase the fuzzy rules and membership functions are automatically determined by the clustering principle in the second phase the parameters of fnn are calculated by the svm with the proposed adaptive fuzzy kernel function in the third phase the relevant fuzzy rules are selected by the proposed reducing fuzzy rule method to investigate the effectiveness of the proposed svfnn classification it is applied to the iris vehicle dna satimage ijcnn1 datasets from the uci repository statlog collection and ijcnn challenge 2001 respectively experimental results show that the proposed svfnn for pattern classification can achieve good classification performance with drastically reduced number of fuzzy kernel functions

incremental evolutionary design of tsk fuzzy controllers this paper proposes a novel method for the incremental design and optimization of first order tagaki sugeno kang tsk fuzzy controllers by means of an evolutionary algorithm starting with a single linear control law the controller structure is gradually refined during the evolution structural augmentation is intertwined with evolutionary adaptation of the additional parameters with the objective not only to improve the control performance but also to maximize the stability region of the nonlinear system from the viewpoint of optimization the proposed method follows a divide and conquer approach additional rules and their parameters are introduced into the controller structure in a neutral fashion such that the adaptations of the less complex controller in the previous stage are initially preserved the proposed scheme is evaluated at the task of tsk fuzzy controller design for the upswing and stabilization of a rotational inverted pendulum in the first case the objective is a time optimal controller that upswings the pendulum in to the upper equilibrium point in shortest time the stabilizing controller is designed as a state optimal controller in a second application the optimization method is applied to the design of a fuzzy controller for vision based mobile robot navigation the results demonstrate that the incremental scheme generates solutions that are similar in control performance to pure parameter optimization of only the gains of a tsk system even more important whereas direct optimization of control systems with more than 35 rules fails to identify a stabilizing control law the incremental scheme optimizes fuzzy state space partitions and gains for hundreds of rules

on the design of linear phase fir integrators for midband frequencies the ideal frequency responses 1 j spl omega an integrator and 1 spl omega an integrating hilbert transformer are usually approximated by iir structures this paper suggests their fir approximations maximal flatness at spl omega spl pi 2 yields exact mathematical formulas for computing the weights the proposed designs are particularly suitable for the frequency range 0 1 spl pi spl les spl omega spl les 0 90 spl pi

a ts type maximizing discriminability based recurrent fuzzy network for classification problems this work proposes a takagi sugeno ts type maximizing discriminability based recurrent fuzzy network mdrfn that can classify highly confusable patterns the discriminative capability plays a significant role in determining classification performance to increase the discriminative capability the proposed mdrfn considers minimum classification error mce and minimum training error mte in mce the weights are updated by maximizing the discrimination among different classes in mte the parameter learning adopts the gradient descent method to reduce the cost function therefore the novelty of mdrfn is that it not only minimizes the cost function but maximizes the discriminative capability as well the effectiveness of the proposed mdrfn is demonstrated by three temporal classification problems in the experiments other rfns including the singleton type recurrent neural fuzzy network srnfn ts type rfn trfn and simple rfn srfn are compared analysis results indicate that the proposed mdrfn exhibits excellent classification performance

generating a hierarchical fuzzy rule based model abstract this study proposes a novel methodology for the extraction of a hierarchical takagi sugeno fuzzy rule based architecture from data this architecture reduces the number and complexity of involved fuzzy rules and in many cases improves the predictive performance of the model the proposed hierarchical architecture takes the form of a cascading topology in which the predicted result computed at the previous layer is considered in the output part of the fuzzy rules we propose a well defined general methodology for the extraction of this hierarchical topology from data and discuss strategies for feature selection and choosing the number of rules at each level the performance of the proposed methodology is demonstrated through extensive experiments including case studies outlining specific behaviors and parameterizations and comparative experiments showing the performance of the proposed architecture compared to a standard flat fuzzy rule based system

a nonlinear multigrid method for the three dimensional incompressible navier stokes equations a nonlinear multigrid method is developed for solving the three dimensional navier stokes equations in conjunction with the artificial compressibility formulation the method is based on the full multigrid fmg full approximation storage fas algorithm and is realized via an unsteady type procedure according to which the equations are not solved exactly on the coarsest grid but some pseudo time iterations are performed on the finer grids and some on the coarsest grid the multigrid method is implemented in conjunction with a third order upwind characteristics based scheme for the discretization of the convection terms and the fourth order runge kutta scheme for time integration the performance of the method is investigated for three dimensional flows in straight and curved channels as well as flow in a cubic cavity the multigrid acceleration is assessed in contrast to the single grid and mesh sequencing algorithms the effects of various multigrid components on the convergence acceleration such as prolongation operators as well as pre and postrelaxation iterations are also investigated

