a new cloud based hp finite element method a hybrid computational method for solving boundary value problems is introduced which combines features of the meshless hp cloud methods with features of conventional finite elements the method admits straightforward nonuniform hp type approximations easy implementation of essential boundary conditions is robust under severe distortions of the mesh and can deliver exponential rates of convergence results of numerical experiments are presented

improving feature space based image segmentation via density modification feature space based approaches have been popularly used to perform low level image analysis in this paper a density modification framework that enhances density map based discriminability of feature values in a feature space is proposed in order to aid feature space based segmentation in images the framework embeds a position dependent property associated with each sample in the feature space of an image into the corresponding density map and hence modifies it the property association and embedding operations in the framework is implemented using a fuzzy set theory based system devised with cues from beam theory of solid mechanics and the appropriateness of this approach is established qualitative and quantitative experimental results of segmentation in images are given to demonstrate the effectiveness of the proposed density modification framework and the usefulness of feature space based segmentation via density modification

on the sharpness of l 2 error estimates of h 1 0 projections onto subspaces of piecewise high order polynomials in a plane polygonal domain consider a poisson problem u f with homogeneous dirichlet boundary condition and the p version finite element solutions of this we give various upper and lower bounds for the error measured in l 2 in the case of a single element i e a convex domain we reduce the question of sharpness of these estimates to the behavior of a certain inf sup constant which is numerically determined and a likely sharp estimate is then conjectured this is confirmed during a series of numerical experiments also for the case of a reentrant comer for a one dimensional analogue problem of rotational symmetry sharp l 2 error estimates are proven directly and via an extension of the classical duality argument here we give sharp l error estimates in some weighted and unweighted norms also

decomposable graphical gaussian model determination we propose a methodology for bayesian model determination in decomposable graphical gaussian models to achieve this aim we consider a hyper inverse wishart prior distribution on the concentration matrix for each given graph to ensure compatibility across models such prior distributions are obtained by marginalisation from the prior conditional on the complete graph we explore alternative structures for the hyperparameters of the latter and their consequences for the model model determination is carried out by implementing a reversible jump markov chain monte carlo sampler in particular the dimension changing move we propose involves adding or dropping an edge from the graph we characterise the set of moves which preserve the decomposability of the graph giving a fast algorithm for maintaining the junction tree representation of the graph at each sweep as state variable we use the incomplete variance covariance matrix containing only the elements for which the corresponding element of the inverse is nonzero this allows all computations to be performed locally at the clique level which is a clear advantage for the analysis of large and complex datasets finally the statistical and computational performance of the procedure is illustrated by mean of both artificial and real datasets

on the finite element method for singularly perturbed reaction diffusion problems in two and one dimensions second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered the behavior of the ordinary galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed

local and pollution error estimation for finite element approximations of elliptic boundary value problems this paper addresses the issue of local elementwise error estimation of finite element approximations of elliptic boundary value problems the characterization of element error as local and pollution error component is presented and the relationship between energy norms of local errors and that predicted by means of a posteriori error estimators is investigated in addition techniques for calculating element indicators of local error pollution error and other error indicators are presented

the multigrid method in solid mechanics part i algorithm description and behaviour a multigrid algorithm is described that can be used to obtain the finite element solution of linear elastic solid mechanics problems the method is applied to some two dimensional problems to evaluate its strengths and weaknesses extensive studies are made to determine the convergence behaviour of the method in general this depends on many factors the number of degrees of freedom in the discretization characteristics of the algorithm poisson s ratio when it is close to 0 5 the amount of bending deformation in the problem under consideration and the degree of non uniformity in the mesh only certain values of the multigrid parameters allow a converged solution to be obtained with a computational effort proportional to the number of degrees of freedom these values include the optimum ones i e those that lead to convergence with the least computational effort the constant of proportionality is only independent of the number of degrees of freedom and still depends on the other factors listed above

fuzzy geometry in image analysis an attempt is made here to demonstrate the way of implementing the concept of fuzzy geometry in image processing analysis problems some new measures are also provided in this context four algorithms are described for object background classification and skeleton extraction in both fuzzy and nonfuzzy crisp forms their effectiveness is demonstrated on various types of images the crisp nonfuzzy outputs of the proposed algorithms are compared with the results of some of the recently developed conventional techniques

estimation of a covariance matrix with zeros summary we consider estimation of the covariance matrix of a multivariate random vector under the constraint that certain covariances are zero we first present an algorithm which we call iterative conditional fitting for computing the maximum likelihood estimate of the constrained covariance matrix under the assumption of multivariate normality in contrast to previous approaches this algorithm has guaranteed convergence properties dropping the assumption of multivariate normality we show how to estimate the covariance matrix in an empirical likelihood approach these approaches are then compared via simulation and on an example of gene expression

