or and neuron in modeling fuzzy set connectives the paper introduces a neural network based model of logical connectives the basic processing unit consists of two types of generic or and and neurons structured into a three layer topology due to the functional integrity we will be referring to it as an or and neuron the specificity of the logical connectives is captured by the or and neuron within its supervised learning further analysis of the connections of the neuron obtained in this way provides a better insight into the nature of the connectives applied in fuzzy sets by emphasizing their features of locality and interactivity afterward we will study several architectures of neural networks comprising these neurons treated as their basic functional components the numerical studies embrace both the structures formed by single or and neurons and aimed at modeling logical connectives including the zimmermann zysno data set 1980 and the networks representing various decision making architectures we will also propose a realization of a pseudo median filter in which the or and neurons play an ultimate role

iteratively regularized gauss newton method for nonlinear inverse problems with random noise we study the convergence of regularized newton methods applied to nonlinear operator equations in hilbert spaces if the data are perturbed by random noise it is shown that the expected square error is bounded by a constant times the minimax rates of the corresponding linearized problem if the stopping index is chosen using a priori knowledge of the smoothness of the solution for unknown smoothness the stopping index can be chosen adaptively based on lepskib s balancing principle for this stopping rule we establish an oracle inequality which implies order optimal rates for deterministic errors and optimal rates up to a logarithmic factor for random noise the performance and the statistical properties of the proposed method are illustrated by monte carlo simulations

convergence rates for tikhonov regularization of a two coefficient identification problem in an elliptic boundary value problem we investigate the convergence rates for tikhonov regularization of the problem of simultaneously estimating the coefficients q and a in the neumann problem for the elliptic equation rm div q nabla u au f rm in omega q partial u partial n g on the boundary partial omega omega subset mathbb r d d geq 1 when u is imprecisely given by z delta in h 1 omega u z delta h 1 omega le delta delta 0 we regularize this problem by minimizing the strictly convex functional of q a begin array lll int limits omega left q nabla u q a z delta 2 a u q a z delta 2 right dx quad rho left q q 2 l 2 omega a a 2 l 2 omega right end array over the admissible set k where 0 is the regularization parameter and q a is an a priori estimate of the true pair q a which is identified and consider the unique solution of these minimization problem as the regularized one to that of the inverse problem we obtain the convergence rate mathcal o sqrt delta as 0 and for the regularized solutions under the simple and weak source condition rm there is a function w in v rm such that u prime q dagger a dagger w q dagger q a dagger a with q dagger a dagger being the q a minimum norm solution of the coefficient identification problem u the frechet derivative of u v the sobolev space on which the boundary value problem is considered our source condition is without the smallness requirement on the source function which is popularized in the theory of regularization of nonlinear ill posed problems furthermore some concrete cases of our source condition are proved to be simply the requirement that the sought coefficients belong to certain smooth function spaces

generalized means as model of compensative connectives abstract empirical investigations have shown that instead of strong logical and and or connectives there is often some degree of compensation in between in this paper the generalized means are proposed as connective operators for fuzzy set theory which easily allow for modelling the degree of compensation in a natural manner including the arithmetic and geometric means as well as the maximum and minimum operators as special cases numerical examples with empirical data form an illustration of this concept some aspects of applications to decision making are discussed

a projection regularized newton method for nonlinear ill posed problems and its application to parameter identification problems with finite element discretization this paper is concerned with nonlinear ill posed operator equations f a y e g parameter identification problems and their approximate solution by a newton type method that is regularized by projecting the linearized equation in each newton step onto a finite dimensional space e g a finite element space and by stopping the newton iteration at an appropriate index we prove convergence as the iteration index n goes to infinity in the noise free case and convergence as the data noise level delta goes to zero in the case of noisy data as well as convergence rates under certain additional conditions

hierarchies consistency analysis for nontransitive problems abstract by assuming different hierarchies or different hierarchy structures for the same problem should yield the same aggregated performance index an approach can be developed for evaluating the performance of multiple level and multiple criteria problems this proposed approach has the advantage that it is completely based on a rational index and no requirement of transitivity is needed furthermore the analyses of different criteria structures give further insights about the problem being considered the disadvantage is of course we must assume that different hierarchies must yield the same aggregate performance index which is not generally true for all problems under all conditions

levenberg marquardt methods for constrained nonlinear equations with strong local convergence properties we consider the problem of finding a solution of a constrained and not necessarily square system of equations i e we consider systems of nonlinear equations and want to find a solution that belongs to a certain feasible set to this end we present two levenberg marquardt type algorithms that differ in the way they compute their search directions the first method solves a strictly convex minimization problem at each iteration whereas the second one solves only one system of linear equations in each step both methods are shown to converge locally quadratically under an error bound assumption that is much weaker than the standard nonsingularity condition both methods can be globalized in an easy way some numerical results for the second method indicate that the algorithm works quite well in practice

