model based gaussian and non gaussian clustering abstract the classification maximum likelihood approach is sufficiently general to encompass many current clustering algorithms including those based on the sum of squares criterion and on the criterion of friedman and rubin 1967 however as currently implemented it does not allow the specification of which features orientation size and shape are to be common to all clusters and which may differ between clusters also it is restricted to gaussian distributions and it does not allow for noise we propose ways of overcoming these limitations a reparameterization of the covariance matrix allows us to specify that some features but not all be the same for all clusters a practical framework for non gaussian clustering is outlined and a means of incorporating noise in the form of a poisson process is described an approximate bayesian method for choosing the number of clusters is given the performance of the proposed methods is studied by simulation with encouraging results the methods are applied to the analysis of a data set arising in the study of diabetes and the results seem better than those of previous analyses rh

a review of a posteriori error estimation techniques for elasticity problems within the framework of linearized elasticity we give an overview of the most popular a posteriori error estimators we discuss their common features the main differences and the underlying principles this will allow us to judge the quality of the different approaches and their suitability for nonlinear problems we also briefly describe a general framework which allows the construction of a posteriori error estimators for nonlinear elasticity

on the greatest distance between two permanental roots of a matrix abstract the permanental spread of a complex square matrix a is defined to be the greatest distance between two roots of the equation per zi a 0 a preliminary study of this number as well as of two related quantities is given in particular we derive upper and lower bounds and deal with comparisons of different bounds finally two inequalities involving the permanental spread are treated

on the accuracy of the gerschgorin circle theorem for bounding the spread ofa symmetric matrix abstract the spread of a matrix with real eigenvalues is the difference between its largest and smallest eigenvalues the gerschgorin circle theorem can be used to bound the extreme eigenvalues of the matrix and hence its spread for nonsymmetric matrices the gerschgorin bound on the spread may be larger by an arbitrary factor than the actual spread even if the matrix is symmetrizable this is not true for real symmetric matrices it is shown that for real symmetric matrices or complex hermitian matrices the ratio between the bound and the spread is bounded by p 1 where p is the maximum number of off diagonal nonzeros in any row of the matrix for full matrices this is just n this bound is not quite sharp for n greater than 2 but examples with ratios of n 1 for all n are given for banded matrices with m nonzero bands the maximum ratio is bounded by m independent of the size of n this bound is sharp provided only that n is at least 2m for sparse matrices p may be quite small and the gerschgorin bound may be surprisingly accurate

positive linear maps and spreads of matrices ii abstract it is shown how positive linear maps can be used to obtain lower bounds for spreads of normal matrices some known results and some new ones are obtained in this way

some inequalities for central moments of matrices abstract in this paper we shall study non commutative central moment inequalities with a focus on whether the commutative bounds are tight in the non commutative case as well we prove that the answer is affirmative for the fourth central moment and several particular results are given in the general case as an application we shall present some lower estimates of the spread of hermitian and normal matrices as well

positive definite completions of partial hermitian matrices abstract the question of which partial hermitian matrices some entries specified some free may be completed to positive definite matrices is addressed it is shown that if the diagonal entries are specified and principal minors composed of specified entries are positive then if the undirected graph of the specified entries is chordal a positive definite completion necessarily exists furthermore if this graph is not chordal then examples exist without positive definite completions in case a positive definite completion exists there is a unique matrix in the class of all positive definite completions whose determinant is maximal and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial hermitian matrix additional observations regarding positive definite completions are made

a simple error estimator and adaptive procedure for practical engineerng analysis a new error estimator is presented which is not only reasonably accurate but whose evaluation is computationally so simple that it can be readily implemented in existing finite element codes the estimator allows the global energy norm error to be well estmated and alos gives a good evaluation of local errors it can thus be combined with a full adaptive process of refinement or more simply provide guidance for mesh redesign which allows the user to obtain a desired accuracy with one or two trials when combined with an automatic mesh generator a very efficient guidance process to analysis is avaiable estimates other than the energy norm have successfully been applied giving for instance a predetermined accuracy of stresses

