a semi parametric bayesian approach to the instrumental variable problem we develop a bayesian semi parametric approach to the instrumental variable problem we assume linear structural and reduced form equations but model the error distributions non parametrically a dirichlet process prior is used for the joint distribution of structural and instrumental variable equations errors our implementation of the dirichlet process prior uses a normal distribution as a base model it can therefore be interpreted as modeling the unknown joint distribution with a mixture of normal distributions with a variable number of mixture components we demonstrate that this procedure is both feasible and sensible using actual and simulated data sampling experiments compare inferences from the non parametric bayesian procedure with those based on procedures from the recent literature on weak instrument asymptotics when errors are non normal our procedure is more efficient than standard bayesian or classical methods

a nonparametric multidimensional latent class irt model in a bayesian framework we propose a nonparametric item response theory model for dichotomously scored items in a bayesian framework the model is based on a latent class lc formulation and it is multidimensional with dimensions corresponding to a partition of the items in homogenous groups that are specified on the basis of inequality constraints among the conditional success probabilities given the latent class moreover an innovative system of prior distributions is proposed following the encompassing approach in which the largest model is the unconstrained lc model a reversible jump type algorithm is described for sampling from the joint posterior distribution of the model parameters of the encompassing model by suitably post processing its output we then make inference on the number of dimensions i e number of groups of items measuring the same latent trait and we cluster items according to the dimensions when unidimensionality is violated the approach is illustrated by two examples on simulated data and two applications based on educational and quality of life data

the matrix stick breaking process flexible bayes meta analysis in analyzing data from multiple related studies it often is of interest to borrow information across studies and to cluster similar studies although parametric hierarchical models are commonly used of concern is sensitivity to the form chosen for the random effects distribution a dirichlet process dp prior can allow the distribution to be unknown while clustering studies however the dp does not allow local clustering of studies with respect to a subset of the coefficients without making independence assumptions motivated by this problem we propose a matrix stick breaking process msbp as a prior for a matrix of random probability measures properties of the msbp are considered and methods are developed for posterior computation using markov chain monte carlo using the msbp as a prior for a matrix of study specific regression coefficients we demonstrate advantages over parametric modeling in simulated examples the methods are further illustrated using a multinational uterotrophic bioassay s

predictive updating methods with application to bayesian classification we propose algorithms based on random draws from predictive distributions of unknown quantities missing values for instance this procedure can either be iterative which is a special variation of the gibbs sampler or be sequential which is a variation of sequential imputation in the latter case one can update the posterior distribution with new observations easily the methods proposed have intuitive statistical implications and can be generalized to accommodate other bayesian like procedures we display some applications of the method in connection with the bayesian bootstrap classification hierarchical models and selection of variables in particular as an application of the method we present a unified treatment of switching regression models driven by a general binary process and we develop a bayesian testing procedure some simulations and a real example are used to illustrate the methods proposed

gibbs sampling methods for stick breaking priors a rich and flexible class of random probability measures which we call stick breaking priors can be constructed using a sequence of independent beta random variables examples of random measures that have this characterization include the dirichlet process its two parameter extension the two parameter poisson dirichlet process finite dimensional dirichlet priors and beta two parameter processes the rich nature of stick breaking priors offers bayesians a useful class of priors for nonparametric problems while the similar construction used in each prior can be exploited to develop a general computational procedure for fitting them in this article we present two general types of gibbs samplers that can be used to fit posteriors of bayesian hierarchical models based on stick breaking priors the first type of gibbs sampler referred to as a polya urn gibbs sampler is a generalized version of a widely used gibbs sampling method currently employed for dirichlet process computing this method applies t

finite element solution of the helmholtz equation with high wave number part i the h version of the fem abstract the paper addresses the properties of finite element solutions for the helmholtz equation the h version of the finite element method with piecewise linear approximation is applied to a one dimensional model problem new results are shown on stability and error estimation of the discrete model in all propositions assumptions are made on the magnitude of hk only where k is the wavelength and h is the stepwidth of the fe mesh previous analytical results had been shown with the assumption that k 2 h is small for medium and high wavenumber these results do not cover the meshsizes that are applied in practical applications the main estimate reveals that the error in h 1 norm of discrete solutions for the helmholtz equation is polluted when k 2 h is not small the error is then not quasioptimal i e the relation of the fe error to the error of best approximation generally depends on the wavenumber k it is noted that the pollution term in the relative error is of the same order as the phase lead of the numerical solution in the result of this analysis thorough and rigorous understanding of error behavior throughout the range of convergence is gained numerical results are presented that show sharpness of the error estimates and highlight some phenomena of the discrete solution behavior the h p version of the fem is studied in part ii

bayesian computation via the gibbs sampler and related markov chain monte carlo methods the use of the gibbs sampler for bayesian computation is reviewed and illustrated in the context of some canonical examples other markov chain monte carlo simulation methods are also briefly described and comments are made on the advantages of sample based approaches for bayesian inference summaries