a triparametric family of cardinality based fuzzy similarity measures previously we introduced a biparametric family of cardinality based fuzzy similarity measures in this paper we generalize this family by adding a third parameter we also study the generalized family for specific values of the third parameter more precisely we show that for particular values of this parameter certain properties can be ensured by imposing constraints on the two remaining parameters to conclude we examine some members of the presented family of fuzzy similarity measures

level set and total variation regularization for elliptic inverse problems with discontinuous coefficients we propose a level set approach for elliptic inverse problems with piecewise constant coefficients the geometry of the discontinuity of the coefficient is represented implicitly by level set functions the inverse problem is solved using a variational augmented lagrangian formulation with total variation regularization of the coefficient the corresponding euler lagrange equation gives the evolution equation for the level set functions and the constant values of the coefficients we use a multiple level set representation which allows the coefficient to have multiple constant regions knowledge of the exact number of regions is not required only an upper bound is needed numerical experiments show that the method can recover coefficients with rather complicated geometries of discontinuities under moderate amount of noise in the observation data the method is also robust with respect to the initial guess for the geometry of the coefficient discontinuities

a variational method for numerical differentiation a new method is given for effecting numerical differentiation by means of an optimization procedure the method is shown to be effective for the differentiation of noisy functions and stability and convergence results for a steepest descent implementation are proved

using fuzzy sets in operational research abstract during the last four decades a large number of mainly mathematical tools have been developed in and for operational research they are primarily devices to find optimal solutions after a problem has been modelled formally hardly any new tools have been developed for proper modelling of problems which are not well structured and which can not easily be cast into classical mathematical models fuzzy set theory as proposed by l zadeh in 1965 promises to bridge part of this gap it is much more adaptable to different problem structures and better suited to model human evaluation and decision making processes than classical mathematics this paper will primarily focus attention on four questions why fuzzy sets in or what are fuzzy sets where can they be used and how are they being used

on a reflexivity preserving family of cardinality based fuzzy comparison measures in this paper we generalize and extend previous work on a triparametric family of cardinality based fuzzy similarity measures more specifically we i generalize this family to a broader family of fuzzy comparison measures ii summarize previously obtained results regarding this family and fill up a few gaps by adding some novel results iii compare our parametric family with an already existing one and iv elaborate on the ranking invariance between certain members of the presented triparametric family of cardinality based fuzzy comparison measures

identification of chemotaxis models with volume filling chemotaxis refers to the directed movement of cells in response to a chemical signal called chemoattractant a crucial point in the mathematical modeling of chemotactic processes is the correct description of the chemotactic sensitivity and of the production rate of the chemoattractant in this paper we investigate the identification of these nonlinear parameter functions in a chemotaxis model with volume filling we also discuss the numerical realization of tikhonov regularization for the stable solution of the inverse problem our theoretical findings are supported by numerical tests

continuity and uniqueness of regularized output least squares optimal estimators the continuity and uniqueness properties of optimal estimators with respect to data are considered for different regularizations it is found that there is a weak stability for optimal estimators as set valued mappings under a weak regularization for stronger regularization results are obtained that give stability in stronger topologies and the finiteness of the set of optimal estimators finally we give conditions that imply uniqueness of optimal estimators and lipschitz continuity with respect to data

convergence rates for total variation regularization of coefficient identification problems in elliptic equations ii abstract we investigate the convergence rates for total variation regularization of the problem of identifying i the coefficient q in the neumann problem for the elliptic equation div q u f in q u n g on ii the coefficient a in the neumann problem for the elliptic equation u a u f in u n g on r d d 1 when u is imprecisely given by z h 1 u z h 1 0 we regularize these problems by correspondingly minimizing the strictly convex functionals 1 2 q u q z 2 d x 1 2 q l 2 2 q and 1 2 u a z 2 d x 1 2 a u a z 2 d x 1 2 a l 2 2 a over admissible sets where u q u a is the solution of the first second neumann boundary value problem 0 is the regularization parameter taking the solutions of these optimization problems as the regularized solutions to the corresponding identification problems we obtain the convergence rates of them to the solution of the inverse problem in the sense of the bregman distance and in the l 2 norm under relatively simple source conditions without the smallness requirement on the source functions

an augmented lagrangian method for identifying discontinuous parameters in elliptic systems the identification of discontinuous parameters in elliptic systems is formulated as a constrained minimization problem combining the output least squares and the equation error method the minimization problem is then proved to be equivalent to the saddle point problem of an augmented lagrangian the finite element method is used to discretize the saddle point problem and the convergence of the discretization is also proved finally an uzawa algorithm is suggested for solving the discrete saddle point problem and is shown to be globally convergent

error estimates for the numerical identification of a variable coefficient error estimates are derived for the approximate identification of an unknown transmissivity coefficient in a partial differential equation describing a model problem in groundwater now the approximation scheme considered determines the coefficient by least squares fitting of the observed pressure data 1 introduction in this paper we wish to present an error analysis of a common numerical scheme used in the identification of parameters in distributed systems specifically we shall concern ourselves here with a model problem in groundwater flow the problem is to identify a spatially varying transmissivity coefficient a x from observations of the piezometric head u x in a two dimensional static aquifer s where a and u are related by the equations 1 div avw ins2