bounds for the spectrum of normal matrices abstract let a be a normal matrix with eigenvalues 1 2 n and let denote the smallest disc containing these eigenvalues we give some inequalities relating the center and radius of to the entries in a when applied to hermitian matrices our results give lower bounds on the spread maxij i j of a when applied to positive definite hermitian matrices they give lower bounds on the kantorovich ratio maxij i j i j

nonlinear 0 1 programming i linearization techniques any real valued nonlinear function in 0 1 variables can be rewritten as a multilinear function we discuss classes of lower and upper bounding linear expressions for multilinear functions in 0 1 variables for any multilinear inequality in 0 1 variables we define an equivalent family of linear inequalities this family contains the well known system of generalized covering inequalities as well as other linear equivalents of the multilinear inequality that are more compact i e of smaller cardinality in a companion paper 7 we discuss dominance relations between various linear equivalents of a multilinear inequality and describe a class of algorithms for multilinear 0 1 programming based on these results

new lower bounds for the randic spread abstract let g v g e g be an n m graph the randic spread of g s r g is defined as the maximum distance of its randic eigenvalues disregarding the randic spectral radius of g in this work we use numerical inequalities and bounds for the matricial spread to obtain relations between this spectral parameter and some structural and algebraic parameters of the underlying graph such as the sequence of vertex degrees the nullity randic index generalized randic indices and its independence number in the last section a comparison is presented for regular graphs

clustering categorical data based on distance vectors we introduce a novel statistical procedure for clustering categorical data based on hamming distance hd vectors the proposed method is conceptually simple and computationally straightforward because it does not require any specific statistical models or any convergence criteria moreover unlike most currently existing algorithms that compute the class membership or membership probability for every data point at each iteration our algorithm sequentially extracts clusters from the given dataset that is at each iteration our algorithm strives to identify only one cluster which will then be deleted from the dataset at the next iteration this procedure repeats until there are no more significant clusters in the remaining data consequently the number of clusters can be determined automatically by the algorithm as for the identification and extraction of a cluster we first locate the cluster center by using a pearson chi squared type statistic on the basis of hd vectors the partition of the datase

korn s inequalities for piecewise h 1 vector fields korn s inequalities for piecewise h 1 vector fields are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

equilibrated patch recovery error estimates simple and accurate upper bounds of the error this paper introduces a new recovery type error estimator ensuring local equilibrium and yielding a guaranteed upper bound of the error the upper bound property requires the recovered solution to be both statically equilibrated and continuous the equilibrium is obtained locally patch by patch and the continuity is enforced by a postprocessing based on the partition of the unity concept this postprocess is expected to preserve the features of the locally equilibrated stress field nevertheless the postprocess phase modifies the equilibrium which is no longer exactly fulfilled a new methodology is introduced that yields upper bound estimates by taking into account this lack of equilibrium this requires computing the 2 norm of the error or relating it with the energy norm the guaranteed upper bounds are obtained by using a pessimistic bound of the error 2 norm derived from an eigenvalue problem nevertheless these bounds are not sharp an additional strategy based on a more accurate assessment of the error 2 norm is introduced providing sharp estimates which are practical upper bounds as it is demonstrated in the numerical tests copyright 2006 john wiley sons ltd

a posteriori error estimation in large strain elasticity using equilibrated local neumann problems abstract a method to obtain a posteriori an accurate measure of the error in finite element computations is to solve a neumann problem with equilibrated tractions for each finite element we extend this approach to large strain elasticity and mixed finite elements which are well suited for nearly incompressible rubber like materials for an l2 projection of the error in the first piola kirchhoff stress tensor we prove that the estimator provides a guaranteed upper bound without multiplicative constants numerical examples confirm the effectiveness for both linear and ogden material

korn s inequalities and their applications in continuum mechanics korn s inequalities have played a central role in the development of linear elasticity not only in connection with the basic theoretical issues such as existence and uniqueness but also in a variety of applications the korn inequalities and other related inequalities for integrals of quadratic functionals also arise in the analysis of viscous incompressible fluid flow the dimensionless optimal constants appearing in these inequalities the korn constants depend only on the shape of the domains of concern information on the geometric dependence of these constants is essential in applications in this review article we summarize the major results on korn s inequalities for bounded domains in two and three dimensions with emphasis on results concerning the korn constants some applications in continuum mechanics are also described