a first order system least squares method for the helmholtz equation we present a first order system least squares fosls method for the helmholtz equation at high wave number k which always leads to a hermitian positive definite algebraic system by utilizing a non trivial solution decomposition to the dual fosls problem which is quite different from that of the standard finite element methods we give an error analysis to the h p version of the fosls method where the dependence on the mesh size h the approximation order p and the wave number k is given explicitly in particular under some assumption of the boundary of the domain the l 2 norm error estimate of the scalar solution from the fosls method is shown to be quasi optimal under the condition that k h p is sufficiently small and the polynomial degree p is at least o log k numerical experiments are given to verify the theoretical results

latent space approaches to social network analysis network models are widely used to represent relational information among interacting units in studies of social networks recent emphasis has been placed on random graph models where the nodes usually represent individual social actors and the edges represent the presence of a specified relation between actors we develop a class of models where the probability of a relation between actors depends on the positions of individuals in an unobserved social space we make inference for the social space within maximum likelihood and bayesian frameworks and propose markov chain monte carlo procedures for making inference on latent positions and the effects of observed covariates we present analyses of three standard datasets from the social networks literature and compare the method to an alternative stochastic blockmodeling approach in addition to improving on model fit for these datasets our method provides a visual and interpretable model based spatial representation of social relationships and improv

markov chains for exploring posterior distributions several markov chain methods are available for sampling from a posterior distribution two important examples are the gibbs sampler and the metropolis algorithm in addition several strategies are available for constructing hybrid algorithms this paper outlines some of the basic methods and strategies and discusses some related theoretical and practical issues on the theoretical side results from the theory of general state space markov chains can be used to obtain convergence rates laws of large numbers and central limit theorems for estimates obtained from markov chain methods these theoretical results can be used to guide the construction of more efficient algorithms for the practical use of markov chain methods standard simulation methodology provides several variance reduction techniques and also give guidance on the choice of sample size and allocation

discontinuous galerkin methods for the helmholtz equation with large wave number this paper develops and analyzes some interior penalty discontinuous galerkin ipdg methods using piecewise linear polynomials for the helmholtz equation with the first order absorbing boundary condition in two and three dimensions it is proved that the proposed discontinuous galerkin methods are stable hence well posed without any mesh constraint for each fixed wave number k optimal order with respect to h error estimate in the broken h 1 norm and suboptimal order estimate in the l 2 norm are derived without any mesh constraint the latter estimate improves to optimal order when the mesh size h is restricted to the preasymptotic regime i e k 2 h gtrsim 1 numerical experiments are also presented to gauge the theoretical result and to numerically examine the pollution effect with respect to k in the error bounds the novelties of the proposed ipdg methods include the following first the methods penalize not only the jumps of the function values across the element edges but also the jumps of the normal and tangential derivatives second the penalty parameters are taken as complex numbers of positive imaginary parts so essentially and practically no constraint is imposed on the penalty parameters since the helmholtz problem is a non hermitian and indefinite linear problem as expected the crucial and the most difficult part of the whole analysis is to establish the stability estimates i e a priori estimates for the numerical solutions to this end the cruxes of our analysis are to establish and to make use of a local version of the rellich identity for the laplacian and to mimic the stability analysis for the pde solutions given in p cummings analysis of finite element based numerical methods for acoustic waves elastic waves and fluid solid interactions in the frequency domain ph d thesis the university of tennessee knoxville tn 2001 p cummings and x feng math models methods appl sci 16 2006 pp 139 160 u hetmaniuk commun math sci 5 2007 pp 665 678

stabilized galerkin approximation of convection diffusion reaction equations discrete maximum principle and convergence we analyze a nonlinear shock capturing scheme for h1 conforming piecewise affine finite element approximations of linear elliptic problems the meshes are assumed to satisfy two standard conditions a local quasi uniformity property and the xu zikatanov condition ensuring that the stiffness matrix associated with the poisson equation is an m matrix a discrete maximum principle is rigorously established in any space dimension for convection diffusion reaction problems we prove that the shock capturing finite element solution converges to that without shock capturing if the cell peclet numbers are sufficiently small moreover in the diffusion dominated regime the difference between the two finite element solutions super converges with respect to the actual approximation error numerical experiments on test problems with stiff layers confirm the sharpness of the a priori error estimates

sweeping preconditioner for the helmholtz equation hierarchical matrix representation the paper introduces the sweeping preconditioner which is highly efficient for iterative solutions of the variable coefficient helmholtz equation including very high frequency problems the first central idea of this novel approach is to construct an approximate factorization of the discretized helmholtz equation by sweeping the domain layer by layer starting from an absorbing layer or boundary condition given this specific order of factorization the second central idea is to represent the intermediate matrices in the hierarchical matrix framework in two dimensions both the construction and the application of the preconditioners are of linear complexity the generalized minimal residual method gmres solver with the resulting preconditioner converges in an amazingly small number of iterations which is essentially independent of the number of unknowns this approach is also extended to the three dimensional case with some success numerical results are provided in both two and three dimensions to demonstrate the efficiency of this new approach