bounds for membership functions a correlation based approach the paper describes a mathematical formulation for defining bounds for s type membership functions using correlation 1 these bounds impose restrictions on variation in the values of membership functions the significance of these bounds in the image processing analysis problems has been extensively studied on various images it has been observed that membership functions satisfying the bounds can provide desirable results it is also found that zadeh s standard s function 2 satisfies these bounds

technique for fractal image compression using genetic algorithm a new method for fractal image compression is proposed using genetic algorithm ga with an elitist model the self transformation property of images is assumed and exploited in the fractal image compression technique the technique described utilizes the ga which greatly decreases the search space for finding the self similarities in the given image this article presents theory implementation and an analytical study of the proposed method along with a simple classification scheme a comparison with other fractal based image compression methods is also reported

analysis and improvements of global local enrichments for the generalized finite element method abstract partition of unity methods like the gfem rely on closed form analytical enrichment functions however only limited a priori knowledge about the solution is available for many problems of engineering relevance control of discretization errors in this class of problems requires mesh refinement which offsets some of the advantages associated with these methods this has led to the development of the gfem with global local enrichments gfem gl which uses ideas of the classical global local finite element method to numerically build enrichment functions for the gfem it involves the solution of local boundary value problems using boundary conditions from the solution of the global problem discretized with a coarse mesh these local solutions are in turn used to enrich the solution space of the global problem with the help of the partition of unity framework this paper presents an a priori error estimate accounting for the effect of inexact boundary conditions applied to local problems two strategies to control the error of the gfem gl solution due to these boundary conditions are also presented the first one is based on the use of a buffer or over sampling zone in the local problems the second strategy investigated is based on multiple global local iterations several numerical experiments performed on three dimensional fracture mechanics problems illustrate the effectiveness of these strategies in controlling the error of the gfem gl solution the problems are selected such that the boundary conditions on the local problems range from smooth to singular functions

on principles for the selection of shape functions for the generalized finite element method abstract effective shape functions for the generalized finite element method should reflect the available information on the solution this information is partially fuzzy because the solution is of course unknown and typically the only available information is the solution s inclusion in a variety of function spaces it is desirable to choose shape functions that perform robustly over a family of relevant situations quantitative notions of robustness are introduced and discussed we show in particular that in one dimension polynomials are robust when the available information consists in inclusions in sobolev type spaces that are x independent

fuzzy homogeneity approach to multilevel thresholding the spatial ambiguity among pixels has inherent vagueness rather than randomness therefore the conventional methods might not work well we propose fuzzy homogeneity vectors to handle the greyness and spatial uncertainties among pixels and to perform multilevel thresholding the experimental results prove that the proposed approach works better than the histogram based algorithms

analysis of periodic schrodinger operators regularity and approximation of eigenfunctions let v be a real valued potential that is smooth everywhere on r3 except at a periodic discrete set s of points where it has singularities of the coulomb type z r we assume that the potential v is periodic with period lattice l we study the spectrum of the schrodinger operator h v acting on the space of bloch waves with arbitrary but fixed wavevector k let t r3 l let u be an eigenfunction of h with eigenvalue and let 0 be arbitrarily small we show that the classical regularity of the eigenfunction u is u h5 2 t in the usual sobolev spaces and u k3 2 m t s in the weighted sobolev spaces the regularity index m can be as large as desired which is crucial for numerical methods for any choice of the bloch wavevector k we also show that h has compact resolvent and hence a complete eigenfunction expansion the case of the hydrogen atom suggests that our regularity results are optimal we present two applications to the numerical approximation of eigenvalues using wave functions and usi

a particle partition of unity method for the solution of elliptic parabolic and hyperbolic pdes in this paper we present a meshless discretization technique for instationary convection diffusion problems it is based on operator splitting the method of characteristics and a generalized partition of unity method we focus on the discretization process and its quality the method may be used as an h version or a p version even for general particle distributions the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid we discuss the implementational aspects of the proposed method furthermore we present the results of numerical examples where we considered instationary convection diffusion instationary diffusion linear advection and elliptic problems

multigrid methods for the computation of singular solutions and stress intensity factors i corner singularities we consider the poisson equation u f with homogeneous dirichlet boundary condition on a two dimensional polygonal domain with cracks multigrid methods for the computation of singular solutions and stress intensity factors using piecewise linear functions are analyzed the convergence rate for the stress intensity factors is mathcal o h 3 2 in whenfel2 and mathcal o h 2 in whenfeh1 the convergence rate in the energy norm is mathcal o h 1 in in the first case and mathcal o h in the second case the costs of these multigrid methods are proportional to the number of elements in the triangulation the general case wherefehm is also discussed

