application of the karhunen loeve procedure for the characterization of human faces the use of natural symmetries mirror images in a well defined family of patterns human faces is discussed within the framework of the karhunen loeve expansion this results in an extension of the data and imposes even and odd symmetry on the eigenfunctions of the covariance matrix without increasing the complexity of the calculation the resulting approximation of faces projected from outside of the data set onto this optimal basis is improved on average

regularized kernel discriminant analysis with a robust kernel for face recognition and verification we propose a robust approach to discriminant kernel based feature extraction for face recognition and verification we show for the first time how to perform the eigen analysis of the within class scatter matrix directly in the feature space this eigen analysis provides the eigenspectrum of its range space and the corresponding eigenvectors as well as the eigenvectors spanning its null space based on our analysis we propose a kernel discriminant analysis kda which combines eigenspectrum regularization with a feature level scheme er kda finally we combine the proposed er kda with a nonlinear robust kernel particularly suitable for face recognition verification applications which require robustness against outliers caused by occlusions and illumination changes we applied the proposed framework to several popular databases yale ar xm2vts and achieved state of the art performance for most of our experiments

superconvergent patch recovery with equilibrium and conjoint interpolant enhancements the superconvergent patch derivative recovery method of zienkiewicz and zhu is enhanced by adding the squares of the residuals of the equilibrium equation and natural boundary conditions in addition a new conjoint polynomial for interpolating the local patch stresses over the element which significantly improves the local projection scheme is presented results show that in the 4 node quadrilateral the equilibrium and boundary condition residuals usually improve accuracy but not the rate of convergence whereas in the 9 node quadrilateral results are mixed the conjoint polynomial always improves the accuracy of the derivative field within the element as compared to the standard nodal interpolation particularly in 4 node quadrilaterals

an a posteriori error estimator for the p and hp versions of the finite element method an a posteriori error estimator is proposed in this paper for the p and hp versions of the finite element method in two dimensional linear elastostatic problems the local error estimator consists in an enhancement of an error indicator proposed by bertoti and szabo int j numer meth engng 1998 42 561 587 which is based on the minimum complementary energy principle in order to obtain the local error estimate this error indicator is corrected by a factor which depends only on the polynomial degree of the element the proposed error estimator shows a good effectivity index in meshes with uniform and non uniform polynomial distributions especially when the global error is estimated furthermore the local error estimator is reliable enough to guide p and hp adaptive refinement strategies

nonsmooth nonnegative matrix factorization nsnmf we propose a novel nonnegative matrix factorization model that aims at finding localized part based representations of nonnegative multivariate data items unlike the classical nonnegative matrix factorization nmf technique this new model denoted nonsmooth nonnegative matrix factorization nsnmf corresponds to the optimization of an unambiguous cost function designed to explicitly represent sparseness in the form of nonsmoothness which is controlled by a single parameter in general this method produces a set of basis and encoding vectors that are not only capable of representing the original data but they also extract highly focalized patterns which generally lend themselves to improved interpretability the properties of this new method are illustrated with several data sets comparisons to previously published methods show that the new nsnmf method has some advantages in keeping faithfulness to the data in the achieving a high degree of sparseness for both the estimated basis and the encoding vectors and in better interpretability of the factors

validation of the a posteriori error estimator based on polynomial preserving recovery for linear elements the purpose of this work is to investigate the quality of the a posteriori error estimator based on the polynomial preserving recovery ppr the main tool in this investigation is the computer based theory also a comparison is made between this estimator and the one based on the superconvergence patch recovery spr the results of this comparison were found to be in favour of the estimator based on the ppr copyright 2004 john wiley sons ltd

nonnegative matrix factorization and i divergence alternating minimization abstract in this paper we consider the nonnegative matrix factorization nmf problem given an elementwise nonnegative matrix v r m n find for assigned k nonnegative matrices w r m k and h r k n such that v wh exact nontrivial nonnegative factorizations do not always exist hence it is interesting to pose the approximate nmf problem the criterion which is commonly employed is i divergence between nonnegative matrices the problem becomes that of finding for assigned k the factorization wh closest to v in i divergence an iterative algorithm em like for the construction of the best pair w h has been proposed in the literature in this paper we interpret the algorithm as an alternating minimization procedure a la csiszar tusnady and investigate some of its stability properties nmf is widespreading as a data analysis method in applications for which the positivity constraint is relevant there are other data analysis methods which impose some form of nonnegativity we discuss here the connections between nmf and archetypal analysis