a unified analysis for conforming and nonconforming stabilized finite element methods using interior penalty we discuss stabilized galerkin approximations in a new framework widening the scope from the usual dichotomy of the discontinuous galerkin method on the one hand and petrov galerkin methods such as the supg method on the other the idea is to use interior penalty terms as a means of stabilizing the finite element method using conforming or nonconforming approximation thus circumventing the need of a petrov galerkin type choice of spaces this is made possible by adding a higher order penalty term giving l2 control of the jumps in the gradients between adjacent elements we consider convection diffusion reaction problems using piecewise linear approximations and prove optimal order a priori error estimates for two different finite element spaces the standard h1 conforming space of piecewise linears and the nonconforming space of piecewise linear elements where the nodes are situated at the midpoint of the element sides the crouzeix raviart element moreover we show how the formulation extends to discontinuous galerkin interior penalty methods in a natural way by domain decomposition using nitsche s method

the nested dirichlet process in multicenter studies subjects in different centers may have different outcome distributions this article is motivated by the problem of nonparametric modeling of these distributions borrowing information across centers while also allowing centers to be clustered starting with a stick breaking representation of the dirichlet process dp we replace the random atoms with random probability measures drawn from a dp this results in a nested dp prior which can be placed on the collection of distributions for the different centers with centers drawn from the same dp component automatically clustered together theoretical properties are discussed and an efficient markov chain monte carlo algorithm is developed for computation the methods are illustrated using a simulation study and an application to quality of care in u s hospitals

wavenumber explicit convergence analysis for galerkin discretizations of the helmholtz equation we develop a stability and convergence theory for a class of highly indefinite elliptic boundary value problems bvps by considering the helmholtz equation at high wavenumber k as our model problem the key element in this theory is a novel k explicit regularity theory for helmholtz bvps that is based on decomposing the solution into two parts the first part has the sobolev regularity properties expected of second order elliptic pdes but features k independent regularity constants the second part is an analytic function for which k explicit bounds for all derivatives are given this decomposition is worked out in detail for several types of bvps namely the helmholtz equation in bounded smooth domains or convex polygonal domains with robin boundary conditions and in exterior domains with dirichlet boundary conditions we present an error analysis for the classical hp version of the finite element method hp fem where the dependence on the mesh width h the approximation order p and the wavenumber k is given explicitly in particular under the assumption that the solution operator for helmholtz problems is polynomially bounded in k it is shown that quasi optimality is obtained under the conditions that kh p is sufficiently small and the polynomial degree p is at least o log k

a semi parametric bayesian approach to generalized linear mixed models summary the linear mixed e ects model with normal errors is a popular model for the analysis of repeated measures and longitudinal data the generalized linear model is useful for data that have non normal errors but where the errors are uncorrelated a descendant of these two models generates a model for correlated data with non normal errors called the generalized linear mixed model glmm frequentist attempts to fit these models generally rely on approximate results and inference relies on asymptotic assumptions recent advances in computing technology have made bayesian approaches to this class of models computationally feasible markov chain monte carlo methods can be used to obtain exact inference for these models as demonstrated by zeger and karim 6 in the linear or generalized linear mixed model the random e ects are typically taken to have a fully parametric distribution such as the normal distribution in this paper we extend the glmm by allowing the random e ects to have a non parametric prior distribution we do this using a dirichlet process prior for the general distribution of the random e ects the approach easily extends to more general population models we perform computations for the models using the gibbs sampler 1998 john wiley sons ltd

a class of discontinuous petrov galerkin methods part iv the optimal test norm and time harmonic wave propagation in 1d the phase error or the pollution effect in the finite element solution of wave propagation problems is a well known phenomenon that must be confronted when solving problems in the high frequency range this paper presents a new method with no phase errors for one dimensional 1d time harmonic wave propagation problems using new ideas that hold promise for the multidimensional case the method is constructed within the framework of the discontinuous petrov galerkin dpg method with optimal test functions we have previously shown that such methods select solutions that are the best possible approximations in an energy norm dual to any selected test space norm in this paper we advance by asking what is the optimal test space norm that achieves error reduction in a given energy norm this is answered in the specific case of the helmholtz equation with l 2 norm as the energy norm we obtain uniform stability with respect to the wave number we illustrate the method with a number of 1d numerical experiments using discontinuous piecewise polynomial hp spaces for the trial space and its corresponding optimal test functions computed approximately and locally a 1d theoretical stability analysis is also developed

finite element solution of the helmholtz equation with high wave number part ii the h p version of the fem in this paper which is part ii in a series of two the investigation of the galerkin finite element solution to the helmholtz equation is continued while part i contained results on the h version with piecewise linear approximation the present part deals with approximation spaces of order p ge 1 as in part i the results are presented on a one dimensional model problem with dirichlet robin boundary conditions in particular there are proven stability estimates both with respect to data of higher regularity and data that is bounded in lower norms the estimates are shown both for the continuous and the discrete spaces under consideration further there is proven a result on the phase difference between the exact and the galerkin finite element solutions for arbitrary p that had been previously conjectured from numerical experiments these results and further preparatory statements are then employed to show error estimates for the galerkin finite element method fem it becomes evident that the error estimate for higher approximation can with certain assumptions on the data be written in the same form as the piecewise linear case namely as the sum of the error of best approximation plus a pollution term that is of the order of the phase difference the paper is concluded with a numerical evaluation