nonlinear 0 1 programming ii dominance relations and algorithms a nonlinear 0 1 program can be restated as a multilinear 0 1 program which in turn is known to be equivalent to a linear 0 1 program with generalized covering g c inequalities in a companion paper 6 we have defined a family of linear inequalities that contains more compact smaller cardinality linearizations of a multilinear 0 1 program than the one based on the g c inequalities in this paper we analyze the dominance relations between inequalities of the above family in particular we give a criterion that can be checked in linear time for deciding whether a g c inequality can be strengthened by extending the cover from which it was derived we then describe a class of algorithms based on these results and discuss our computational experience we conclude that the g c inequalities can be strengthened most of the time to an extent that increases with problem density in particular the algorithm using the strengthening procedure outperforms the one using only g c inequalities whenever the number of nonlinear terms per constraint exceeds about 12 15 and the difference in their performance grows with the number of such terms

error estimate procedure in the finite element method and applications statically and kinematically admissible fields are explicitly derived from the finite element solution of the primal form of linear models the error on constitutive law for these fields yields an expression of the finite element error moreover the contribution of each element to this error allows to implement an automatic mesh refinement procedure leading to a uniform distribution of a given accuracy

anisotropic discretization and model error estimation in solid mechanics by local neumann problems first a survey of existing residuum based error estimators and error indicators is given generally residual error estimators which have at least upper bound in contrast to indicators can be locally computed from residua of equilibrium and stress jumps at element interfaces using dirichlet or neumann conditions for element patches or individual elements rem another equivalent method for error estimation can be derived from a posteriori computed improved boundary tractions which provide exact equilibrium of elements they are computed from local neumann problems and yield improved local solution by testing with higher test functions this method is called posterior equilibrium method pem and yields better efficiency indices than rem for the investigated problem even in case of locking also the important feature of model adaptivity via hierarchical error estimation can only be realized by this strategy global neumann type estimators and examples are presented for elastic and elastoplastic deformations controlling equilibrium in tangential space

laplacian spread of graphs lower bounds and relations with invariant parameters abstract the spread of an n n complex matrix b with eigenvalues 1 2 n is defined by s b max i j i j where the maximum is taken over all pairs of eigenvalues of b let g be a graph on n vertices the concept of laplacian spread of g is defined by the difference between the largest and the second smallest laplacian eigenvalue of g in this work by combining old techniques of interlacing eigenvalues and rank 1 perturbation matrices new lower bounds on the laplacian spread of graphs are deduced some of them involving invariant parameters of graphs as it is the case of the bandwidth independence number and vertex connectivity

constant free explicit error estimator with sharp upper error bound property for adaptive fe analysis in elasticity and fracture summary the explicit residual based error estimator originally proposed by babuska and miller 1987 for adaptive finite element analysis with application to problems of linear elasticity and fracture is known to be one of the most simple and inexpensive error estimators indeed it provides a theoretically guaranteed upper bound on a discretization error measured in the energy norm and requires small nearly negligible post processing computational effort the main issue with this classical estimator however is that an upper error bound is explicitly computable up to an unknown multiplicative constant in this paper we track the source of this generic constant revise the original derivation procedure and derive analytically four pre computable constants that constitute an upper error bound resulting in what we then call the constant free error estimator of the babuska miller type the performance of this constant free estimator as well as an adaptive fem based on it are illustrated on regular and singular benchmark problems and on numerical examples featuring crack propagation the special property of the estimator is a sharp upper bound of the error effectivity indices in the range of 1 2 2 0 are obtained what is treated as practically very acceptable in terms of efficiency judged against simplicity and inexpensiveness the proposed constant free explicit estimator is superior to for example corresponding implicit residual estimators and may be seen as a very competitive one in comparison with other known yet more intricate and complex error estimation techniques copyright 2014 john wiley sons ltd

some inequalities for positive linear maps it has long been known that an analogue of jensen s inequality holds for positive unital linear maps on matrix algebras provided that instead of ordinary convex functions one restricts to matrix convex functions we show that this restriction is not necessary in the case of 2 2 matrices a noncommutative analogue of the variance is studied and a basic inequality with several applications is established