gaussian wave packet transform based numerical scheme for the semi classical schrodinger equation with random inputs abstract in this work we study the semi classical limit of the schrodinger equation with random inputs and show that the semi classical schrodinger equation produces o e oscillations in the random variable space with the gaussian wave packet transform the original schrodinger equation is mapped to an ordinary differential equation ode system for the wave packet parameters coupled with a partial differential equation pde for the quantity w in rescaled variables further we show that the w equation does not produce e dependent oscillations and thus it is more amenable for numerical simulations we propose multi level sampling strategy in implementing the gaussian wave packet transform where in the most costly part i e simulating the w equation it is sufficient to use e independent samples we also provide extensive numerical tests as well as meaningful numerical experiments to justify the properties of the numerical algorithm and hopefully shed light on possible future directions

a generalized kernel approach to dissimilarity based classification usually objects to be classified are represented by features in this paper we discuss an alternative object representation based on dissimilarity values if such distances separate the classes well the nearest neighbor method offers a good solution however dissimilarities used in practice are usually far from ideal and the performance of the nearest neighbor rule suffers from its sensitivity to noisy examples we show that other more global classification techniques are preferable to the nearest neighbor rule in such cases for classification purposes two different ways of using generalized dissimilarity kernels are considered in the first one distances are isometrically embedded in a pseudo euclidean space and the classification task is performed there in the second approach classifiers are built directly on distance kernels both approaches are described theoretically and then compared using experiments with different dissimilarity measures and datasets including degraded data simulating the problem of missing values

nonlinear non negative component analysis algorithms in this paper general solutions for nonlinear non negative component analysis for data representation and recognition are proposed motivated by a combination of the non negative matrix factorization nmf algorithm and kernel theory which has lead to a recently proposed nmf algorithm in a polynomial feature space we propose a general framework where one can build a nonlinear non negative component analysis method using kernels the so called projected gradient kernel non negative matrix factorization pgknmf in the proposed approach arbitrary positive definite kernels can be adopted while at the same time it is ensured that the limit point of the procedure is a stationary point of the optimization problem moreover we propose fixed point algorithms for the special case of gaussian radial basis function rbf kernels we demonstrate the power of the proposed methods in face and facial expression recognition applications

patch recovery based on superconvergent derivatives and equilibrium in this paper a postprocessing technique is developed for determining first order derivatives fluxes stresses at nodal points based on derivatives in superconvergent points it is an extension of the superconvergent patch recovery technique presented by zienkiewicz and zhu in contrast to that technique all flux or stress components are interpolated at the same time coupled by equilibrium equations at the superconvergent points the equilibrium equations and use of one order higher degree of interpolation polynomials of stress give a dramatic decrease in error of recovered derivatives even at boundaries

exploiting discriminant information in nonnegative matrix factorization with application to frontal face verification in this paper two supervised methods for enhancing the classification accuracy of the nonnegative matrix factorization nmf algorithm are presented the idea is to extend the nmf algorithm in order to extract features that enforce not only the spatial locality but also the separability between classes in a discriminant manner the first method employs discriminant analysis in the features derived from nmf in this way a two phase discriminant feature extraction procedure is implemented namely nmf plus linear discriminant analysis lda the second method incorporates the discriminant constraints inside the nmf decomposition thus a decomposition of a face to its discriminant parts is obtained and new update rules for both the weights and the basis images are derived the introduced methods have been applied to the problem of frontal face verification using the well known xm2vts database both methods greatly enhance the performance of nmf for frontal face verification

projected gradient methods for nonnegative matrix factorization nonnegative matrix factorization nmf can be formulated as a minimization problem with bound constraints although bound constrained optimization has been studied extensively in both theory and practice so far no study has formally applied its techniques to nmf in this letter we propose two projected gradient methods for nmf both of which exhibit strong optimization properties we discuss efficient implementations and demonstrate that one of the proposed methods converges faster than the popular multiplicative update approach a simple matlab code is also provided