weighted sobolev spaces and regularity for polyhedral domains we prove a regularity result for the poisson problem u f u p g on a polyhedral domain p r3 using the babuska kondratiev spaces kam p these are weighted sobolev spaces in which the weight is given by the distance to the set of edges 4 33 in particular we show that there is no loss of kam regularity for solutions of strongly elliptic systems with smooth coefficients we also establish a trace theorem for the restriction to the boundary of the functions in kam p

quasi optimal rates of convergence for the generalized finite element method in polygonal domains we consider a mixed boundary value interface problem for the elliptic operator p i j i a i j j u f on a polygonal domain r 2 with straight sides we endowed the boundary of partially with dirichlet boundary conditions u 0 on d and partially with neumann boundary conditions i j i a i j j u 0 on n the coefficients a i j are piecewise smooth with jump discontinuities across the interface which is allowed to have singularities and cross the boundary of in particular we consider triple junctions and even multiple junctions our main result is to construct a sequence of generalized finite element spaces s n that yield h m quasi optimal rates of convergence m 1 for the galerkin approximations u n s n of the solution u more precisely we prove that u u n c dim s n m 2 f h m 1 where c depends on the data for the problem but not on f u or n and dim s n our construction is quite general and depends on a choice of a good sequence of approximation spaces s n on a certain subdomain w that is at some distance to the vertices in case the spaces s n are generalized finite element spaces then the resulting spaces s n are also generalized finite element spaces

gaussian multiresolution models exploiting sparse markov and covariance structure in this paper we consider the problem of learning gaussian multiresolution mr models in which data are only available at the finest scale and the coarser hidden variables serve to capture long distance dependencies tree structured mr models have limited modeling capabilities as variables at one scale are forced to be uncorrelated with each other conditioned on other scales we propose a new class of gaussian mr models in which variables at each scale have sparse conditional covariance structure conditioned on other scales our goal is to learn a tree structured graphical model connecting variables across scales which translates into sparsity in inverse covariance while at the same time learning sparse structure for the conditional covariance not its inverse within each scale conditioned on other scales this model leads to an efficient new inference algorithm that is similar to multipole methods in computational physics we demonstrate the modeling and inference advantages of our approach over methods that use mr tree models and single scale approximation methods that do not use hidden variables

stability and convergence of fully discrete galerkin fems for the nonlinear thermistor equations in a nonconvex polygon in this paper we establish the unconditional stability and optimal error estimates of a linearized backward euler galerkin finite element method fem for the time dependent nonlinear thermistor equations in a two dimensional nonconvex polygon due to the nonlinearity of the equations and the non smoothness of the solution in a nonconvex polygon the analysis is not straightforward while most previous efforts for problems in nonconvex polygons mainly focused on linear models our theoretical analysis is based on an error splitting proposed in 30 31 together with rigorous regularity analysis of the nonlinear thermistor equations and the corresponding iterated time discrete elliptic system in a nonconvex polygon with the proved regularity we establish the stability in l infty l infty l l and the convergence in l infty l 2 l l2 for the fully discrete finite element solution without any restriction on the time step size the approach used in this paper may also be applied to other nonlinear parabolic systems in nonconvex polygons numerical results confirm our theoretical analysis and show clearly that no time step condition is needed

a particle partition of unity method part ii efficient cover construction and reliable integration in this paper we present a meshfree discretization technique based only on a set of irregularly spaced points x i in mathbb r d and the partition of unity approach in this sequel to m griebel and m a schweitzer siam j sci comput 22 2000 pp 853 890 we focus on the cover construction and its interplay with the integration problem arising in a galerkin discretization we present a hierarchical cover construction algorithm and a reliable decomposition quadrature scheme here we decompose the integration domains into disjoint cells on which we employ local sparse grid quadrature rules to improve computational efficiency the use of these two schemes already reduces the operation count for the assembly of the stiffness matrix significantly now the overall computational costs are dominated by the number of the integration cells we present a regularized version of the hierarchical cover construction algorithm which reduces the number of integration cells even further and subsequently improves the computational efficiency in fact the computational costs during the integration of the nonzeros of the stiffness matrix are comparable to that of a finite element method yet the presented method is completely independent of a mesh moreover our method is applicable to general domains and allows for the construction of approximations of any order and regularity