meta theorems on inequalities for scalar fuzzy set cardinalities we present meta theorems stating general conditions ensuring that certain inequalities for cardinalities of ordinary sets are preserved under fuzzification when adopting a scalar approach to fuzzy set cardinality the conditions pertain to the commutative conjunctor used for modelling fuzzy set intersection in particular this conjunctor should fulfil a number of bell type inequalities the advantage of these meta theorems is that repetitious calculations can be avoided this is illustrated in the demonstration of the lukasiewicz transitivity of fuzzified versions of the simple matching coefficient and the jaccard coefficient or equivalently the triangle inequality of the corresponding dissimilarity measures

ranking engineering design concepts using a fuzzy outranking preference model abstract the conceptual design evaluation is important since the poor selection of a design concept can rarely be compensated at later design stages due to subjective and incomplete design information collected at the early design stage it is difficult to select the best design concepts from a number of alternatives to tackle this problem an outranking preference model based on the possibility theory is developed in this paper the fuzzy outranking relation is developed to model the imprecise preference relation between each pair of design concepts a design concept outranked others if and only if there is sufficient evidence to support that the concept is superior or at least equal to the others according to the fuzzy outranking relation identified between each pair of design concepts three types of indices are developed to determine the non dominated design concepts for continuous improvement or further development at later design stages moreover the sensitivity analysis is used to examine the robustness of the result the fuzzy outranking preference model developed is more suitable to be used for concept selection in the imprecise and uncertain design environment

towards a general convergence theory for inexact newton regularizations we develop a general convergence analysis for a class of inexact newton type regularizations for stably solving nonlinear ill posed problems each of the methods under consideration consists of two components the outer newton iteration and an inner regularization scheme which applied to the linearized system provides the update in this paper we give a novel and unified convergence analysis which is not confined to a specific inner regularization scheme but applies to a multitude of schemes including landweber and steepest decent iterations iterated tikhonov method and method of conjugate gradients

regularization for curve representations uniform convergence for discontinuous solutions of ill posed problems this paper is concerned with a new approach for regularizing problems with discontinuous solutions regularization for curve representations the idea of this approach is to represent a discontinous function as a curve with parameterization a t b t a combination with nonlinear tikhonov regularization then yields uniform convergence of the regularized solutions the method is applied to deblurring and denoising problems in signal processing a numerical example for a deblurring problem is presented

on equivalence classes of fuzzy connectives the case of fuzzy integrals an equivalence relation between connectives or operators in the framework of multiple criteria decision making is introduced two operators are said to be equivalent if they lead to the same ranking of alternatives some general results to find equivalence classes of operators are given through the concept of level surfaces in a second part these results are applied to the case of discrete fuzzy integrals which are considered here as n place operators first some general results on the situation of fuzzy integrals among fuzzy operators are given and then the equivalence classes of fuzzy integrals

on the recovery of multiple flow parameters from transient head data the problem of estimating groundwater flow parameters from head measurements and other ancillary data is fundamental to the process of modelling a groundwater system we consider here a new method that allows for the simultaneous computation of multiple parameters as the unique minimum of a convex functional

a fuzzy extension of saaty s priority theory we present a fuzzy method for choosing among a number of alternatives under conflicting decision criteria a fuzzy version of saaty s pairwise comparison method 1980 extended by de graan 1980 and lootsma 1981 each ratio expressing the relative significance of a pair of factors is displayed in a matrix from which suitable weights can be extracted since these ratios are essentially fuzzy they express the opinion of a decision maker on the importance of a pair of factors we have adapted the above mentioned method in such a way that decision makers are asked to express their opinions in fuzzy numbers with triangular membership functions we apply the method at two distinct levels first to find fuzzy weights for the decision criteria and second to find fuzzy weights for the alternatives under each of the decision criteria by a suitable combination of these results we obtain fuzzy scores for the alternatives as well as their sensitivities using these fuzzy scores the decision makers should be able to make a choice for one of the alternatives

new classes of threshold aggregation functions based upon the tsallis q exponential with applications to perceptual computing we introduce two new classes of single parameter aggregation functions based upon the tsallis q exponential qe function of nonextensive statistical mechanics these aggregation functions denoted qe aggregation facilitate simple modeling of the common human reasoning trait of threshold inference where either 1 at least one input must exceed a threshold in order to achieve a nonzero aggregation output or 2 if any one of the inputs exceeds a different threshold the aggregation output takes its maximum value we illustrate the thresholding behavior of these functions on interval type 2 fuzzy inputs using an example known in the literature as the investment judgment advisor we believe that the new qe class of aggregation operators will prove useful in extending the range of options available for the design of perceptual computing systems