the superconvergent patch recovery and a posteriori error estimates part 1 the recovery technique this is the first of two papers concerning superconvergent recovery techniques and a posteriori error estimation in this paper a general recovery technique is developed for determining the derivatives stresses of the finite element solutions at nodes the implementation of the recovery technique is simple and cost effective the technique has been tested for a group of widely used linear quadratic and cubic elements for both one and two dimensional problems numerical experiments demonstrate that the recovered nodal values of the derivatives with linear and cubic elements are superconvergent one order higher accuracy is achieved by the procedure with linear and cubic elements but two order higher accuracy is achieved for the derivatives with quadratic elements in particular an o h4 convergence of the nodal values of the derivatives for a quadratic triangular element is reported for the first time the performance of the proposed technique is compared with the widely used smoothing procedure of global l2 projection and other methods it is found that the derivatives recovered at interelement nodes by using l2 projection are also superconvergent for linear elements but not for quadratic elements numerical experiments on the convergence of the recovered solutions in the energy norm are also presented higher rates of convergence are again observed the results presented in this part of the paper indicate clearly that a new powerful and economical process is now available which should supersede the currently used post processing procedures applied in most codes

a recovery explicit error estimator in energy norm for linear elasticity significant research effort has been devoted to produce one sided error estimates for finite element analyses in particular to provide upper bounds of the actual error typically this has been achieved using residual type estimates one of the most popular and simpler in terms of implementation techniques used in commercial codes is the recovery based error estimator this technique produces accurate estimations of the exact error but is not designed to naturally produce upper bounds of the error in energy norm some attempts to remedy this situation provide bounds depending on unknown constants here a new step towards obtaining error bounds from the recovery based estimates is proposed the idea is 1 to use a locally equilibrated recovery technique to obtain an accurate estimation of the exact error 2 to add an explicit type error bound of the lack of equilibrium of the recovered stresses in order to guarantee a bound of the actual error and 3 to efficiently and accurately evaluate the constants appearing in the bounding expressions thus providing asymptotic bounds the numerical tests with h adaptive refinement process show that the bounding property holds even for coarse meshes providing upper bounds in practical applications

an enhanced strain error estimator for galerkin meshfree methods based on stabilized conforming nodal integration gradient averaging type a posteriori error estimators applied to the finite element method enjoy great popularity in the engineering community this is mainly because they are easy in their construction and computer implementation and usually provide constant free and sharp error estimates on the expense of losing the bounding property however it proves difficult to transfer this error estimation procedure to galerkin meshfree methods one reason is that the meshfree gradient approximation is generally rather accurate per se and difficult to improve moreover in many galerkin meshfree methods the shape functions are no interpolants as required in the original idea of constructing the recovered and thus smooth gradient field in this paper a novel error estimation procedure is presented that is based on the enhanced assumed strain eas method and applied to the reproducing kernel particle method rkpm as a representative of galerkin meshfree methods the error estimator is naturally tailored to galerkin meshfree methods if stabilized conforming nodal integration scni is employed which provides two gradient fields a globally smooth one the compatible strain and a cellwise constant one the enhanced assumed strain it is shown that the difference the enhanced strain can be used to derive an error estimator that follows the notion of gradient averaging type error estimators and resolves its issues when applied to galerkin meshfree methods in addition the enhanced strain error estimator is even easier to implement and computationally less expensive than gradient averaging type error estimators numerical examples of engineering interest illustrate the performance of the enhanced strain error estimator presented in this paper

on random quadratic bottleneck assignment problems the relative error between best and worst solution of quadratic bottleneck assignment problems with cost coefficientsdijpq aipbjq is considered whereaip is either arbitrarily given or corresponds to a distance in the plane it is shown that the relative error is bounded by a function m tending to zero with probability tending to one asm provided the data are uniformly distributed this implies that any algorithm for the above mentioned problems yields asymptotically an arbitrarily small relative error with probability tending to one

local error estimates of fem for displacements and stresses in linear elasticity by solving local neumann problems in this paper we present two types of local error estimators for the primal finite element method fem by duality arguments they are first derived from the explicit residual error estimation method rem and then as a new contribution from the implicit posterior equilibrium method pem using improved boundary tractions gained by local post processing with local neumann problems with applications in elastic problems for the displacements a local error estimator with an upper bound is derived and also a local estimator for stresses furthermore for better numerical efficiency the residua are projected energy invariant onto reference elements where the local neumann problems have to be solved comparative examples between rem and pem type local estimators show superior effectivity indices for the latter one copyright 2001 john wiley sons ltd