histogram thresholding by minimizing graylevel fuzziness the problem of histogram sharpening and thresholding by minimizing greyness ambiguity using the measures of fuzziness in a set is considered this work provides a complete mathematical formulation of the said problem in order that both fuzzy and crisp segmentation result the criteria regarding the choices of membership functions and the window sizes are established they involve symmetry in ambiguity around the crossover point and bound functions for restricting the variation of membership function values the segmentation thus obtained provides not only flexibility in methodology but also effective results experimental results further establish the same conclusions

an adaptable threshold detector a data set often comprises some data classes for example a gray scale image may consist of some objects each of which has similar pixels gray scales the threshold obtained by otsu s thresholding method otm is biased towards certain data class with larger variance or larger number of data when the variances or the numbers of data among classes are quite different in this paper adaptable threshold detector atd is proposed to improve the effectiveness of otm in determining proper thresholds by dividing class variance by class interval atd is more versatile at selecting application dependent thresholds by changing two parameter values which describe the relative importance among data size standard deviation and class interval of a class in this paper atd is applied to crop the expected objects from images to verify its effect upon thresholding experimental results demonstrate that atd is able to perform better than otm in segmenting objects from images besides excelling over the valley emphasis method vem and the minimum class variance thresholding method mcvtm atd is also suitable for separating objects from serialized video images i e computerized tomography

a robust procedure for gaussian graphical model search from microarray data with p larger than n learning of large scale networks of interactions from microarray data is an important and challenging problem in bioinformatics a widely used approach is to assume that the available data constitute a random sample from a multivariate distribution belonging to a gaussian graphical model as a consequence the prime objects of inference are full order partial correlations which are partial correlations between two variables given the remaining ones in the context of microarray data the number of variables exceed the sample size and this precludes the application of traditional structure learning procedures because a sampling version of full order partial correlations does not exist in this paper we consider limited order partial correlations these are partial correlations computed on marginal distributions of manageable size and provide a set of rules that allow one to assess the usefulness of these quantities to derive the independence structure of the underlying gaussian graphical model furthermore we introduce a novel structure learning procedure based on a quantity obtained from limited order partial correlations that we call the non rejection rate the applicability and usefulness of the procedure are demonstrated by both simulated and real data

the generalized finite element method this paper describes a pilot design and implementation of the generalized finite element method gfem as a direct extension of the standard finite element method sfem or fem which makes possible the accurate solution of engineering problems in complex domains which may be practically impossible to solve using the fem the development of the gfem is illustrated for the laplacian in two space dimensions in domains which may include several hundreds of voids and or cracks for which the construction of meshes used by the fem is practically impossible the two main capabilities are 1 it can construct the approximation using meshes which may overlap part or all of the domain boundary 2 it can incorporate into the approximation handbook functions which are known analytically or are generated numerically and approximate well the solution of the boundary value problem in the neighborhood of corner points voids cracks etc the main tool is a special integration algorithm which we call the fast remeshing approach which is robust and works for any domain with arbitrary complexity the incorporation of the handbook functions into the gfem is done by employing the partition of unity method pum the presented formulations and implementations can be easily extended to the multi material medium where the voids are replaced by inclusions of various shapes and sizes and to the case of the elasticity problem this work can also be understood as a pilot study for the feasibility and demonstration of the capabilities of the gfem which is needed before analogous implementations are attempted in the three dmensional and nonlinear cases which are the cases of main interest for future work

thresholding using two dimensional histogram and fuzzy entropy principle this paper presents a thresholding approach by performing fuzzy partition on a two dimensional 2 d histogram based on fuzzy relation and maximum fuzzy entropy principle the experiments with various gray level and color images have demonstrated that the proposed approach outperforms the 2 d nonfuzzy approach and the one dimensional 1 d fuzzy partition approach

a nonconforming generalized finite element method for transmission problems we obtain quasi optimal rates of convergence for transmission interface problems on domains with smooth curved boundaries using a nonconforming generalized finite element method gfem more precisely we study the strongly elliptic problem pu sum partial j a ij partial i u f in a smooth bounded domain omega with dirichlet boundary conditions the coefficients a ij are piecewise smooth possibly with jump discontinuities along a smooth closed surface gamma called the interface which does not intersect the boundary of the domain we consider a sequence of approximation spaces s mu satisfying two conditions 1 nearly zero boundary and interface matching and 2 approximability which are similar to those in babuska nistor and tarfulea j comput appl math 218 2008 pp 175 183 then if u mu in s mu mu ge 1 is a sequence of galerkin approximations of the solution u to the interface problem the approximation error u u mu hat h 1 omega is