identification of the coefficient in elliptic equations this paper seeks to identify the space distributed diffusion coefficient a in the steady state diffusion equation operatorname div a operatorname grad u f in a bounded region of the real plane u f and the flux at the boundary are observed this problem is solved by a least square method choosing for the cost function the equation error defined in a natural way by the riesz representation theorem the method is stable if a smoothing term is added to the cost function tikhonov regularisation this paper shows that selection strategies of the smoothing coefficient exist that insure convergence under the only condition that a be sufficiently smooth the case is also considered where a is piecewise smooth and it is shown how to extend the previous formulation in both cases computational results are presented

lever aggregation operators we make use of the physical concept of lever to generalize well known operators based on weights all these operators can be understood as levers in equilibrium with weights masses located at the points in 0 1 that opinions represent

effects of membership function parameters on the performance of a fuzzy signal detector this paper describes a signal detection algorithm based on fuzzy logic the detector combines evidence provided by two waveform features and explicitly considers uncertainty in the detection decision the detector classifies waveforms including a signal not including a signal or being uncertain in which case no conclusion regarding presence or absence of a signal is drawn piecewise linear membership functions are used and a method to describe the membership functions in terms of two parameters is developed the performance of the detector is compared to a bayesian maximum likelihood detector using brainstem auditory evoked potential signals in simulated noise and the effects of the steepness slope and overlap of the membership functions on detector performance are evaluated by varying the membership function steepness and overlap the fuzzy detector can almost completely eliminate classification errors at the cost of a large number of uncertain classifications or it can be made to perform similarly to the bayesian detector

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

asa and its application to multi criteria decision making case study aggregation plays an important role in many applications related to the development of intelligent systems in a fuzzy environment the existing aggregation operators are generally the t norm t conorm mean operators yager s operator and operator however these aggregation operators do not properly reflect the situation in the aggregation process that is these types of aggregation operators are independent of the aggregation situation in order to solve these problems we propose a new aggregation method to reflect the situation in the aggregation process it is the aggregation based on situation assessment asa method it consists of the situation assessment model sam and the asa algorithm in this asa method the sam is utilized to reflect the situation in the aggregation process this model generates the parameter which is controlled by decision maker it indicates the current degree of aggregation situation therefore our method can be adapted to certain situations by using the parameter in this paper asa method is applied to the multi criteria decision making environment

an extended rule based inference for general decision making problems complex decision making problems are usually modeled in terms of a number of decisive variables that are related hierarchically pieces of evidence are aggregated in a bottom up way to determine the final decision goal for dealing with such problems ishizuka et al 5 6 developed a rule based inference procedure by using fuzzy sets to represent imprecise data and employed the dempster shafer theory to aggregate evidence for structural damage assessment this paper highlights the necessity of applying the rules with all possible consequent clauses in the procedure with the considerations of information quality and quantity when applying the procedure to more general decision making problems the use of such rules can provide high quality information which contains much less ignorance in the resultant information also it can afford more information due to the decrease in total ignorance a simple example with two cases is used to compare the results based on the two different presentations of rules in the inference procedure for illustrating the importance of considering the rules proposed in this paper

regularizing newton kaczmarz methods for nonlinear ill posed problems we introduce a class of stabilizing newton kaczmarz methods for nonlinear ill posed problems and analyze their convergence and regularization behavior as usual for iterative methods for solving nonlinear ill posed problems conditions on the nonlinearity or the derivatives have to be imposed in order to obtain convergence as we shall discuss in general and in some specific examples the nonlinearity conditions obtained for the newton kaczmarz methods are less restrictive than those for previously existing iteration methods and can be verified for several practical applications we also discuss the discretization and efficient numerical solution of the linear problems arising in each step of a newton kaczmarz method and we carry out numerical experiments for two model problems

the levenberg marquardt method applied to a parameter estimation problem arising from electrical resistivity tomography abstract an efficient and robust electrical resistivity tomographic inversion algorithm based on the levenberg marquardt method is considered to obtain quantities like grain size spatial scale and particle size distribution of mineralized rocks the corresponding model in two dimensions is based on the maxwell equations and leads to a partial differential equation with mixed dirichlet neumann boundary conditions the forward problem is solved numerically with the finite difference method however the inverse problem at hand is a classical nonlinear and ill posed parameter estimation problem linearizing and applying the tikhonov regularization method yields an iterative scheme the levenberg marquardt method several large systems of equations have to be solved efficiently in each iteration step which is accomplished by the conjugate gradient method without setting up the corresponding matrix instead fast matrix vector multiplications are performed directly therefore the derivative and its adjoint for the parameter to solution map are needed numerical results demonstrate the performance of our method as well as the possibility to reconstruct some of the desired parameters

