on the poincare friedrichs inequality for piecewise h 1 functions in anisotropic discontinuous galerkin finite element methods the purpose of this paper is to propose a proof for the poincare friedrichs inequality for piecewise h 1 functions on anisotropic meshes by verifying suitable assumptions involved in the newly proposed proof we show that the poincare friedrichs inequality for piecewise h 1 functions holds independently of the aspect ratio which characterizes the shape regular condition in finite element analysis in addition under the maximum angle condition we establish the poincare friedrichs inequality for the crouzeix raviart non conforming linear finite element counterexamples show that the maximum angle condition is only sufficient

shrinkage priors for bayesian prediction we investigate shrinkage priors for constructing bayesian predictive distributions it is shown that there exist shrinkage predictive distributions asymptotically dominating bayesian predictive distributions based on the jeffreys prior or other vague priors if the model manifold satisfies some differential geometric conditions kullback leibler divergence from the true distribution to a predictive distribution is adopted as a loss function conformal transformations of model manifolds corresponding to vague priors are introduced we show several examples where shrinkage predictive distributions dominate bayesian predictive distributions based on vague priors

information geometry on the curved q exponential family with application to survival data analysis abstract information geometry and statistical manifold have attracted wide attention in the past few decades the amari chentsov structure plays a central role both in optimization and statistical inference because of the invariance under sufficient statistics this paper discusses the amari chentsov structure on statistical manifolds induced by the curved q exponential family with type i censored data the reliability function and cumulant generating function of the family are obtained where the cumulant generating function contains a random variable depends on the experimental design after the construction of the statistical manifold the geometric quantities i e fisher metric amari tensor and connection coefficients are investigated we find that the affine connection is not equal to 0 which is different from the existing results the relationship among the geometric quantities based on the q exponential family and curved q exponential family is discussed the relationship between the curvature on the manifold in the normal and tangent directions is obtained employing the maximum likelihood and bayesian methods the parameter vector and its dual are estimated the main results are illustrated with the two parameter q exponential distribution and a real data analysis based on a terminal human cancer data set

estimation of the mean of a multivariate normal distribution estimation of the means of independent normal random variables is considered using sum of squared errors as loss an unbiased estimate of risk is obtained for an arbitrary estimate and certain special classes of estimates are then discussed the results are applied to smoothing by use of moving averages and to trimmed analogs of the james stein estimate a suggestion is made for calculating approximate confidence sets for the mean vector centered at an arbitrary estimate

interpolation error estimates in w 1 p for degenerate q 1 isoparametric elements optimal order error estimates in h 1 for the q 1 isoparametric interpolation were obtained in acosta and duran siam j numer anal37 18 36 1999 for a very general class of degenerate convex quadrilateral elements in this work we show that the same conlusions are valid in w 1 p for 1 p 3 and we give a counterexample for the case p 3 showing that the result cannot be generalized for more regular functions despite this fact we show that optimal order error estimates are valid for any p 1 keeping the interior angles of the element bounded away from 0 and independently of the aspect ratio we also show that the restriction on the maximum angle is sharp for p 3

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

bayesian prediction with approximate frequentist validity we characterize priors which asymptotically match the posterior coverage probability of a bayesian prediction region with the corresponding frequentist coverage probability this is done considering both posterior quantiles and highest predictive density regions with reference to a future observation the resulting priors are shown to be invariant under reparameterization the role of jeffreys prior in this regard is also investigated it is further shown that for any given prior it may be possible to choose an interval whose bayesian predictive and frequentist coverage probabilities are asymptotically matched

the maximum likelihood prior consider an estimate theta of a parameter theta based on repeated observations from a family of densities f theta evaluated by the kullback leibler loss function k theta theta int log f theta f theta f theta the maximum likelihood prior density if it exists is the density for which the corresponding bayes estimate is asymptotically negligibly different from the maximum likelihood estimate the bayes estimate corresponding to the maximum likelihood prior is identical to maximum likelihood for exponential families of densities in predicting the next observation the maximum likelihood prior produces a predictive distribution that is asymptotically at least as close in expected truncated kullback leibler distance to the true density as the density indexed by the maximum likelihood estimate it frequently happens in more than one dimension that maximum likelihood corresponds to no prior density and in that case the maximum likelihood estimate is asymptotically inadmissible and may be improved upon by using the estimate corresponding to a least favorable prior as in brown the asymptotic risk for an arbitrary estimate near maximum likelihood is given by an expression involving derivatives of the estimator and of the information matrix admissibility questions for these near ml estimates are determined by the existence of solutions to certain differential equations