analysis and applications of a generalized finite element method with global local enrichment functions abstract this paper presents a procedure to build enrichment functions for partition of unity methods like the generalized finite element method and the hp cloud method the procedure combines classical global local finite element method concepts with the partition of unity approach it involves the solution of local boundary value problems using boundary conditions from a global problem defined on a coarse discretization the local solutions are in turn used to enrich the global space using the partition of unity framework the computations at local problems can be parallelized without difficulty allowing the solution of large problems very efficiently the effectiveness of the approach in terms of convergence rates and computational cost is investigated in this paper we also analyze the effect of inexact boundary conditions applied to local problems and the size of the local domains on the accuracy of the enriched global solution key aspects of the computational implementation in particular the numerical integration of generalized fem approximations built with global local enrichment functions are presented the method is applied to fracture mechanics problems with multiple cracks in the domain our numerical experiments show that even on a serial computer the method is very effective and allows the solution of complex problems our analysis also demonstrates that the accuracy of a global problem defined on a coarse mesh can be controlled using a fixed number of global degrees of freedom and the proposed global local enrichment functions

generalized finite element method using mesh based handbooks application to problems in domains with many voids this paper describes a new version of the generalized finite element method originally developed int j numer methods engrg 47 2000 1401 comput methods appl mech engrg 181 2000 43 the design and implementation of the generalized finite element method ph d thesis texas am comput methods appl mech engrg 190 2001 4081 which is well suited for problems set in domains with a large number of internal features e g voids inclusions cracks etc the main idea is to employ handbook functions constructed on subdomains resulting from the mesh discretization of the problem domain the proposed new version of the gfem is shown to be robust with respect to the spacing of the features and is capable of achieving high accuracy on meshes which are rather coarse relative to the distribution of the features

improving the rate of convergence of high order finite elements on polygons and domains with cusps let u and uv v be the solution and respectively the discrete solution of the non homogeneous dirichlet problem u f on u 0 for any m and any bounded polygonal domain we provide a construction of a new sequence of finite dimensional subspaces vn such that where f hm 1 is arbitrary and c is a constant that depends only on and not on n we do not assume u hm 1 the existence of such a sequence of subspaces was first proved in a ground breaking paper by babuska 8 our method is different it is based on the homogeneity properties of sobolev spaces with weights and the well posedness of non homogeneous dirichlet problem in suitable sobolev spaces with weights for which we provide a new proof and which is a substitute of the usual shift theorems for boundary value problems in domains with smooth boundary our results extended right away to domains whose boundaries have conical points we also indicate some of the changes necessary to deal with domains with cusps our numerical computation are in agreement with our theoretical results

joint estimation of multiple graphical models gaussian graphical models explore dependence relationships between random variables through the estimation of the corresponding inverse covariance matrices in this paper we develop an estimator for such models appropriate for data from several graphical models that share the same variables and some of the dependence structure in this setting estimating a single graphical model would mask the underlying heterogeneity while estimating separate models for each category does not take advantage of the common structure we propose a method that jointly estimates the graphical models corresponding to the different categories present in the data aiming to preserve the common structure while allowing for differences between the categories this is achieved through a hierarchical penalty that targets the removal of common zeros in the inverse covariance matrices across categories we establish the asymptotic consistency and sparsity of the proposed estimator in the high dimensional case and illustrate its performance on a number of simulated networks an application to learning semantic connections between terms from webpages collected from computer science departments is included

on testing marginal versus conditional independence we consider testing marginal independence versus conditional independence in a trivariate gaussian setting the two models are non nested and their intersection is a union of two marginal independences we consider two sequences of such models one from each type of independence that are closest to each other in the kullback leibler sense as they approach the intersection they become indistinguishable if the signal strength as measured by the product of two correlation parameters decreases faster than the standard parametric rate under local alternatives at such rate we show that the asymptotic distribution of the likelihood ratio depends on where and how the local alternatives approach the intersection to deal with this non uniformity we study a class of envelope distributions by taking pointwise suprema over asymptotic cumulative distribution functions we show that these envelope distributions are well behaved and lead to model selection procedures with rate free uniform error guarantees and near optimal power to control the error even when the two models are indistinguishable rather than insist on a dichotomous choice the proposed procedure will choose either or both models