sampling based approaches to calculating marginal densities abstract stochastic substitution the gibbs sampler and the sampling importance resampling algorithm can be viewed as three alternative sampling or monte carlo based approaches to the calculation of numerical estimates of marginal probability distributions the three approaches will be reviewed compared and contrasted in relation to various joint probability structures frequently encountered in applications in particular the relevance of the approaches to calculating bayesian posterior densities for a variety of structured models will be discussed and illustrated

assessing a mixture model for clustering with the integrated completed likelihood we propose an assessing method of mixture model in a cluster analysis setting with integrated completed likelihood for this purpose the observed data are assigned to unknown clusters using a maximum a posteriori operator then the integrated completed likelihood icl is approximated using the bayesian information criterion bic numerical experiments on simulated and real data of the resulting icl criterion show that it performs well both for choosing a mixture model and a relevant number of clusters in particular icl appears to be more robust than bic to violation of some of the mixture model assumptions and it can select a number of dusters leading to a sensible partitioning of the data

pre asymptotic error analysis of h p interior penalty discontinuous galerkin methods for the helmholtz equation with large wave number in this paper we shall consider to improve the pre asymptotic stability and error estimates of some h p interior penalty discontinuous galerkin h p ipdg methods for the helmholtz equation with the first order absorbing boundary condition in two and three dimensions given in feng and wu 2011 the proposed h p ipdg methods are defined using a sesquilinear form which is not only mesh dependent or h dependent but also degree dependent or p dependent by using a modified duality argument given in zhu and wu 2013 pre asymptotic error estimates are improved for the proposed h p ipdg methods under the condition of k h p c 0 p k 1 p 1 in this paper where c 0 is some constant independent of k h p and the penalty parameters it is shown that the pollution error of the method in the broken h 1 norm is o k 2 p 1 h 2 p if p o 1 which coincides with existent dispersion analyses for the dg method on cartesian grids numerical tests are provided to verify the theoretical findings and to illustrate great capability of the ipdg method in reducing the pollution effect

a discontinuous finite element formulation for helmholtz equation abstract a discontinuous finite element formulation for helmholtz problems is presented with c 0 continuity across inter element boundaries enforced in a weak sense depending on two parameters and whose choice is crucial for the performance of the proposed method these parameters are chosen through one dimension numerical experiments aiming at minimizing the pollution error optimal values of these parameters are then adopted in more general situations the accuracy and stability of the proposed formulation for linear and bilinear shape functions are demonstrated in several numerical examples in one and two dimensions

an interproduct competition model incorporating branding hierarchy and product similarities using store level data we develop and implement a bayesian semiparametric model of demand under interproduct competition that enables us to assess the respective contributions of brand sku stock keeping unit hierarchy and interproduct similarity to explaining and predicting demand to incorporate brand sku hierarchy effects we use bayesian hierarchical clustering inherent in a nested dirichlet process to simultaneously partition brands and skus conditional on brands into groups of similarity clusters we examine cluster memberships and postprocess the markov chain monte carlo output to infer cluster properties by accounting for parameter uncertainty our proposed approach lends to a spatial competition interpretation in latent attribute space and helps uncover the extent to which competition across skus in the latent attribute space is local or global in a related vein we discuss the implications of well defined groups of similar skus as subcategory or submarket boundaries in latent attribute space we empirically test our model using aggregate beer category sales data from a midsize u s retail chain we find that branding hierarchy effects dominate those from product similarity we find that the model partitions the 15 brands in the data into 4 brand clusters and the 96 skus into 25 sku clusters conditional on brand cluster membership in estimating a set of models of spatial interproduct competition we find that sku competition is more local than global in that only subsets of products compete within groups of comparable products finally we discuss the substantive implications of our results this paper was accepted by pradeep chintagunta marketing

convergence analysis for finite element discretizations of the helmholtz equation with dirichlet to neumann boundary conditions a rigorous convergence theory for galerkin methods for a model helmholtz problem in bbb r d d in 1 2 3 is presented general conditions on the approximation properties of the approximation space are stated that ensure quasi optimality of the method as an application of the general theory a full error analysis of the classical hp version of the finite element method is presented for the model problem where the dependence on the mesh width h the approximation order p and the wave number k is given explicitly in particular it is shown that quasi optimality is obtained under the conditions that kh p is sufficiently small and the polynomial degree p is at least o log k