three dimensional recovery of stress intensity factors and energy release rates from surface full field displacements the identification of stress intensity factors sifs from full field displacement measurements by the optic method is accelerating rapidly following the development of precise digital cameras and image correlation algorithms nevertheless as only surface displacements are available most of the identification methods used up to now have been restricted to problems that are invariant with respect to the normal direction of the free surface and must rely on the plane elasticity hypothesis the problem of sifs identification is tackled here in a full three dimensional framework by first deriving a data completion method in elasticity to determine the elastic displacement field inside the solid on the basis of surface displacements the method solves the cauchy problem for the lame operator after which usual numerical methods for computing sifs and energy release rates can be used numerical applications in three dimensional elasticity are described first for a cracked specimen subjected to 3d loadings and secondly for a heterogeneous specimen in a quasi plane situation

inadmissibility of the best equivariant predictive density in the unknown variance case where is an estimator of and the expectation is taken with respect to random quantities usedin the construction of this phenomenon the stein e ect was rst brought to lightin the context of parameter estimation for such a problem many classes of estimatorsdominating the average have been proposed in parallel a similar phenomenon has been observed for predictive density estimation for instance let us observe nindependent d dimensional vectors x

error estimates for cq_1 isoparametric elements satisfying a weak angle condition we prove optimal order error estimates in the h1 norm for the cq 1 isoparametric interpolation on convex quadrilateral elements under rather weak hypotheses improving previously known results choose one diagonal and divide the element into two triangles we show that if the chosen diagonal is the longest one then the constant in the error estimate depends only on the maximum angle of the two triangles otherwise the constant depends on that maximum angle and on the ratio between the two diagonals in particular we obtain the optimal order error estimate under the maximum angle condition as in the case of triangular elements consequently the error estimate is uniformly valid for a rather general class of degenerate quadrilaterals

simultaneous prediction of independent poisson observables simultaneous predictive distributions for independent poisson observables are investigated a class of improper prior distributions for poisson means is introduced the bayesian predictive distributions based on priors from the introduced class are shown to be admissible under the kullback leibler loss a bayesian predictive distribution based on a prior in this class dominates the bayesian predictive distribution based on the jeffreys prior

lagrange and average interpolation over 3d anisotropic elements abstract an average interpolation is introduced for 3 rectangles and tetrahedra and optimal order error estimates in the h1 norm are proved the constant in the estimate depends weakly improving the results given in duran math comp 68 1999 187 199 on the uniformity of the mesh in each direction for tetrahedra the constant also depends on the maximum angle of the element on the other hand merging several known results acosta and duran siam j numer anal 37 1999 18 36 duran math comp 68 1999 187 199 krizek siam j numer anal 29 1992 513 520 al shenk math comp 63 1994 105 119 we prove optimal order error for the p 1 lagrange interpolation in w1 p p 2 with a constant depending on p as well as the maximum angle of the element again under the maximum angle condition optimal order error estimates are obtained in the h1 norm for higher degree interpolations

error estimates on anisotropic elements for functions in weighted sobolev spaces in this paper we prove error estimates for a piecewise q 1 average interpolation on anisotropic rectangular elements i e rectangles with sides of different orders in two and three dimensions our error estimates are valid under the condition that neighboring elements have comparable size this is a very mild assumption that includes more general meshes than those allowed in previous papers in particular strong anisotropic meshes arising naturally in the approximation of problems with boundary layers fall under our hypotheses moreover we generalize the error estimates allowing on the right hand side some weighted sobolev norms this extension is of interest in singularly perturbed problems finally we consider the approximation of functions vanishing on the boundary by finite element functions with the same property a point that was not considered in previous papers on average interpolations for anisotropic elements as an application we consider the approximation of a singularly perturbed reaction diffusion equation and show that as a consequence of our results almost optimal order error estimates in the energy norm valid uniformly in the perturbation parameter can be obtained