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

meshfree particle methods for thin plates abstract in this paper we are concerned with meshfree particle methods for the solutions of the classical plate model the vertical displacement of a thin plate is governed by a fourth order elliptic equation and thus the approximation functions for numerical solutions are required to have continuous partial derivatives hence the conventional finite element method has difficulties to solve the fourth order problems meshfree methods have the advantage of constructing smooth approximation functions however most of the earlier works on meshfree methods for plate problems used either moving least squares method with penalty method or coupling fem with meshfree method to deal with essential boundary conditions in this paper by using generalized product partition of unity introduced by oh et al we introduce meshfree particle methods in which approximation functions have high order polynomial reproducing property and the kronecker delta property we also prove error estimates for the proposed meshfree methods moreover to demonstrate the effectiveness of our method results of the proposed method are compared with existing results for various shapes of plates with variety of boundary conditions and loads

fitting truncated geometric distributions in large scale real world networks degree distribution of nodes especially a power law degree distribution has been regarded as one of the most significant structural characteristics of social and information networks however it is observed here that for many large scale real world networks the power law does not fit properly because of the presence of large fluctuations and sparsity in upper and lower tails of the distribution here we have proposed to fit the truncated geometric distribution on three distinct and non overlapping parts of the degree frequency table extensive experiments on twenty three 23 real world networks revealed that the proposed model fitted better than the power law and other distributions

model selection and estimation in the gaussian graphical model summary we propose penalized likelihood methods for estimating the concentration matrix in the gaussian graphical model the methods lead to a sparse and shrinkage estimator of the concentration matrix that is positive definite and thus conduct model selection and estimation simultaneously the implementation of the methods is nontrivial because of the positive definite constraint on the concentration matrix but we show that the computation can be done effectively by taking advantage of the efficient maxdet algorithm developed in convex optimization we propose a bic type criterion for the selection of the tuning parameter in the penalized likelihood methods the connection between our methods and existing methods is illustrated simulations and real examples demonstrate the competitive performance of the new methods

elliptical graphical modelling we propose elliptical graphical models based on conditional uncorrelatedness as a gen eralization of gaussian graphical models by letting the population distribution be elliptical instead of normal allowing the tting of data with arbitrarily heavy tails we study the class of propor tionally ane equivariant scatter estimators and show how they can be used to perform elliptical graphical modelling leading to a new class of partial correlation estimators and analogues of the classical deviance test general expressions for the asymptotic variance of partial correlation esti mators unconstrained and under decomposable models are given and the asymptotic chi square approximation of the pseudo deviance test statistic is proved the feasibility of our approach is demonstrated by a simulation study using among others tyler s scatter estimator which is distribution free within the elliptical model our approach provides a robustication of gauss ian graphical modelling the latter is likelihood based and known to be very sensitive to model misspecication and outlying observations 1 introduction and notation the statistical theory of undirected graphical models for continuous variables is usually based on the assumption of multivariate normality in practice data may deviate from the normal model in various ways outliers and heavy tails pose a problem of particular gravity they frequently occur and the normal likelihood methods such as the sample covariance matrix are very susceptible to them our objective is to deal with heavy tailed data and to safeguard graphical modelling against the impact of faulty outliers we restrict our attention to the case where we have only continuous variables and only undirected edges joint multivariate normality is often assumed in this situation and the statistical methodology is called gaussian graphical modelling we propose the class of elliptical distributions as a more general model and call our approach elliptical graphical modelling the lack of robustness of gaussian graphical modelling has been noted by several authors four proposals of robust approaches to gaussian graphical modelling are known to us becker 2005 and gottard and pacillo 2010 suggest replacing the sample covariance matrix by the reweighted minimum covariance determinant estimator miyamura and kano 2006 propose an alternative m type estimation and finegold drton arxiv 1009 3669 consider robustied versions of the graphical lasso by friedman et al 2008 this article delivers a systematic treatment of the plug in approach used in the rst two ref erences we show that the sample covariance matrix may be replaced by any ane equivariant

a two level finite element discretization of the streamfunction formulation of the stationary quasi geostrophic equations of the ocean in this paper we proposed a two level finite element discretization of the nonlinear stationary quasi geostrophic equations which model the wind driven large scale ocean circulation optimal error estimates for the two level finite element discretization were derived numerical experiments for the two level algorithm with the argyris finite element were also carried out the numerical results verified the theoretical error estimates and showed that for the appropriate scaling between the coarse and fine mesh sizes the two level algorithm significantly decreases the computational time of the standard one level algorithm

edge exclusion tests for graphical model selection complex gaussian vectors and time series we consider the problem of inferring the conditional independence graph cig of a stationary multivariate gaussian time series a p variate gaussian time series graphical model associated with an undirected graph with p vertices is defined as the family of time series that obey the conditional independence restrictions implied by the edge set of the graph in some existing methods partial coherence has been used as a test statistic for graphical model selection to test inclusion exclusion of a given edge in the graph the test is applied at distinct frequencies requiring multiple tests and leading to a loss in test power the nonparametric method of matsuda uses the kullback leibler divergence measure to define a test statistic which does not need multiple testing to test between two competing models in this paper we propose a generalized likelihood ratio test glrt based edge exclusion test statistic that also does not need multiple testing it is computationally significantly faster than the method of matsuda and simulations show that we achieve comparable power levels our proposed approach is based on a novel formulation of a glrt based edge exclusion test for p variate complex gaussian graphical models cggms this result is also of independent interest the computational complexity of the proposed statistic is mathcal o p 3 compared to mathcal o p 5 for the existing result we also apply our time series graphical model selection method to a foreign exchange data set consisting of monthly trends of foreign exchange rates of 10 countries