stochastic relaxation gibbs distributions and the bayesian restoration of images we make an analogy between images and statistical mechanics systems pixel gray levels and the presence and orientation of edges are viewed as states of atoms or molecules in a lattice like physical system the assignment of an energy function in the physical system determines its gibbs distribution because of the gibbs distribution markov random field mrf equivalence this assignment also determines an mrf image model the energy function is a more convenient and natural mechanism for embodying picture attributes than are the local characteristics of the mrf for a range of degradation mechanisms including blurring nonlinear deformations and multiplicative or additive noise the posterior distribution is an mrf with a structure akin to the image model by the analogy the posterior distribution defines another imaginary physical system gradual temperature reduction in the physical system isolates low energy states annealing or what is the same thing the most probable states under the gibbs distribution the analogous operation under the posterior distribution yields the maximum a posteriori map estimate of the image given the degraded observations the result is a highly parallel relaxation algorithm for map estimation we establish convergence properties of the algorithm and we experiment with some simple pictures for which good restorations are obtained at low signal to noise ratios

edge stabilization for galerkin approximations of convection diffusion reaction problems n this paper we recall a stabilization technique for finite element methods for convection diffusion reaction equations originally proposed by j douglas jr and t dupont in computing methods in applied sciences second internat sympos versailles 1975 207 216 lecture notes in phys 58 springer berlin 1976 mr0440955 55 13823 the method uses least square stabilization of the gradient jumps across element boundaries we prove that the method is stable in the hyperbolic limit and prove optimal a priori error estimates we address the question of monotonicity of discrete solutions and present some numerical examples illustrating the theoretical results

ipcdgm and multiscale ipdpgm for the helmholtz problem with large wave number abstract in this paper based on the two nested spaces of piecewise linear polynomials on fine and coarse meshes with sizes h and h respectively which are continuous in macro elements but discontinuous across interior edges faces of them an interior penalty continuous discontinuous galerkin method ipcdgm and a multiscale interior penalty discontinuous petrov galerkin method msipdpgm are proposed to solve the helmholtz problem with large wave number k and homogeneous robin boundary condition this paper devotes to analyzing the preasymptotic error of the two methods separately in order to reduce the pollution errors efficiently the two methods not only include the penalty terms on jumps of function on coarse mesh edges faces but also add the penalty terms on jumps of first order normal derivatives on fine mesh edges faces the error of the ipcdg solution in the broken h 1 norm is bounded by o k h k 3 h 2 by splitting into coarse and fine scales and basing on the ipcdg variational formulation the msipdpgm s trial and test spaces are constructed with the macro corrected bases due to the exponential decay of the correctors the corrected bases are obtained by solving the local problems on localized subdomains of size l h l being the oversampling parameter the preasymptotic error analysis of msipdpgm shows that if k h and log k l fall below some constants and if the fine mesh size h is sufficient small the errors of numerical solutions in the broken h 1 norm can be dominated by o h 3 without pollution effect numerical tests are provided to verify the theoretical findings and advantages of the two methods

absolutely stable local discontinuous galerkin methods for the helmholtz equation with large wave number two local discontinuous galerkin ldg methods using some non standard numerical uxes are developed for the helmholtz equation with the rst order absorbing boundary condition in the high frequency regime it is shown that the proposed ldg methods are absolutely stable hence well posed with respect to both the wave number and the mesh size optimal order with respect to the mesh size error estimates are proved for all wave numbers in the preasymptotic regime to analyze the proposed ldg methods they are recasted and treated as non conforming mixed nite element methods the crux of the analysis is to establish a generalized inf sup condition which holds without any mesh constraint for each ldg method the generalized inf sup conditions then easily infer the desired absolute stability of the proposed ldg methods in return the stability results not only guarantee the well posedness of the ldg methods but also play a crucial role in the derivation of the error estimates numerical experiments which conrm the theoretical results and compare the proposed two ldg methods are also presented in the paper

bayesian analysis of mixtures of factor analyzers for bayesian inference on the mixture of factor analyzers natural conjugate priors on the parameters are introduced and then a gibbs sampler that generates parameter samples following the posterior is constructed in addition a deterministic estimation algorithm is derived by taking modes instead of samples from the conditional posteriors used in the gibbs sampler this is regarded as a maximum a posteriori estimation algorithm with hyperparameter search the behaviors of the gibbs sampler and the deterministic algorithm are compared on a simulation experiment

hierarchical dirichlet processes we consider problems involving groups of data where each observation within a group is a draw from a mixture model and where it is desirable to share mixture components between groups we assume that the number of mixture components is unknown a priori and is to be inferred from the data in this setting it is natural to consider sets of dirichlet processes one for each group where the well known clustering property of the dirichlet process provides a nonparametric prior for the number of mixture components within each group given our desire to tie the mixture models in the various groups we consider a hierarchical model specifically one in which the base measure for the child dirichlet processes is itself distributed according to a dirichlet process such a base measure being discrete the child dirichlet processes necessarily share atoms thus as desired the mixture models in the different groups necessarily share mixture components we discuss representations of hierarchical dirichlet processes