the maximum angle condition for mixed and nonconforming elements application to the stokes equations for the lagrange interpolation it is known that optimal order error estimates hold for elements satisfying the maximum angle condition the objective of this paper is to obtain similar results for the raviart thomas interpolation arising in the analysis of mixed methods we prove that optimal order error estimates hold under the maximum angle condition for this interpolation both in two and three dimensions and moreover that this condition is indeed necessary to have these estimates error estimates for the mixed approximation of second order elliptic problems and for the nonconforming piecewise linear approximation of the stokes equations are derived from our results

bayesian duality and risk analysis on the statistical manifold of exponential family with censored data abstract information geometry has been attracted wide attentions in the past few decades this paper focuses on the bayesian duality on a statistical manifold derived from the exponential family with data from life tests based on life testing data the statistical manifold is constructed with a new cumulant generating function the bregman divergence between two parameter points is studied the dual coordinate system and dual function are obtained then the dualistic structure on the manifold is discussed the results show that the maximum likelihood estimate can be obtained by minimizing the bregman divergence induced from the dual function the bayesian analysis and prediction are investigated based on informative and non informative priors consider the gamma distribution as an example the closed form representations of the dual coordinate system and dual function are obtained a real data set is employed to illustrate the methodologies and experimental designs developed in this paper

on the maximum angle condition for linear tetrahedral elements synge s maximum angle condition for triangular elements is generalized to tetrahedral elements for the generalized condition it is proved that tetrahedra may degenerate in a certain way and the error of the standard linear interpolation remains o h in the w p 1 omega norm for sufficiently smooth functions and p in 1 infty

a uniformly accurate finite element method for the reissner mindlin plate a simple finite element method for the reissner mindlin plate model in the prim itive variables is presented and analyzed the method uses nonconforming linear finite elements for the transverse displacement and conforming linear finite elements enriched by bubbles for the rotation with the computation of the element stiffness matrix modified by the inclusion of a simple elementwise averaging it is proved that the method converges with optimal order uniformly with respect to thickness

the mortar finite element method with lagrange multipliers the present paper deals with a variant of a non conforming domain decomposition technique the mortar finite element method in the opposition to the original method this variant is never conforming because of the relaxation of the matching constraints at the vertices and the edges in 3d of subdomains it is shown that written under primal hybrid formulation the approximation problem issued from a discretization of a second order elliptic equation in 2d is nonetheless well posed and provides a discrete solution that satisfies optimal error estimates with respect to natural norms finally the parallelization advantages consequence of this variant are also addressed

existence of ℋ matrix approximants to the inverse fe matrix of elliptic operators with l coefficients this article deals with the existence of blockwise low rank approximants so called matrices to inverses of fem matrices in the case of uniformly elliptic operators with l coefficients unlike operators arising from boundary element methods for which the matrix theory has been extensively developed the inverses of these operators do not benefit from the smoothness of the kernel function however it will be shown that the corresponding green functions can be approximated by degenerate functions giving rise to the existence of blockwise low rank approximants of fem inverses numerical examples confirm the correctness of our estimates as a side product we analyse the matrix property of the inverse of the fe mass matrix

stabilization mechanisms in discontinuous galerkin finite element methods in this paper we propose a new general framework for dg methods which allows to uncover a basic mechanism that ensures suitable stability properties of the methods we show that such a mechanism is essentially the same for apparently unrelated stabilizations such as jump penalty upwinding and hughes franca type residual based stabilizations

an energy error based method for the resolution of the cauchy problem in 3d linear elasticity a new method is described for the problem of expanding known displacement fields at the boundary of a solid together with the surface tractions on it towards the solid interior up to the inaccessible part of the boundary the solid is supposedly linearly elastic with known elastic moduli but not necessarily homogeneous nor isotropic the problem is the cauchy problem for the lame operator a new form of this cauchy problem suited for applications associated with surface tangential fields measurements is also stated and studied the method is based on the splitting of the elastic fields into two separate solutions of wellposed problems the gap between these fields is subsequently minimised with respect to the unknown boundary data in order to produce the desired expanded elastic fields the gap used here is an energy error associated with the elastic energy of the system various 3d applications are given including non linear boundary conditions on the unreachable boundary