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

convex and semi nonnegative matrix factorizations we present several new variations on the theme of nonnegative matrix factorization nmf considering factorizations of the form x fgt we focus on algorithms in which g is restricted to containing nonnegative entries but allowing the data matrix x to have mixed signs thus extending the applicable range of nmf methods we also consider algorithms in which the basis vectors of f are constrained to be convex combinations of the data points this is used for a kernel extension of nmf we provide algorithms for computing these new factorizations and we provide supporting theoretical analysis we also analyze the relationships between our algorithms and clustering algorithms and consider the implications for sparseness of solutions finally we present experimental results that explore the properties of these new methods

η superconvergence of finite element approximations in the interior of general meshes of triangles abstract in this paper we introduce a new definition of superconvergence tne superconvergence which generalizes the classical idea of superconvergence to general meshes we show that this new definition can be employed to determine the regions of least error in any element in the interior of any grid by using a computer based approach we present numerical results for the standard displacement finite element method for the scalar equation of orthotropic heat conduction for meshes of conforming triangles of degree p 1 p 5 and elements in the interior of the mesh the results demonstrate that unlike classical superconvergence superconvergence is applicable to the complex grids which are employed in practical engineering computations

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

nonlinear component analysis as a kernel eigenvalue problem a new method for performing a nonlinear form of principal component analysis is proposed by the use of integral operator kernel functions one can efficiently compute principal components in high dimensional feature spaces related to input space by some nonlinear map for instance the space of all possible five pixel products in 16 16 images we give the derivation of the method and present experimental results on polynomial feature extraction for pattern recognition

principal component analysis with complex kernel the widely linear model nonlinear complex representations via the use of complex kernels can be applied to model and capture the nonlinearities of complex data even though the theoretical tools of complex reproducing kernel hilbert spaces crkhs have been recently successfully applied to the design of digital filters and regression and classification frameworks there is a limited research on component analysis and dimensionality reduction in crkhs the aim of this brief is to properly formulate the most popular component analysis methodology i e principal component analysis pca in crkhs in particular we define a general widely linear complex kernel pca framework furthermore we show how to efficiently perform widely linear pca in small sample sized problems finally we show the usefulness of the proposed framework in robust reconstruction using euler data representation

nonnegative matrix factorization in polynomial feature space plenty of methods have been proposed in order to discover latent variables features in data sets such approaches include the principal component analysis pca independent component analysis ica factor analysis fa etc to mention only a few a recently investigated approach to decompose a data set with a given dimensionality into a lower dimensional space is the so called nonnegative matrix factorization nmf its only requirement is that both decomposition factors are nonnegative to approximate the original data the minimization of the nmf objective function is performed in the euclidean space where the difference between the original data and the factors can be minimized by employing l 2 norm in this paper we propose a generalization of the nmf algorithm by translating the objective function into a hilbert space also called feature space under nonnegativity constraints with the help of kernel functions we developed an approach that allows high order dependencies between the basis images while keeping the nonnegativity constraints on both basis images and coefficients two practical applications namely facial expression and face recognition show the potential of the proposed approach

a posteriori error estimation for the semidiscrete finite element method of parabolic differential equations in this paper three a posteriori error estimators of the error in the semidiscrete finite element solution discrete in space and continuous in time of parabolic partial differential equations are analyzed this approach is based on a posteriori error estimators for the elliptic pdes it is proven that guaranteed resp asymptotically exact a posteriori error estimator for the elliptic problem yields the guaranteed resp asymptotically exact estimator for the parabolic problem

the superconvergent patch recovery spr and adaptive finite element refinement abstract the error estimation processes which have been developed by the authors depend on the accuracy with which stresses gradients can be recovered the recovery methods available currently such as the l 2 projection and averaging are inaccurate and an improved process is needed this paper presents a very much improved recovery process yielding superconvergent values throughout the domain details of this method named the superconvergent patch recovery are presented with test results showing its efficiency examples of adaptive h based refinement are given showing high local and overall efficiency indices for error estimation