transitivity preserving fuzzification schemes for cardinality based similarity measures abstract a family of fuzzification schemes is proposed that can be used to transform cardinality based similarity measures for ordinary sets into similarity measures for fuzzy sets in a finite universe the family is based on rules for fuzzy set cardinality and for the standard operations on fuzzy sets in particular the fuzzy set intersections are pointwisely generated by frank t norms the fuzzification schemes are applied to a variety of previously studied rational cardinality based similarity measures for ordinary sets and it is demonstrated that transitivity is preserved in the fuzzification process

fuzzy rule generation methods for high level computer vision abstract in many decision making systems involving multiple sources the decisions made may be considered as the result of a rule based system in which the decision rules are usually enumerated by experts or generated by a learning process in this paper we discuss the various issues involved in the generation of fuzzy rules automatically from training data for high level computer vision features are treated as linguistic variables that appear in the antecedent clauses of the rules we present methods to generate the corresponding linguistic labels values and their membership functions rules are generated by constructing a minimal approximate fuzzy aggregation network and then training the network using gradient descent methods several examples are given

resolution of regularized output least squares estimation procedures the regularized output least squares estimation technique is considered as an example of an estimation procedure in order to evaluate sensitivity of mappings that arise in the context of identical twin test problems for such test problems one specifies a reasonable test parameter to generate data from which the test parameter is to be recovered by means of an estimation procedure the properties of the mapping taking the test coefficient to the optimal estimator are studied conditions are determined under which the mapping is well defined as a function of the set of admissible parameters into itself resolution properties are related to the properties of the derivative of this mapping examples are considered using this information to formulate what is meant to be an optimal observer and to determine the optimal location of observers

soft learning vector quantization and clustering algorithms based on ordered weighted aggregation operators this paper presents the development and investigates the properties of ordered weighted learning vector quantization lvq and clustering algorithms these algorithms are developed by using gradient descent to minimize reformulation functions based on aggregation operators an axiomatic approach provides conditions for selecting aggregation operators that lead to admissible reformulation functions minimization of admissible reformulation functions based on ordered weighted aggregation operators produces a family of soft lvq and clustering algorithms which includes fuzzy lvq and clustering algorithms as special cases the proposed lvq and clustering algorithms are used to perform segmentation of magnetic resonance mr images of the brain the diagnostic value of the segmented mr images provides the basis for evaluating a variety of ordered weighted lvq and clustering algorithms

generalized fuzzy indices for similarity matching the purpose of this paper is to introduce a new family of fuzzy similarity indices referred to as the generalised tversky index gti the development of gti is based on the theoretical findings by amos tversky regarding the human perception of similarity between different objects as formulated by the features contrast model fc although gti was developed starting from tversky s fc it is shown that it provides a fuzzy extension and generalization of several widely used similarity indices like the jaccard and dice coefficients the mathematical properties of two members of the gti family namely tim and tip are studied and their interpretation of similarity is explained by comparison to other conventional indices

optimal encoding of graph homomorphism energy using fuzzy information aggregation operators abstract the attributed relational graph matching arg strategy is a well known approach to object pattern recognition in the past for the parallel solution of arg matching problem an overall objective function was constructed using linearly weighted information aggregation function and one set of parameter values was chosen for all models by trial and error for the parameters in the function in this paper the compatibility between every pair of model and scene attributes is interpreted as a fuzzy value and subsequently the nonlinear fuzzy information aggregation operators are used to fuse the information captured in the chosen attributes to learn the parameters in the fuzzy information aggregation operators the learning from samples strategy is used the learning of weight parameters is formulated as an optimisation problem and solved using the gradient projection algorithm based learning procedure the learning procedure implicitly evaluates ambiguity robustness and discriminatory power of the relational attributes chosen for graph matching and assigns weights appropriately to the chosen attributes the learning procedure also enables us to compute a distinct set of optimal parameters for every model to reflect the characteristics of the model so that the homomorphic arg matching problem can be optimally encoded in the energy function for the model experimental results are presented to illustrate effectiveness and necessity of the parameter estimation and learning procedures

similarity measures with complex multimedia data we see the emergence of database systems in which the fundamental operation is similarity assessment before database issues can be addressed it is necessary to give a definition of similarity as an operation we develop a similarity measure based on fuzzy logic that exhibits several features that match experimental findings in humans the model is dubbed fuzzy feature contrast ffc and is an extension to a more general domain of the feature contrast model due to tversky 1977 we show how the ffc model can be used to model similarity assessment from fuzzy judgment of properties and we address the use of fuzzy measures to deal with dependencies among the properties

similarity and dissimilarity measures between fuzzy sets a formal relational study the paper deals with the well known notion of dis similarity measures between fuzzy sets we provide three separate lists of axioms that fit with the respective notions of general comparison measure similarity measure and dissimilarity measure then we review some of the most important axiomatic definitions of dis similarity measures in the literature by referring to the axioms in those lists satisfied by each specific definition this common framework will make our study about the formal relationships among different axiomatic definitions much easier some of them which are apparently different do in fact share many commonalities we provide a self contained picture of these relationships by providing formal results and counterexamples that reflect which of the dis similarity definitions in the literature are connected by implication relations and which of them are not we finalize the paper with an in depth study about the notion of duality between similarity and dissimilarity measures as well as with some concluding remarks