low level interpretability and high level interpretability a unified view of data driven interpretable fuzzy system modelling this paper aims at providing an in depth overview of designing interpretable fuzzy inference models from data within a unified framework the objective of complex system modelling is to develop reliable and understandable models for human being to get insights into complex real world systems whose first principle models are unknown because system behaviour can be described naturally as a series of linguistic rules data driven fuzzy modelling becomes an attractive and widely used paradigm for this purpose however fuzzy models constructed from data by adaptive learning algorithms usually suffer from the loss of model interpretability model accuracy and interpretability are two conflicting objectives so interpretation preservation during adaptation in data driven fuzzy system modelling is a challenging task which has received much attention in fuzzy system modelling community in order to clearly discriminate the different roles of fuzzy sets input variables and other components in achieving an interpretable fuzzy model a taxonomy of fuzzy model interpretability is first proposed in terms of low level interpretability and high level interpretability in this paper the low level interpretability of fuzzy models refers to fuzzy model interpretability achieved by optimizing the membership functions in terms of semantic criteria on fuzzy set level while the high level interpretability refers to fuzzy model interpretability obtained by dealing with the coverage completeness and consistency of the rules in terms of the criteria on fuzzy rule level some criteria for low level interpretability and high level interpretability are identified respectively different data driven fuzzy modelling techniques in the literature focusing on the interpretability issues are reviewed and discussed from the perspective of low level interpretability and high level interpretability furthermore some open problems about interpretable fuzzy models are identified and some potential new research directions on fuzzy model interpretability are also suggested

an extension on statistical comparisons of classifiers over multiple data sets for all pairwise comparisons in a recently published paper in jmlr dem sar 2006 recommends a set of non parametric statistical tests and procedures which can be safely used for comparing the performance of classifiers over multiple data sets after studying the paper we realize that the paper correctly introduces the basic procedures and some of the most advanced ones when comparing a control method however it does not deal with some advanced topics in depth regarding these topics we focus on more powerful proposals of statistical procedures for comparing n n classifiers moreover we illustrate an easy way of obtaining adjusted and comparable p values in multiple comparison procedures

a novel two grid method for semilinear elliptic equations a new finite element discretization technique based on two coarse and fine subspaces is presented for a semilinear elliptic boundary value problem the solution of a nonlinear system on the fine space is reduced to the solution of two small one linear and one nonlinear systems on the coarse space and a linear system on the fine space it is shown both theoretically and numerically that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation as a result the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

fuzzy approximation relations on fuzzy n cell number space and their applications in classification in this paper the problems of fuzzy binary relations on fuzzy n cell number space and their applications are investigated firstly we have defined some fuzzy approximation relations on fuzzy n cell number space and studied their properties secondly as application we have developed an algorithmic version of classification in an imprecise or uncertain environment by using the fuzzy approximation relations practical examples are provided to show the application and rationality of the proposed techniques

rigorous quantitative analysis of multigrid i constant coefficients two level cycle with l 2 norm exact numerical convergence factors for any multigrid cycle can be predicted by local mode fourier analysis for general linear elliptic partial differential equation pde systems with piecewise smooth coefficients in general domains discretized by uniform grids it is proved that in the limit of small meshsizes these predicted factors are indeed obtained provided the cycle is supplemented with a proper processing at and near the boundaries that processing it is proved costs negligible extra computer work apart from mode analysis a coarse grid approximation cga condition is introduced which is both necessary and sufficient for the multigrid algorithm to work properly the present part studies the l 2 convergence in one cycle for equations with constant coefficients in the sequel brandt rigorous quantitative analysis of multigrid ii extensions and practical implications manuscript extensions are discussed to many cycles asymptotic convergence to more levels with arbitrary cycle ty