a robust multilevel method for hybridizable discontinuous galerkin method for the helmholtz equation a robust multilevel preconditioner based on the hybridizable discontinuous galerkin method for the helmholtz equation with high wave number is presented in this paper there are two keys in our algorithm one is how to choose a suitable intergrid transfer operator and the other is using gmres smoothing on the coarse grids the multilevel method is performed as a preconditioner in the outer gmres iteration to give a quantitative insight of our algorithm we use local fourier analysis to analyze the convergence property of the proposed multilevel method numerical results show that for fixed wave number the convergence of the algorithm is mesh independent moreover the performance of the algorithm depends relatively mildly on wave number

a generalized finite element method for solving the helmholtz equation in two dimensions with minimal pollution when using the galerkin fem for solving the helmholtz equation in two dimensions the error of the corresponding solution differs substantially from the error of the best approximation and this effect increases with higher wave number k in this paper we will design a generalized finite element method gfem for the helmholtz equation such that the pollution effect is minimal

staggered discontinuous galerkin methods for the helmholtz equation with large wave number abstract in this paper we investigate staggered discontinuous galerkin method for the helmholtz equation with large wave number on general polygonal meshes the method is highly flexible by allowing rough grids such as the trapezoidal grids and highly distorted grids and at the same time is numerical flux free furthermore it allows hanging nodes which can be simply treated as additional vertices by exploiting a modified duality argument the stability and convergence can be proved under the condition that h c 0 1 1 m 1 where m is the polynomial order is the wave number h is the mesh size and c 0 is a positive constant independent of h error estimates for both the scalar and vector variables in l 2 norm are established several numerical experiments are tested to verify our theoretical results and to present the capability of our method for capturing singular solutions

application of an ultra weak variational formulation of elliptic pdes to the two dimensional helmholtz problem a new technique to solve elliptic linear pdes called ultra weak variational formulation uwvf in this paper is introduced in b despres c r acad sci paris 318 1994 pp 939 944 this paper is devoted to an evaluation of the potentialities of this technique it is applied to a model wave problem the two dimensional helmholtz problem the new method is presented in three parts following the same style of presentation as the classical one of the finite elements method even though they are definitely conceptually different methods the first part is committed to the variational formulation and to the continuous problem the second part defines the discretization process using a galerkin procedure the third part actually studies the efficiency of the technique from the order of convergence point of view this is achieved using theoretical proofs and a series of numerical experiments in particular it is proven and shown the order of convergence is lower bounded by a linear function of the number of degrees of freedom an application to scattering problems is presented in a fourth part

continuous interior penalty finite element method for oseen s equations in this paper we present an extension of the continuous interior penalty method of douglas and dupont interior penalty procedures for elliptic and parabolic galerkin methods in computing methods in applied sciences lecture notes in phys 58 springer verlag berlin 1976 pp 207 216 to oseen s equations the method consists of a stabilized galerkin formulation using equal order interpolation for pressure and velocity to counter instabilities due to the pressure velocity coupling or due to a high local reynolds number we add a stabilization term giving l2 control of the jump of the gradient over element faces edges in two dimensions to the standard galerkin formulation boundary conditions are imposed in a weak sense using a consistent penalty formulation due to nitsche we prove energy type a priori error estimates independent of the local reynolds number and give some numerical examples recovering the theoretical results

preasymptotic error analysis of cip fem and fem for helmholtz equation with high wave number part ii hp version in this paper which is the second in a series of two the preasymptotic error analysis of the continuous interior penalty finite element method cip fem and the fem for the helmholtz equation in two and three dimensions is continued while part i contained results on the linear cip fem and fem the present part deals with approximation spaces of order p ge 1 by using a modified duality argument preasymptotic error estimates are derived for both methods under the condition of frac kh p le c 0 frac p k frac 1 p 1 where k is the wave number h is the mesh size and c 0 is a constant independent of k h p and the penalty parameters it is shown that the pollution errors of both methods in h 1 norm are o k 2p 1 h 2p if p o 1 and are o frac k p 2 frac kh sigma p 2p if the exact solution u in h 2 omega which coincide with existent dispersion analyses for the fem on cartesian grids here sigma is a constant independent of k h p and the penalty par

accurate finite difference methods for time harmonic wave propagation finite difference methods for solving problems of time harmonic acoustics are developed and analyzed multi dimensional inhomogeneous problems with variable possibly discontinuous coefficients are considered accounting for the effects of employing non uniform grids a weighted average representation is less sensitive to transition in wave resolution due to variable wave numbers or non uniform grids than the standard pointwise representation further enhancement in method performance is obtained by basing the stencils on generalizations of pade approximation or generalized definitions of the derivative reducing spurious dispersion anisotropy and reflection and by improving the representation of source terms the resulting schemes have fourth order accurate local truncation error on uniform grids and third order in the non uniform case guidelines for discretization pertaining to grid orientation and resolution are presented