induced 2 tuple linguistic generalized aggregation operators and their application in decision making the induced 2 tuple linguistic generalized ordered weighted averaging 2 tilgowa operator is presented this new aggregation operator extends previous approaches by using generalized means order inducing variables in the reordering of the arguments and linguistic information represented with the 2 tuple linguistic approach its main advantage is that it includes a wide range of linguistic aggregation operators thus its analyses can be seen from different perspectives and we obtain a much more complete picture of the situation considered and are able to select the alternative that best fits with our interests or beliefs we further generalize the 2 tilgowa by using quasi arithmetic means and choquet integrals the result is the quasi 2 tilowa operator and the 2 tuple linguistic induced quasi arithmetic choquet integral aggregation we conclude this paper by analysing the applicability of this new approach in a multi person linguistic decision making problem concerning product management

the existence of noise terms for systems of inhomogeneous differential and integral equations in this paper we develop a framework for the necessary condition for the existence of noise terms for systems of inhomogeneous differential and integral equations we show that the noise terms are conditional and are generated for inhomogeneous equations if specific criteria is justified we numerically test our approach for a variety of systems of inhomogeneous problems

fuzzy integral in multicriteria decision making abstract this paper presents a synthesis on the use of fuzzy integral as an aggregation operator in multicriteria decision making definitions essential properties are given and compared to those of usual aggregation operators a quick survey on applications is given

uniqueness for an elliptic inverse problem for the elliptic equation nabla cdot p x nabla v lambda q x v f x in omega subset bbr n the problem of determining when one or more of the coefficient functions p q and f are defined uniquely by a knowledge of one or more of the solution functions v v p q f lambda is considered

a convergence analysis of nonlinear implicit iterative method for nonlinear ill posed problems abstract implicit iterative method acquires good effect in solving linear ill posed problems we have ever applied the idea of implicit iterative method to solve nonlinear ill posed problems under the restriction that is appropriate large we proved the monotonicity of iterative error and obtained the convergence and stability of iterative sequence numerical results show that the implicit iterative method for nonlinear ill posed problems is efficient in this paper we analyze the convergence and stability of the corresponding nonlinear implicit iterative method when k are determined by hanke criterion

measuring similarity and ordering based on interval type 2 fuzzy numbers this paper extends a preference degree to rank interval type 2 fuzzy numbers a general method to measure the similarity between interval type 2 fuzzy numbers is also introduced using some theorems and lemmas it is proven that the extended similarity measures satisfy many common and desired properties based on the common axiomatic definitions introduced for similarity measures the main properties of the extended preference degree are studied in addition several possible applications of the proposed methods are illustrated by some application examples in pattern recognition and multicriteria group decision making moreover the proposed methods are examined in order to compare them with other existing methods and the feasibility and effectiveness of the proposed methods are cleared via some lemmas and numerical comparisons

review of parameter identification procedures in groundwater hydrology the inverse problem the purpose of this survey is to review parameter identification procedures in groundwater hydrology and to examine computational techniques which have been developed to solve the inverse problem parameter identification methods are classified under the error criterion used in the formulation of the inverse problem the problem of ill posedness in connection with the inverse problem is addressed typical inverse solution techniques are highlighted the review also includes the evaluation of methods used for computing the sensitivity matrix statistics which can be used to estimate the parameter uncertainty are outlined attempts have been made to compare and contrast representative inverse procedures and direction for future research is suggested

parameter identification by regularization for surface representation via the moving grid approach we consider the identification of a diffusion parameter in a second order elliptic equation in two dimensions by interior measurements the diffusion parameter is assumed to have discontinuities for its reconstruction we propose regularization algorithms with an adaptive grid the grid is adapted according to a measure of the smoothness of the regularized solution for the numerical computation we compare several iterative methods such as the minimal error method the steepest descent method and an inexact iteratively regularized gauss newton method the computations show that these algorithms can effectively identify the discontinuities

functorial homotopy decompositions of looped co h spaces in recent work of the first and third authors functorial coalgebra decompositions of tensor algebras were geometrically realized to give functorial homotopy decompositions of loop suspensions later work by all three authors generalized this to functorial decompositions of looped coassociative co h spaces in this paper we use different methods which allow for the coassociative hypothesis to be removed

variational method for multiple parameter identification in elliptic pdes abstract in the present paper we investigate the inverse problem of identifying simultaneously the diffusion matrix source term and boundary condition in the neumann boundary value problem for an elliptic partial differential equation pde from a measurement data which is weaker than required of the exact state a variational method based on energy functions with tikhonov regularization is here proposed to treat the identification problem we discretize the pde with the finite element method and prove the convergence as well as analyze error bounds of this approach