a recovery type error estimator for the extended finite element method based on singular smooth stress field splitting a new stress recovery procedure that provides accurate estimations of the discretization error for linear elastic fracture mechanic problems analyzed with the extended finite element method xfem is presented the procedure is an adaptation of the superconvergent patch recovery spr technique for the xfem framework it is based on three fundamental aspects a the use of a singular smooth stress field decomposition technique involving the use of different recovery methods for each field standard spr for the smooth field and reconstruction of the recovered singular field using the stress intensity factor k for the singular field b direct calculation of smoothed stresses at integration points using conjoint polynomial enhancement and c assembly of patches with elements intersected by the crack using different stress interpolation polynomials at each side of the crack the method was validated by testing it on problems with an exact solution in mode i mode ii and mixed mode and on a problem without analytical solution the results obtained showed the accuracy of the proposed error estimator copyright 2008 john wiley sons ltd

a method for enforcing integrability in shape from shading algorithms an approach for enforcing integrability a particular implementation of the approach an example of its application to extending an existing shape from shading algorithm and experimental results showing the improvement that results from enforcing integrability are presented a possibly nonintegrable estimate of surface slopes is represented by a finite set of basis functions and integrability is enforced by calculating the orthogonal projection onto a vector subspace spanning the set of integrable slopes the integrability projection constraint was applied to extending an iterative shape from shading algorithm of m j brooks and b k p horn 1985 experimental results show that the extended algorithm converges faster and with less error than the original version good surface reconstructions were obtained with and without known boundary conditions and for fairly complicated surfaces

a posteriori estimation of the error in the error estimate in this paper we address the problem of a posteriori estimation of the error in the error estimate we consider the case of estimates for the error in the derivatives the strains or the stresses which are constructed in terms of locally computed element error indicators of the element residual or the least squares recovery type the estimates of the error in the error estimate have the same structure as the original error estimates and are determined by locally averaging recycling the original error indicators the most accurate indicators of the error in the error indicators are obtained by employing a harmonic basis in the recycling of the indicators namely a basis which locally satisfies the partial differential equation and the boundary conditions

independent component analysis of gabor features for face recognition we present an independent gabor features igfs method and its application to face recognition the novelty of the igf method comes from 1 the derivation of independent gabor features in the feature extraction stage and 2 the development of an igf features based probabilistic reasoning model prm classification method in the pattern recognition stage in particular the igf method first derives a gabor feature vector from a set of downsampled gabor wavelet representations of face images then reduces the dimensionality of the vector by means of principal component analysis and finally defines the independent gabor features based on the independent component analysis ica the independence property of these gabor features facilitates the application of the prm method for classification the rationale behind integrating the gabor wavelets and the ica is twofold on the one hand the gabor transformed face images exhibit strong characteristics of spatial locality scale and orientation selectivity these images can thus produce salient local features that are most suitable for face recognition on the other hand ica would further reduce redundancy and represent independent features explicitly these independent features are most useful for subsequent pattern discrimination and associative recall experiments on face recognition using the face recognition technology feret and the orl datasets where the images vary in illumination expression pose and scale show the feasibility of the igf method in particular the igf method achieves 98 5 correct face recognition accuracy when using 180 features for the feret dataset and 100 accuracy for the orl dataset using 88 features

non negative matrix factorization with sparseness constraints non negative matrix factorization nmf is a recently developed technique for finding parts based linear representations of non negative data although it has successfully been applied in several applications it does not always result in parts based representations in this paper we show how explicitly incorporating the notion of sparseness improves the found decompositions additionally we provide complete matlab code both for standard nmf and for our extension our hope is that this will further the application of these methods to solving novel data analysis problems

feature space interpretation of svms with indefinite kernels kernel methods are becoming increasingly popular for various kinds of machine learning tasks the most famous being the support vector machine svm for classification the svm is well understood when using conditionally positive definite cpd kernel functions however in practice non cpd kernels arise and demand application in svm the procedure of plugging these indefinite kernels in svm often yields good empirical classification results however they are hard to interpret due to missing geometrical and theoretical understanding in this paper we provide a step toward the comprehension of svm classifiers in these situations we give a geometric interpretation of svm with indefinite kernel functions we show that such svm are optimal hyperplane classifiers not by margin maximization but by minimization of distances between convex hulls in pseudo euclidean spaces by this we obtain a sound framework and motivation for indefinite svm this interpretation is the basis for further theoretical analysis e g investigating uniqueness and for the derivation of practical guidelines like characterizing the suitability of indefinite svm