an asymptotic universal focal decomposition for non isochronous potentials galileo in the xvii century observed that the small os cillations of a pendulum seem to have constant period in fact the taylor expansion of the period map of the pendulum is constant up to second order in the initial angular velocity around the stable equi librium it is well known that for small oscillations of the pendulum and small intervals of time the dynamics of the pendulum can be ap proximated by the dynamics of the harmonic oscillator we study the dynamics of a family of mechanical systems that includes the pendulum at small neighbourhoods of the equilibrium but after long intervals of time so that the second order term of the period map can no longer be neglected we analyze such dynamical behaviour through a renor malization scheme acting on the dynamics of this family of mechanical systems the main theorem states that the asymptotic limit of this renormalization scheme is universal it is the same for all the elements in the considered class of mechanical systems as a consequence we obtain an universal asymptotic focal decomposition for this family of mechanical systems this paper is intended to be the first of a series of articles aiming at a semiclassical quantization of systems of the pendu lum type as a natural application of the focal decomposition associated to the two point boundary value problem

a bayesian approach to finite mixture models in bioassay via data augmentation and gibbs sampling and its application to insecticide resistance after continued treatment with an insecticide within the population of the susceptible insects resistant strains will occur it is important to know whether there are any resistant strains what the proportions are and what the median lethal doses are for the insecticide lwin and martin 1989 biometrics 45 721 732 propose a probit mixture model and use the em algorithm to obtain the maximum likelihood estimates for the parameters this approach has difficulties in estimating the confidence intervals and in testing the number of components we propose a bayesian approach to obtaining the credible intervals for the location and scale of the tolerances in each component and for the mixture proportions by using data augmentation and gibbs sampler we use bayes factor for model selection and determining the number of components we illustrate the method with data published in lwin and martin 1989

markov chain monte carlo estimation of classical and dynamic switching and mixture models bayesian estimation of a very general model class where the distribution of the observations depends on a latent process taking values in a discrete state space is discussed in this article this model class covers finite mixture modeling markov switching autoregressive modeling and dynamic linear models with switching the consequences the unidentifiability of this type of model has on markov chain monte carlo mcmc estimation are explicitly dealt with joint bayesian estimation of all latent variables model parameters and parameters that determine the probability law of the latent process is carried out by a new mcmc method called permutation sampling the permutation sampler first samples from the unconstrained posterior which often can be done in a convenient multimove manner and then applies a permutation of the current labeling of the states of the latent process in a first run the random permutation sampler used selected the permutation randomly the mcmc output of the random permutation s

analysis of a spectral galerkin approximation to the helmholtz equation in exterior domains an error analysis is presented for the spectral galerkin method to the helmholtz equation in 2 and 3 dimensional exterior domains the problem in unbounded domains is first reduced to a problem on a bounded domain via the dirichlet to neumann operator and then a spectral galerkin method is employed to approximate the reduced problem the error analysis is based on exploring delicate asymptotic behaviors of the hankel functions and on deriving a priori estimates with explicit dependence on the wave number for both the continuous and the discrete problems explicit error bounds with respect to the wave number are derived and some illustrative numerical examples are also presented

bayesian density estimation and inference using mixtures abstract we describe and illustrate bayesian inference in models for density estimation using mixtures of dirichlet processes these models provide natural settings for density estimation and are exemplified by special cases where data are modeled as a sample from mixtures of normal distributions efficient simulation methods are used to approximate various prior posterior and predictive distributions this allows for direct inference on a variety of practical issues including problems of local versus global smoothing uncertainty about density estimates assessment of modality and the inference on the numbers of components also convergence results are established for a general class of normal mixture models

convergence analysis of an adaptive continuous interior penalty finite element method for the helmholtz equation abstract we are concerned with an adaptive continuous interior penalty finite element method for the helmholtz equation a convergence result with explicit constraints on the wave number k and the initial mesh size h 0 is derived in particular it is shown that if k 1 s h 0 s is sufficiently small where 1 2 s 1 depends on the regularity of the weak solution of the model problem then the adaptive continuous interior penalty finite element method is a contraction with respect to the sum of the energy error and a scaled error estimator numerical experiments are provided to verify the theoretical findings and show advantages of the adaptive continuous interior penalty finite element method

computational aspects of the ultra weak variational formulation the ultra weak variational formulation uwvf approach has been proposed as an effective method for solving helmholtz problems with high wave numbers however for coarse meshes the method can suffer from instability in this paper we consider computational aspects of the ultra weak variational formulation for the inhomogeneous helmholtz problem we introduce a method to improve the uwvf scheme and we compare iterative solvers for the resulting linear system computations for the acoustic transmission problem in 2d show that the new approach enables helmholtz problems to be solved on a relatively coarse mesh for a wide range of wave numbers

practical bayesian density estimation using mixtures of normals abstract mixtures of normals provide a flexible model for estimating densities in a bayesian framework there are some difficulties with this model however first standard reference priors yield improper posteriors second the posterior for the number of components in the mixture is not well defined if the reference prior is used third posterior simulation does not provide a direct estimate of the posterior for the number of components we present some practical methods for coping with these problems finally we give some results on the consistency of the method when the maximum number of components is allowed to grow with the sample size