the optimal convergence rate of a c1 finite element method for non smooth domains we establish optimal up to arbitrary e 0 convergence rates for a finite element formulation of a model second order elliptic boundary value problem in a weightedh 2 sobolev space with 5th degree argyris elements this formulation arises while generalizing to the case of non smooth domains an unconditionally stable scheme developed by liu et al j g liu j liu r l pego stability and convergence of efficient navier stokes solvers via a commutator estimate comm pure appl math 60 2007 pp 1443 1487 for the navier stokes equations we prove the optimality for both quasiuniform and graded mesh refinements and provide numerical results that agree with our theoretical predictions

nonparametric bayes inference on conditional independence in many application areas a primary focus is on assessing evidence in the data refuting the assumption of independence of y and x conditionally on z with y response variables x predictors of interest and z covariates ideally one would have methods available that avoid parametric assumptions allow y x z to be random variables on arbitrary spaces with arbitrary dimension and accommodate rapid consideration of different candidate predictors as a formal decision theoretic approach has clear disadvantages in this context we instead rely on an encompassing nonparametric bayes model for the joint distribution of y x and z with conditional mutual information used as a summary of the strength of conditional dependence we construct a functional of the encompassing model and empirical measure for estimation of conditional mutual information the implementation relies on a single markov chain monte carlo run under the encompassing model with conditional mutual information for candidate models calculated as a byproduct we provide an asymptotic theory supporting the approach and apply the method to variable selection the methods are illustrated through simulations and criminology applications

image segmentation using fuzzy homogeneity criterion abstract the approach proposed here is using fuzzy homogeneity vectors and the fuzzy cooccurrence matrix to segment images homogeneity vectors could be used to represent the homogeneous feature between a pixel and its neighbors and the degree of homogeneity could be determined by using a fuzzy membership function combining the homogeneity vectors and the fuzzy membership function we could extract the feature of an image which determines the fuzzy region width with the fuzzy region width we could use the fuzzy cooccurrence matrix to measure the fuzzy geometry properties of an image which is the fuzzy entropy values that can be employed for determinine the thresholds to segment images the advantages of this approach are as follows first homogeneity vectors take account of the spatial gray tone dependence thus using homogeneity vectors has a better noise tolerance second from the extracted feature the fuzzy region width could be determined automatically third because the fuzzy region width is decided by the feature of the image it could be adjusted according to the nature of the image fourth the dimension of the fuzzy cooccurrence matrix representing the image is significantly decreased but the properties of images are faithfully preserved a large number of experiments have been carried out on different kinds of images and good results have been achieved by the proposed method this method will have wide application in image processing

analytic regularity of stokes flow on polygonal domains in countably weighted sobolev spaces we investigate the analytic regularity of the stokes problem in a polygonal domain r2 with straight sides and piecewise analytic data we establish a shift theorem in weighted sobolev spaces of arbitrary order with explicit control of the order dependence of the constants the shift theorem in the framework of countably weighted sobolev spaces implies in particular interior analyticity and gevrey type analytic regularity near the corners

interior energy error estimates for the weak galerkin finite element method consider the poisson equation in a polytopal domain omega subset mathbb r d d 2 3 as the model problem we study interior energy error estimates for the weak galerkin finite element approximation to elliptic boundary value problems in particular we show that the interior error in the energy norm is bounded by three components the best local approximation error the error in negative norms and the trace error on the element boundaries this implies that the interior convergence rate can be polluted by solution singularities on the domain boundary even when the solution is smooth in the interior region numerical results are reported to support the theoretical findings to the best of our knowledge this is the first local energy error analysis that applies to general meshes consisting of polytopal elements and hanging nodes

ancestral graph markov models this paper introduces a class of graphical independence models that is closed under marginalization and conditioning but that contains all dag independence models this class of graphs called maximal ancestral graphs has two attractive features there is at most one edge between each pair of vertices every missing edge corresponds to an independence relation these features lead to a simple parameterization of the corresponding set of distributions in the gaussian case