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

newton s method for large bound constrained optimization problems we analyze a trust region version of newton s method for bound constrained problems our approach relies on the geometry of the feasible set not on the particular representation in terms of constraints the convergence theory holds for linearly constrained problems and yields global and superlinear convergence without assuming either strict complementarity or linear independence of the active constraints we also show that the convergence theory leads to an efficient implementation for large bound constrained problems

kernel independent component analysis we present a class of algorithms for independent component analysis ica which use contrast functions based on canonical correlations in a reproducing kernel hilbert space on the one hand we show that our contrast functions are related to mutual information and have desirable mathematical properties as measures of statistical dependence on the other hand building on recent developments in kernel methods we show that these criteria and their derivatives can be computed efficiently minimizing these criteria leads to flexible and robust algorithms for ica we illustrate with simulations involving a wide variety of source distributions showing that our algorithms outperform many of the presently known algorithms

the pre image problem in kernel methods in this paper we address the problem of finding the pre image of a feature vector in the feature space induced by a kernel this is of central importance in some kernel applications such as on using kernel principal component analysis pca for image denoising unlike the traditional method in which relies on nonlinear optimization our proposed method directly finds the location of the pre image based on distance constraints in the feature space it is noniterative involves only linear algebra and does not suffer from numerical instability or local minimum problems evaluations on performing kernel pca and kernel clustering on the usps data set show much improved performance

a posteriori estimation and adaptive control of the pollution error in the h version of the finite element method abstract we studied the pollution error in the h version of the finite element method and its effect on the quality of the local error indicators resp the quality of the derivatives recovered by local postprocessing in the interior of the mesh here we show that it is possible to construct a posteriori estimates of the pollution error in a patch of elements by employing the local error indicators over the entire mesh we also give an adaptive algorithm for the local control of the pollution error in a patch of elements of interest

on the convergence of multiplicative update algorithms for nonnegative matrix factorization nonnegative matrix factorization nmf is useful to find basis information of nonnegative data currently multiplicative updates are a simple and popular way to find the factorization however for the common nmf approach of minimizing the euclidean distance between approximate and true values no proof has shown that multiplicative updates converge to a stationary point of the nmf optimization problem stationarity is important as it is a necessary condition of a local minimum this paper discusses the difficulty of proving the convergence we propose slight modifications of existing updates and prove their convergence techniques invented in this paper may be applied to prove the convergence for other bound constrained optimization problems

nonlinear nonnegative matrix factorization based on mercer kernel construction generalizations ofnonnegative matrix factorization nmf in kernel feature space such as projected gradient kernel nmf pgknmf and polynomial kernel nmf pnmf have been developed for face and facial expression recognition recently however these existing kernel nmf approaches cannot guarantee the nonnegativity of bases in kernel feature space and thus are essentially semi nmf methods in this paper we show that nonlinear semi nmf cannot extract the localized components which offer important information in object recognition therefore nonlinear nmf rather than semi nmf is needed to be developed for extracting localized component as well as learning the nonlinear structure in order to address the nonlinear problem of nmf and the semi nonnegative problem of the existing kernel nmf methods we develop the nonlinear nmf based on a self constructed mercer kernel which preserves the nonnegative constraints on both bases and coefficients in kernel feature space experimental results in face and expressing recognition show that the proposed approach outperforms the existing state of the art kernel methods such as kpca gda pnmf and pgknmf