continuous generalized owa operator and its application to decision making in this paper we present a new class of operators called the continuous generalized ordered weighted averaging c gowa operators which extends the continuous ordered weighted averaging operator to adapt to uncertain and complex situations in decision making we use differentiated aggregation method and produce special forms of c gowa operator in addition we apply the c gowa operator to the aggregation of multiple interval arguments and obtain a wide range of aggregation operators we especially discuss the controlling parameter and the selection of basic unit interval monotonic function introduced in the information aggregation to make it more feasible we further generalize the previous approaches by using choquet integral and quasi arithmetic means obtaining the combined continuous generalized choquet integral aggregation operator and the combined continuous quasi arithmetic choquet integral aggregation operator we also present some further extension by using hierarchical aggregation rules obtaining the ordered hierarchies of the combined c gowa operator meanwhile some desirable properties of these operators are investigated finally we give a numerical example to illustrate the application of these operators to group decision making with interval arguments

basic concepts for a theory of evaluation the aggregative operator abstract starting with an explication of the aggregative concept and deducing a general structure which satisfies a number of minimal requirements properties of clustering the main features of a new mathematical theory called theory of evaluation are developed the theory sheds new light on such well known concepts as membership conjunction and disjunction and seems to be a very promising tool to handle representation problems as they grow from the fields of theory of fuzzy set and its many applications of human decision making and of multicriteria analysis

parameter identification for elliptic problems abstract the problem of computing a principal coefficient function p in the differential equation p x u f x r m m 1 on a bounded region from a knowledge of the solution function u and the right hand side f where u f are known only approximately and p may have mild discontinuities is solved by minimization of an associated functional

an efficient linear solver for nonlinear parameter identification problems in this paper we study some efficient numerical methods for parameter identifications in elliptic systems the proposed numerical methods are conducted iteratively and each iteration involves only solving positive definite linear algebraic systems although the original inverse problems are ill posed and highly nonlinear the positive definite systems can be naturally preconditioned with their corresponding block diagonal matrices numerical experiments are presented to illustrate the efficiency of the proposed algorithms

a combinatorial approach to the exponents of moore spaces in this article we give a combinatorial approach to the exponents of moore spaces our result states that the projection of the p r 1 power map of the loop space of the 2n 1 dimensional mod p r moore space to its atomic piece containing the bottom cell t 2n 1 p r is null homotopic for n 1 p 3 and r 1 this result strengthens the classical result that omega t 2n 1 p r has an exponent p r 1

compensatory operators in fuzzy linear programming with multiple objectives previously h j zimmermann 11 presented a fuzzy approach to solve the linear vectormaximum problem in the mentioned approach the author uses the minimum or the product operators to translate the semantic meaning of the and the aim of this paper is to reconsider this approach by using operators which allow some degree of compensation between aggregated membership functions firstly the c operator already defined in 12 is used to combine our fuzzy objectives it is shown that solutions obtained by this way are always efficient unfortunately the substitute problem is not generally computationally feasible we suggest then the min bounded sum operator which is a compensatory one has some desired properties and is supported by empirical arguments it allows a considerable decrease of the computational burden in the substitute problem and leads to a solution which is attractive from the stand point of efficiency

the functor a min on p local spaces in a previous paper the authors gave the finest functorial decomposition of the loop suspension of a p torsion suspension the purpose of this paper is to generalize this theorem to the loop suspension of arbitrary p local path connected spaces

for numerical differentiation dimensionality can be a blessing finite difference methods such as the mid point rule have been applied successfully to the numerical solution of ordinary and partial differential equations if such formulas are applied to observational data in order to determine derivatives the results can be disastrous the reason for this is that measurement errors and even rounding errors in computer approximations are strongly amplified in the differentiation process especially if small step sizes are chosen and higher derivatives are required a number of authors have examined the use of various forms of averaging which allows the stable computation of low order derivatives from observational data the size of the averaging set acts like a regularization parameter and has to be chosen as a function of the grid size h in this paper it is initially shown how first and higher order single variate numerical differentiation of higher dimensional observational data can be stabilized with a reduced loss of accuracy than occurs for the corresponding differentiation of one dimensional data the result is then extended to the multivariate differentiation of higher dimensional data the nature of the trade off between convergence and stability is explicitly characterized and the complexity of various implementations is examined

inverse problem of groundwater modeling by iteratively regularized gauss newton method with a nonlinear regularization term abstract a nonlinear minimization problem f d u min u u is a typical mathematical model of various applied inverse problems in order to solve this problem numerically in the lack of regularity we introduce iteratively regularized gauss newton procedure with a nonlinear regularization term irgn nrt the new algorithm combines two very powerful features iterative regularization and the most general stabilizing term that can be updated at every step of the iterative process the convergence analysis is carried out in the presence of noise in the data and in the modified source condition numerical simulations for a parameter identification ill posed problem arising in groundwater modeling demonstrate the efficiency of the proposed method