information geometry of generalized bayesian prediction using alpha divergences as loss functions in this paper the methods of information geometry are employed to investigate a generalized bayes rule for prediction taking alpha divergences as the loss functions optimality and asymptotic properties of the generalized bayesian predictive densities are considered we show that the bayesian predictive densities minimize a generalized bayes risk we also find that the asymptotic expansions of the densities are related to the coefficients of the alpha connections of a statistical manifold in addition we discuss the difference between two risk functions of the generalized bayesian predictions based on different priors finally using the non informative priors i e jeffreys and reference priors uniform prior and conjugate prior two examples are presented to illustrate the main results

poincare friedrichs inequalities for piecewise h 1 functions poincare friedrichs inequalities for piecewise h1 functions are established they can be applied to classical nonconforming finite element methods mortar methods and discontinuous galerkin methods

on the equivalence of regularity criteria for triangular and tetrahedral finite element partitions in this note we examine several regularity criteria for families of simplicial finite element partitions in r d email protected 2 3 these are usually required in numerical analysis and computer implementations we prove the equivalence of four different definitions of regularity often proposed in the literature the first one uses the volume of simplices the others involve the inscribed and circumscribed ball conditions and the minimal angle condition

on asymptotic properties of predictive distributions summary second order asymptotic theory of predictive distributions is investigated it is shown that estimative distributions with asymptotically efficient estimators can be improved to predictive distributions that do not belong to the model the model is assumed to be a multidimensional curved exponential family predictive distributions can be constructed by shifting estimative distributions in a direction orthogonal to the model the average kullback leibler divergence from the true distribution to a predictive distribution is represented as the sum of two components one of them depends on the choice of the estimative distribution and the other depends on the shift orthogonal to the model the optimal shift orthogonal to the model is obtained a quantity that can be regarded as the mixture mean curvature of the model in the space of all probability distributions is introduced the difference in average kullback leibler divergence between an estimative distribution and the optimal predictive distribution obtained by shifting it can be regarded as the mixture mean curvature of the model an asymptotic expression for bayesian predictive distributions is also obtained

improved minimax predictive densities under kullback leibler loss let x n p u x i and y n p v y i be independentp dimensional multivariate normal vectors with common unknown mean based on only observing x x we consider the problem of obtaining a predictive density p y x for y that is close to p y as measured by expected kullback leibler loss a natural procedure for this problem is the formal bayes predictive density p u y x under the uniform prior u 1 which is best invariant and minimax we show that any bayes predictive density will be minimax if it is obtained by a prior yielding a marginal that is superharmonic or whose square root is superharmonic this yields wide classes of minimax procedures that dominate p u y x including bayes predictive densities under superharmonic priors fundamental similarities and differences with the parallel theory of estimating a multivariate normal mean under quadratic loss are described

a shrinkage predictive distribution for multivariate normal observables summary we investigate shrinkage methods for constructing predictive distributions we consider the multivariate normal model with a known covariance matrix and show that there exists a shrinkage predictive distribution dominating the bayesian predictive distribution based on the vague prior when the dimension is not less than three kullback leibler divergence from the true distribution to a predictive distribution is adopted as a loss function somiie key w ords invariance james stein estimator kullback leibler divergence stein s prior vague prior

a simple predictive density based on the p formula the predictive density proposed by harris 1989 is based on integrating the density for a new observation with respect to the estimated sampling distribution of the maximum likelihood estimator of the unknown parameter this has good properties but is rather complicated to compute even for simple models an approximation to the harris proposal is considered which consists of approximating the sampling distribution of the maximum likelihood estimator of the unknown parameter by barndorff nielsen s 1983 p formula and then using a laplace approximation with o n 1 correction terms for integrating out the parameter the result can generally be expressed in terms of standard likelihood derivatives and takes a quite simple form for exponential families and for location models