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

validation of a posteriori error estimators by numerical approach abstract a numerical methodology which determines the quality or robustness of a posteriori error estimators is described the methodology accounts precisely for the factors which affect the quality of error estimators for finite element solutions of linear elliptic problems namely the local geometry of the grid and the structure of the solution the methodology can be employed to check the robustness of any estimator for the complex grids which are used in engineering computations

enhanced superconvergent patch recovery incorporating equilibrium and boundary conditions patch recovery based on superconvergent derivatives and equilibrium spre an enhancement of the superconvergent patch recovery spr is studied for linear elasticity problems the paper also presents a further improvement for recovery of derivatives near boundaries spreb where either tractions or displacements are prescribed this is made by inclusion of weighted residual errors at boundary points in the patch recovery a pronounced improvement in the post processed gradients of the finite element solution is observed by this method

large scale graph regularized non negative matrix factorization with cal ell _1 normalization based on kullback leibler divergence we propose a novel algorithm for graph regularized non negative matrix factorization nmf with l1 normalization based on the kullback leibler divergence the l1 normalization is imposed to overcome the scaling ambiguity in earlier work on graph regularized nmf gnmf in d cai x he j han and t huang graph regularized non negative matrix factorization for data representation ieee trans pattern anal mach intell vol 33 pp 1548 1560 2011 the algorithm only involves element wise iterative updating to ensure both non negativity of the solution and convergence its element wise structure makes the proposed algorithm suitable for large scale problems experiments on spoken pattern discovery on the tidigits database and on image clustering of the pie dataset show that the algorithm outperforms the previous one with a better accuracy and a lower computational complexity

manifold regularized discriminative nonnegative matrix factorization with fast gradient descent nonnegative matrix factorization nmf has become a popular data representation method and has been widely used in image processing and pattern recognition problems this is because the learned bases can be interpreted as a natural parts based representation of data and this interpretation is consistent with the psychological intuition of combining parts to form a whole for practical classification tasks however nmf ignores both the local geometry of data and the discriminative information of different classes in addition existing research results show that the learned basis is unnecessarily parts based because there is neither explicit nor implicit constraint to ensure the representation parts based in this paper we introduce the manifold regularization and the margin maximization to nmf and obtain the manifold regularized discriminative nmf md nmf to overcome the aforementioned problems the multiplicative update rule mur can be applied to optimizing md nmf but it converges slowly in this paper we propose a fast gradient descent fgd to optimize md nmf fgd contains a newton method that searches the optimal step length and thus fgd converges much faster than mur in addition fgd includes mur as a special case and can be applied to optimizing nmf and its variants for a problem with 165 samples in r1600 fgd converges in 28 s while mur requires 282 s we also apply fgd in a variant of md nmf and experimental results confirm its efficiency experimental results on several face image datasets suggest the effectiveness of md nmf

a procedure for a posteriori error estimation for h p finite element methods abstract a new approach to a posteriori error estimation is outlined which is applicable to general h p finite element approximations of general classes of boundary value problems the approach makes use of duality arguments and is based on the element residual method erm important aspects of the method are that it provides a systematic approach toward deriving element boundary conditions for the erm it leads to an upper bound for the global error in an appropriate energy norm and it is valid for non uniform and irregular h p meshes in the present exposition a brief outline of the theoretical foundations of the method is given together with the results of its application to several representative problems these results show that the approach is applicable to general linearly elliptic systems including unsymmetrical operators and that the method is valid for broad classes of linear and non linear problems

kernel discriminant analysis for positive definite and indefinite kernels kernel methods are a class of well established and successful algorithms for pattern analysis thanks to their mathematical elegance and good performance numerous nonlinear extensions of pattern recognition techniques have been proposed so far based on the so called kernel trick the objective of this paper is twofold first we derive an additional kernel tool that is still missing namely kernel quadratic discriminant kqd we discuss different formulations of kqd based on the regularized kernel mahalanobis distance in both complete and class related subspaces secondly we propose suitable extensions of kernel linear and quadratic discriminants to indefinite kernels we provide classifiers that are applicable to kernels defined by any symmetric similarity measure this is important in practice because problem suited proximity measures often violate the requirement of positive definiteness as in the traditional case kqd can be advantageous for data with unequal class spreads in the kernel induced spaces which cannot be well separated by a linear discriminant we illustrate this on artificial and real data for both positive definite and indefinite kernels