ranking scalar products to improve bounds for the quadratic assignment problem abstract the eigenvalue bound for the quadratic assignment problem qap is successively improved by considering a set of k best scalar products related to the qap an efficient procedure is proposedto find such a set of k best scalar products a class of qaps is described for which this procedure in general improves existing lower bounds and at the same time generates good suboptimal solutions the method leaves the user with a large flexibility in controlling the quality of the bound however since the method is sensitive to input data it should only be used in combination with other bounding rules

some a posteriori error estimators for elliptic partial differential equations we present three new a posteriori error estimators in the energy norm for finite element solutions to elliptic partial differential equations the estimators are based on solving local neumann problems in each element the estimators differ in how they enforce consistency of the neumann problems we prove that as the mesh size decreases under suitable assumptions two of the error estimators approach upper bounds on the norm of the true error and all three error estimators are within multiplicative constants of the norm of the true error we present numerical results in which one of the error estimators appears to converge to the norm of the true error

a heuristic for quadratic boolean programs with applications to quadratic assignment problems abstract a heuristic for quadratic boolean programs is presented computational tests with quadratic assignment problems qap showed that it finds very good suboptimal solutions in moderate time and behaves computationally stable in the appendix a fortran program for qap is listed which improves an earlier code published by burkard and derigs

a thermodynamically motivated simulation procedure for combinatorial optimization problems abstract exchange algorithms are an important class of heuristics for hard combinatorial optimization problems as e g salesman problems or quadratic assignment problems in kirkpatrick s and cerny s exchange algorithms for the travelling salesman problem and placement problem they propose to perform an exchange not only if the objective function value decreases by this exchange but also in certain cases if the objective function value increases an exchange increasing the objective function value is performed stochastically depending on the size of the increment computational tests with quadratic assignment problems revealed an excellent behaviour in such an approach suboptimal solutions differing 1 2 from the best known solutions are obtained by a simple program in short time by starting this program several times with different starting values all known minimal objective function values were reached thus this approach is well suited also for smaller computers and leads in short time to acceptable solutions

quadratic assignment problems abstract this paper surveys quadratic assignment problems qap at first several applications of this problem class are described and mathematical formulations of qaps are given then some exact solution methods and good heuristics are outlined their computational behaviour is illustrated by numerical results further recent results on the asymptotic probabilistic behaviour of qaps are outlined

goal oriented a posteriori error estimates in linear elastic fracture mechanics goal oriented a posteriori error estimators are presented in this paper for the error obtained while approximately evaluating the j integral in linear elastic fracture mechanics using the finite element method the j integral provides a criterion for crack propagation of a pre existing crack therefore its reliable and accurate evaluation is of considerable importance in engineering applications to prevent failure of structures this can be achieved by goal oriented adaptive finite element methods based on the well established strategy of solving an auxiliary dual problem in order to control the error of a non linear functional here the j integral in this paper two principally different goal oriented adaptive strategies are presented the first strategy rests upon equilibrated residual error estimators based on local neumann problems whereas the second one makes use of averaging techniques the paper is concluded by a numerical example that illustrates our theoretical results

lower bounds for the spread of a hermitian matrix abstract some simple lower bounds for the spread of a hermitian matrix are derived these bounds are explicit functions of the entries of the matrix one of which is sharper than a recent result due to e r barnes and a j hoffman comparisons are made with several known results

adaptive finite element methods in computational mechanics we present a general approach to adaptivity for finite element methods and give applications to linear elasticity non linear elasto plasticity and nonlinear conservation laws including numerical results

characterizations and lower bounds for the spread of a normal matrix abstract the spread of an n n matrix a with eigenvalues 1 n is defined by spr a max j k j k we prove that if a is normal then spr a max x ax y ay x y c n x y 1 max x ay y ax x y c n x y 1 re x y 0 max x ay y ax x y c n x y 1 x y 0 max spr z a z a 2 z c z 1 2 max x a 2 x x ax 2 x a ax x ax 2 1 2 x c n x 1 we also present several lower bounds for spr a given by these characterizations