parallel multigrid computation of unsteady incompressible viscous flows using a matrix free implicit method and high resolution characteristics based scheme abstract a three dimensional parallel unstructured non nested multigrid solver for solutions of unsteady incompressible viscous flow is developed and validated the finite volume navier stokes solver is based on the artificial compressibility approach with a high resolution method of characteristics based scheme for handling convection terms the unsteady flow is calculated with a matrix free implicit dual time stepping scheme the parallelization of the multigrid solver is achieved by multigrid domain decomposition approach mg dd using single program multiple data spmd and multiple instruction multiple data mimd programming paradigm there are two parallelization strategies proposed in this work first strategy is a one level parallelization strategy using geometric domain decomposition technique alone second strategy is a two level parallelization strategy that consists of a hybrid of both geometric domain decomposition and data decomposition techniques message passing interface mpi and openmp standard are used to communicate data between processors and decompose loop iterations arrays respectively the parallel multigrid code is used to simulate both steady and unsteady incompressible viscous flows over a circular cylinder and a lid driven cavity flow a maximum speedup of 22 5 could be achieved on 32 processors for instance the lid driven cavity flow of re 1000 the results obtained agree well with numerical solutions obtained by other researchers as well as experimental measurements a detailed study of the time step size and number of pseudo sub iterations per time step required for simulating unsteady flow are presented in this paper

neuro fuzzy systems for function approximation abstract we present a neuro fuzzy architecture for function approximation based on supervised learning the learning algorithm is able to determine the structure and the parameters of a fuzzy system the approach is an extension to our already published nefcon and nefclass models which are used for control or classification purposes the proposed extended model which we call nefprox is more general and can be used for any application based on function approximation

particle swarm optimization based on intermediate disturbance strategy algorithm and its application in multi threshold image segmentation abstract particle swarm optimization pso algorithm simulates social behavior among individuals or particles flying through multidimensional search space for enhancing the local search ability of pso and guiding the search a region that had most number of the particles was defined and analyzed in detail inspired by the ecological behavior we presented a pso algorithm with intermediate disturbance searching strategy idpso which enhances the global search ability of particles and increases their convergence rates the experimental results on comparing the idpso to ten known pso variants on 16 benchmark problems demonstrated the effectiveness of the proposed algorithm furthermore we applied the idpso algorithm to multilevel image segmentation problem for shortening the computational time experimental results of the new algorithm on a variety of images showed that it can effectively segment an image faster

genetic programming for model selection of tsk fuzzy systems this paper compares a genetic programming gp approach with a greedy partition algorithm lolimot for structure identification of local linear neuro fuzzy models the crisp linear conclusion part of a takagi sugeno kang tsk fuzzy rule describes the underlying model in the local region specified in the premise the objective of structure identification is to identify an optimal partition of the input space into gaussian axis orthogonal fuzzy sets the linear parameters in the rule consequent are then estimated by means of a local weighted least squares algorithm lolimot is an incremental tree construction algorithm that partitions the input space by axis orthogonal splits in each iteration it greedily adds the new model that minimizes the classification error gp performs a global search for the optimal partition tree and is therefore able to backtrack in case of sub optimal intermediate split decisions we compare the performance of both methods for function approximation of a highly non linear two dimensional test function and an engine characteristic map

geometric multigrid with applications to computational fluid dynamics an overview is given of the development of the geometric multigrid method with emphasis on applications in computational fluid dynamics over the last ten years both compressible and incompressible flow problems and their corresponding multigrid solution methods are discussed the state of the art is described with respect to methods employed in industry as well as the multigrid efficiency obtained in academic applications

multidimensional upwinding for incompressible flows based on characteristics in this paper the multidimensional characteristic based upwind scheme mcb which has been recently introduced by the authors is applied to two another benchmark problems namely flow in a channel with a backward facing step and two dimensional steady and unsteady flows past a circular cylinder extension of mcb scheme for calculating convective fluxes on non cartesian grids is presented here for the flow over backward facing step obtained results were compared against well known experimental data and the results show high accuracy of mcb scheme and faster convergence rate with respect to conventional cb scheme in the case of flow over circular cylinder the flow at steady and transient regimes is investigated by mcb scheme again the results obtained by mcb are compared to the other results in the literature and show good agreement with them also rapid convergence rate of mcb was observed in this case too it is concluded that the genuinely multidimensional characteristic based mcb scheme has been introduced earlier by the authors is a robust and powerful scheme for modeling incompressible viscous flows for achieving the high accuracy and remarkable advantage in convergence rate with respect to conventional characteristic based schemes