multimodality of the likelihood in the bivariate seemingly unrelated regressions model summary we analyse the simplest two equation seemingly unrelated regressions model and demonstrate that its likelihood may have up to five stationary points and thus there may be up to three local modes consequently the estimates obtained via iterative estimation methods may depend on starting values we further show that the probability of multi modality vanishes asymptotically monte carlo simulations suggest that multimodality rarely occurs if the seemingly unrelated regressions model is true but can become more frequent if the model is misspecified the existence of multimodality in the likelihood for seemingly unrelated regressions models contradicts several claims in the literature

regularity estimates for elliptic boundary value problems in besov spaces we consider the dirichlet problem for poisson s equation on a nonconvex plane polygonal domain new regularity estimates for its solution in terms of besov and sobolev norms of fractional order are proved the analysis is based on new interpolation results and multilevel representations of norms on sobolev and besov spaces the results can be extended to a large class of elliptic boundary value problems some new sharp finite element error estimates are deduced

well posedness and regularity for the elasticity equation with mixed boundary conditions on polyhedral domains and domains with cracks we prove a regularity result for the anisotropic linear elasticity equation p u rm div left boldmath mathsf c cdot nabla u right f with mixed displacement and traction boundary conditions on a curved polyhedral domain omega subset mathbb r 3 in weighted sobolev spaces mathcal k m 1 a 1 omega for which the weight is given by the distance to the set of edges in particular we show that there is no loss of mathcal k m a regularity our curved polyhedral domains are allowed to have cracks we establish a well posedness result when there are no neighboring traction boundary conditions and a 0 that depends on p on the boundary conditions and on the domain our results extend to other strongly elliptic systems and higher dimensions

objective bayesian search of gaussian directed acyclic graphical models for ordered variables with non local priors directed acyclic graphical dag models are increasingly employed in the study of physical and biological systems to model direct influences between variables identifying the graph from data is a challenging endeavor which can be more reasonably tackled if the variables are assumed to satisfy a given ordering in this case we simply have to estimate the presence or absence of each potential edge working under this assumption we propose an objective bayesian method for searching the space of gaussian dag models which provides a rich output from minimal input we base our analysis on non local parameter priors which are especially suited for learning sparse graphs because they allow a faster learning rate relative to ordinary local parameter priors when the true unknown sampling distribution belongs to a simple model we implement an efficient stochastic search algorithm which deals effectively with data sets having sample size smaller than the number of variables and apply our method to a variety of simulated and real data sets our approach compares favorably in terms of the roc curve for edge hit rate versus false alarm rate to current state of the art frequentist methods relying on the assumption of ordered variables under this assumption it exhibits a competitive advantage over the pc algorithm which can be considered as a frequentist benchmark for unordered variables importantly we find that our method is still at an advantage for learning the skeleton of the dag when the ordering of the variables is only moderately mis specified prospectively our method could be coupled with a strategy to learn the order of the variables thus dropping the known ordering assumption

correlation between two fuzzy membership functions the need for a measure of correlation between two membership functions is stated the properties that the correlation measure may possess are examined a measure of correlation is defined on the basis of those properties the definition is extended to the cases when a the domain is finite and b the domain is a subset of r n the problems encountered while extending the definition are also discussed

asymptotic exactness of an a posteriori error estimator based on the equilibrated residual method in this paper we analyze an a posteriori error estimator based on the equilibrated residual method we prove that this estimator is asymptotically exact in the energy norm for regular solutions and o h 1 alpha alpha 0 meshes numerical examples are included to illustrate the theoretical results

model selection for gaussian concentration graphs gga multivariate gaussian graphical markov model for an undirected graph g also called a covariance selection model or concentration graph model is defined in terms of the markov properties i e conditional independences associated with g which in turn are equivalent to specified zeros among the set of pairwise partial correlation coefficients by means of fisher s z transformation and sidak s correlation inequality conservative simultaneous confidence intervals for the entire set of partial correlations can be obtained leading to a simple method for model selection that controls the overall error rate for incorrect edge inclusion the simultaneous p values corresponding to the partial correlations are partitioned into three disjoint sets a significant set s an indeterminate set i and a nonsignificant set n our model selection method selects two graphs a graph g si whose edges correspond to the set s i and a more conservative graph g s whose edges correspond to s only similar considerations apply to covariance graph models which are defined in terms of marginal independence rather than conditional independence the method is applied to some well known examples and to simulated data

on the sharpness of certain local estimates for projections into finite element spaces influence of a re entrant corner in a plane polygonal domain with a reentrant corner consider a homogeneous dirichlet problem for poisson s equation au f with f smooth and the corresponding galerkin finite element solutions in a family of piecewise polynomial spaces based on quasi uniform uniformly regular triangulations with the diameter of each element compara