descent gradient methods for nonsmooth minimization problems in ill posed problems descent gradient methods are the most frequently used algorithms for computing regularizers of inverse problems they are either directly applied to the discrepancy term which measures the difference between operator evaluation and data or to a regularized version incorporating suitable penalty terms in its basic form gradient descent methods converge slowly we aim at extending different optimization schemes which have been recently introduced for accelerating these approaches by addressing more general penalty terms in particular we use a general setting in infinite hilbert spaces and examine accelerated algorithms for regularization methods using total variation or sparsity constraints to illustrate the efficiency of these algorithms we apply them to a parameter identification problem in an elliptic partial differential equation using total variation regularization

numerical approximation of an sqp type method for parameter identification this paper deals with the numerical approximation of the levenberg marquardt sqp lmsqp method for parameter identification problems which has been presented and analyzed in m burger and w muhlhuber inverse problems 18 2002 pp 943 969 it is shown that a galerkin type discretization leads to a convergent approximation and that the indefinite system arising from the karush kuhn tucker kkt system is well posed in addition we present a multilevel version of the levenberg marquardt method and discuss the simultaneous solution of the discretized kkt system by preconditioned iteration methods for indefinite problems from a discussion of the numerical effort we conclude that these approaches may lead to a considerable speed up with respect to standard iterative regularization methods that eliminate the underlying state equation the numerical efficiency of the lmsqp method is confirmed by numerical examples

empiric considerations on the fuzzy metric truth approach the fuzzy metric truth approach has as its objective computing with words in the full sense of this expression in order to achieve this aim a logico methodological user interface based on the semantic differential technique theory of truthlikeness and fuzzy systems has been constructed in the previous articles this article considers empiric aspects of the fuzzy metric truth approach by simulating and testing three typical data sets the system outputs of the user interface are compared with the desired outputs in the light of two main criteria the root mean square and the correlation coefficient the user interface is also compared with the algorithms which are used within non linear regression analysis the mamdani method and sugeno takagi method the results show that particularly in the case of the first two data sets the interface seems satisfactory and justifiable

a saddle point variational formulation for projection regularized parameter identification this paper is concerned with the ill posed problem of identifying a parameter in an elliptic equation and its solution applying regularization by projection as the theory has shown the ansatz functions for the parameter have to be sufficiently smooth in this paper we show that these for a practical implementation unrealistic smoothness assumptions can be circumvented by reformulating the problem under consideration as a mixed variational equation we prove convergence as the discretization gets finer in the noise free case and convergence as the data noise level delta goes to zero in the case of noisy data as well as convergence rates under additional smoothness conditions

decisions and evaluations by hierarchical aggregation of information usually in preparing a decision a hierarchy of criteria for evaluating alternatives can he agreed upon by the decision makers the problem addressed in this paper is the appropriate modelling of such evaluations and their aggregation in order to arrive at a proper overall evaluation function by which the decision alternatives can be ordered easily representing the criteria subjective categories by fuzzy sets it is possible to use aggregation operators as models for the amalgamation of those criteria in the framework of an empirical research project a hierarchy of criteria was developed for the evaluation of credit worthiness of customers 50 credit applications were evaluated by a number of credit managers of banks and their judgement could be appropriately modelled by a four level hierarchy of criteria

identification of parameters in distributed parameter systems by regularization identification of spatially varying parameters in distributed parameter systems from noisy data is an ill posed problem the concept of regularization widely used in solving linear fredholm integral equations is developed for the identification of parameters in distributed parameter systems a general regularization identification theory is first presented and then applied to the identification of parabolic systems the performance of the regularization identification method is evaluated by numerical experiments on the identification of a spatially varying diffusivity in the diffusion equation

complex structured decision making model a hierarchical frame work for complex structured data we introduce a hierarchical framework we call complex structured decision making model for complexly structured knowledge representation in intelligent decision making we show that our model extends non hierarchical flat decision making models to hierarchical decision making models that are similar to comprehensible human decision making processes further we make an argument that hierarchial representation of knowledge in a complex structured decision making model simplifies the approximation of aggregation functions to easily adapt to the underline relation of the system additionally using a real world complex structured data set we show that hierarchically organized fuzzy integrals e g choquet integral and sugeno integral and fuzzy signatures outperform these non hierarchical fuzzy integrals

an inverse problem for the steady state diffusion equation we consider the problem of identifying a spatially varying diffusion coefficient using an observed solution to the forward problem under appropriate conditions this problem can be treated as a first order hyperbolic equation in the unknown coefficient we develop some continuous dependence results for this problem and also propose a particularly advantageous set of test conditions for observing the forward solution

suspension splittings and james hopf invariants james constructed a functorial homotopy decomposition x w1 n 1 x n for path connected pointed cw complexes x we generalize this to a functorial decomposition of a where a is any functorial retract of a looped co h space this is used to construct hopf invariants in a more general context in addition when a y is the loops on a co h space we show that the wedge summands of y further functorially decompose by using an action of an appropriate symmetric group