new bounds for the signless laplacian spread abstract let g be an undirected simple graph the signless laplacian spread of g is defined as the maximum distance of pairs of its signless laplacian eigenvalues this paper establishes some new bounds both lower and upper for the signless laplacian spread several of these bounds depend on invariant parameters of the graph we also use a minmax principle to find several lower bounds for this spectral invariant

on goal oriented error estimation for elliptic problems application to the control of pointwise errors a method to estimate and control pointwise errors in finite element approximations of elliptic problems is presented as an application of the more general theory of goal oriented error estimation in the latter the accuracy of numerical approximations is assessed in terms of measures that are of practical interest to engineers rather than the classical global energy norm goal oriented error estimation requires the solution of a problem dual to the original problem and the computation of several global error estimates the performance of the method is carefully tested in the particular case of estimation and control of pointwise errors on a one dimensional problem

error controlled adaptive goal oriented modeling and finite element approximations in elasticity abstract in our earlier work e stein s ohnimus coupled model and solution adaptivity in the finite element method comput methods appl mech engrg 150 1997 327 350 e stein s ohnimus anisotropic discretization and model error estimation in solid mechanics by local neumann problems comput methods appl mech engrg 176 1999 363 385 e stein m ruter s ohnimus adaptive finite element analysis and modelling of solids and structures findings problems and trends int j numer methods engrg 60 2004 103 138 residual based combined model and finite element discretization error estimators as a basis for adaptivity were derived that rely on the solutions of neumann problems on the element level in particular these estimators control the errors obtained while expanding the model hierarchically from 1d to 2d and from 2d to 3d elastic theories in the global energy norm in this paper we extend both types of estimates to goal oriented a posteriori error estimation within the framework of nonlinear structural mechanics structural model adaptivity is characterized by the fact that the dimensions in general the constitutive equations and especially the related subdomains and thus the test and solution spaces of the process dependent models considered are different this has the important consequence that consistent prolongations of kinematic and static nodal quantities of finite elements of the coarser model m to those of the refined model m 1 are necessary therefore such prolongation operators are discussed in detail furthermore duality techniques are used for the model and discretization error estimates using system and load dependent quantities of interest in this paper we also consider different finite element meshes for the primal and the dual problem finally some numerical examples demonstrate the expected effects realizing verification and even validation in structural mechanics i e approaching a new paradigm in computational mechanics for achieving reliability and efficiency

estimation of interpolation error constants for the p0 and p1 triangular finite elements abstract we give some fundamental results on the error constants for the piecewise constant interpolation function and the piecewise linear one over triangles for the piecewise linear one we mainly analyze the conforming case but some results are also given for the non conforming case we obtain explicit relations for the dependence of such error constants on the geometric parameters of triangles in particular we explicitly determine the babuska aziz constant which plays an essential role in the interpolation error estimation of the linear triangular finite element the equation for determination is the transcendental equation tan 0 so that the solution can be numerically obtained with desired accuracy and verification such highly accurate approximate values for the constant as well as estimates for other constants can be widely used for a priori and a posteriori error estimations in adaptive computation and numerical verification of finite element solutions

incremental learning based multiobjective fuzzy clustering for categorical data the problem of clustering categorical data where attribute values cannot be naturally ordered as numerical values has gained more importance in recent time due to the special properties of categorical attributes the clustering of categorical data seems to be more complicated than that of numerical data although a few clustering algorithms that optimize single clustering objective have been proposed it has found that such single measure may not be appropriate for all kind of datasets hence in this article an incremental learning based multiobjective fuzzy clustering for categorical data is proposed for this purpose a multiobjective modified differential evolution based fuzzy clustering algorithm is developed thereafter it integrates with the well known supervised classifier called random forest using incremental learning to propose the aforementioned technique here the multiobjective algorithm produces a set of optimal clustering solutions known as pareto optimal solutions by optimizing two conflicting objectives simultaneously subsequently through incremental learning using random forest classifier final solution is evolved from the ensemble pareto optimal solutions the results of the proposed method are demonstrated quantitatively and visually in comparison with widely used state of the art methods for six synthetic and four real life datasets finally statistical test is conducted to show the superiority of the results produced by the proposed method