a model study of the quality of a posteriori error estimators for linear elliptic problems error estimation in the interior of patchwise uniform grids of triangles abstract this paper is the first in a series in which we discuss computational methodologies for checking the quality of a posteriori error estimators for finite element approximations of linear elliptic problems in this first part we study the asymptotic properties of error estimators in the interior of patchwise uniform grids of triangles a completely numerical methodology for the analysis of the quality of estimators is presented results from the application of the methodology to the study of the quality of several well known error estimators are reported in subsequent papers we shall discuss methods to study the properties of estimators for meshes of quadrilaterals non uniform grids at boundaries grid interfaces and near singular points

the superconvergent patch recovery and a posteriori error estimates part 2 error estimates and adaptivity in this second part of the paper the issue of a posteriori error estimation is discussed in particular we derive a theorem showing the dependence of the effectivity index for the zienkiewicz zhu error estimator on the convergence rate of the recovered solution this shows that with superconvergent recovery the effectivity index tends asymptotically to unity the superconvergent recovery technique developed in the first part of the paper1 is the used in the computation of the zienkiewicz zhu error estimator to demonstrate accurate estimation of the exact error attainable numerical tests are shown for various element types illustrating the excellent effectivity of the error estimator in the energy norm and pointwise gradient stress error estimation several examples of the performance of the error estimator in adaptive mesh refinement are also presented

pollution error in the h version of the finite element method and the local quality of the recovered derivatives abstract in this paper we address the quality of the solution derivatives which are recovered from finite element solutions by local averaging schemes as an example we consider the zienkiewicz zhu superconvergent patch recovery scheme the zz spr scheme and we study its accuracy in the interior of the mesh for finite element approximations of solutions of laplace s equation in polygonal domains we will demonstrate the following 1 in general the accuracy of the derivatives recovered by the zz spr or any other local averaging scheme may not be higher than the accuracy of the derivatives computed directly from the finite element solution 2 if the mesh is globally adaptive i e it is nearly equilibrated in the energy norm then we can practically always gain in accuracy by employing the recovered derivatives instead of the derivatives computed directly from the finite element solution 3 it is possible to guarantee that the recovered solution derivatives have higher accuracy than the derivatives computed directly from the finite element solution in any patch of elements of interest by employing a mesh which is adaptive only with respect to the patch of interest i e it is nearly equilibrated in a weighted energy norm 4 in practice we are often interested in obtaining highly accurate derivatives or heat fluxes stresses etc only in a few critical regions which are identified by a preliminary analysis a grid which is adaptive only with respect to the critical regions of interest may be much more economical for this purpose because it may achieve the desired accuracy by employing substantially fewer degrees of freedom than a globally adaptive grid which achieves comparable accuracy in the critical regions

input space versus feature space in kernel based methods this paper collects some ideas targeted at advancing our understanding of the feature spaces associated with support vector sv kernel functions we first discuss the geometry of feature space in particular we review what is known about the shape of the image of input space under the feature space map and how this influences the capacity of sv methods following this we describe how the metric governing the intrinsic geometry of the mapped surface can be computed in terms of the kernel using the example of the class of inhomogeneous polynomial kernels which are often used in sv pattern recognition we then discuss the connection between feature space and input space by dealing with the question of how one can given some vector in feature space find a preimage exact or approximate in input space we describe algorithms to tackle this issue and show their utility in two applications of kernel methods first we use it to reduce the computational complexity of sv decision functions second we combine it with the kernel pca algorithm thereby constructing a nonlinear statistical denoising technique which is shown to perform well on real world data

a model study of the quality ofa posteriori error estimators for finite element solutions of linear elliptic problems with particular reference to the behavior near the boundary

a unified approach to a posteriori error estimation using element residual methods this paper deals with the problem of obtaining numerical estimates of the accuracy of approximations to solutions of elliptic partial differential equations it is shown that by solving appropriate local residual type problems one can obtain upper bounds on the error in the energy norm moreover in the special case of adaptiveh p finite element analysis the estimator will also give a realistic estimate of the error a key feature of this is the development of a systematic approach to the determination of boundary conditions for the local problems the work extends and combines several existing methods to the case of fullh p finite element approximation on possibly irregular meshes with elements of non uniform degree as a special case the analysis proves a conjecture made by bank and weiser some a posteriori error estimators for elliptic partial differential equations math comput 44 283 301 1985