on accuracy conditions for the numerical computation of waves abstract the helmholtz equation k 2 n 2 u f with a variable index of refraction n and a suitable radiation condition at infinity serves as a model for a wide variety of wave propagation problems such problems can be solved numerically by first truncating the given unbounded domain imposing a suitable outgoing radiation condition on an artificial boundary and then solving the resulting problem on the bounded domain by direct discretization for example using a finite element method in practical applications the mesh size h and the wave number k are not independent but are constrained by the accuracy of the desired computation it will be shown that the number of points per wavelength measured by kh 1 is not sufficient to determine the accuracy of a given discretization for example the quantity k 3 h 2 is shown to determine the accuracy in the l 2 norm for a second order discretization method applied to several propagation models

continuous interior penalty hp finite element methods for advection and advection diffusion equations a continuous interior penalty hp finite element method that penalizes the jump of the discrete solution across mesh interfaces is introduced error estimates are obtained for first order and advection dominated transport operators the analysis relies on three technical results that are of independent interest an hp inverse trace inequality a local discontinuous to continuous hp interpolation result and hp error estimates for continuous l2 orthogonal projections

the discontinuous enrichment method we propose a finite element based discretization method in which the standard polynomial field is enriched within each element by a nonconforming field that is added to it the enrichment contains free space solutions of the homogeneous differential equation that are not represented by the underlying polynomial field continuity of the enrichment across element interfaces is enforced weakly by lagrange multipliers in this manner we expect to attain high coarse mesh accuracy without significant degradation of conditioning assuring good performance of the computation at any mesh resolution examples of application to acoustics and advection diffusion are presented

finite element method and its analysis for a nonlinear helmholtz equation with high wave numbers the well posedness of a nonlinear helmholtz equation with an impedance boundary condition is established for high frequencies in two and three dimensions stability estimates are derived with expli

bayesian analysis of cross section and clustered data treatment models this paper is concerned with the problem of determining the effect of a categorical treatment variable on a response given that the treatment is non randomly assigned and the response on any given subject is observed for one setting of the treatment we consider classes of models that are designed for such problems these models are subjected to a fully bayesian analysis based on markov chain monte carlo methods the analysis of the treatment effect is then based on amongst other things the posterior distribution of the potential outcomes counter factuals at the subject level which is obtained as a by product of the mcmc simulation procedure the analysis is extended to models with categorical treatments and binary and clustered outcomes the problem of model comparisons is also considered different aspects of the methodology are illustrated through two data examples

how large a shift is needed in the shifted helmholtz preconditioner for its effective inversion by multigrid the shifted helmholtz operator has received a lot of attention over the past decade as a preconditioner for the iterative solution of the helmholtz equation the idea is that if one uses a small complex shift the shifted helmholtz operator is still close to the original helmholtz operator and could thus be an effective preconditioner it was shown in m j gander i g graham and e a spence numer math 53 2015 pp 573 579 that the shift can be at most o k to prove rigorously wave number independent convergence of the preconditioned system solved with gmres provided the preconditioner is inverted exactly in practice however the preconditioner is inverted only approximately and if one shifts enough this can be done effectively by standard multigrid methods we show in this paper that for a finite element discretization the shift has to be at least o k 2 to be able to invert the shifted helmholtz preconditioner using multigrid there is therefore a gap between being a good precondi

a dispersion minimizing finite difference scheme for the helmholtz equation based on point weighting abstract in this paper we develop a new dispersion minimizing finite difference scheme for the helmholtz equation with perfectly matched layer pml in two dimensional domain which is a second order 9 point scheme to discretize the second derivative operator we employ a linear combination of a point and its neighboring grid points to replace each of the five points in the traditional central difference scheme based on minimizing the numerical dispersion the combination weights are determined by minimizing the numerical dispersion with a flexible selection strategy the new scheme is simple rotation free and pointwise consistent with the equation which is different from the classical rotated 9 point difference scheme obtained by combining the cartesian coordinate system and the rotated system moreover it is a robust scheme even if the step sizes of different directions are not equal convergence analysis and dispersion analysis are given several numerical examples are presented to illustrate the numerical convergence and effectiveness of the new scheme

preasymptotic error analysis of higher order fem and cip fem for helmholtz equation with high wave number a preasymptotic error analysis of the finite element method fem and some continuous interior penalty finite element method cip fem for the helmholtz equation in two and three dimensions is proposed h 1 and l 2 error estimates with explicit dependence on the wave number k are derived in particular it is shown that if k 2p 1 h 2p is sufficiently small then the pollution errors of both methods in h 1 norm are bounded by o k 2p 1 h 2p which coincides with the phase error of the fem obtained by existent dispersion analyses on cartesian grids where h is the mesh size and p is the order of the approximation space and is fixed the cip fem extends the classical one by adding more penalty terms on jumps of higher up to p th order normal derivatives in order to reduce efficiently the pollution errors of higher order methods numerical tests are provided to verify the theoretical findings and to illustrate the great capability of the cip fem in reducing the pollution effect

estimation of finite mixture distributions through bayesian sampling summary a formal bayesian analysis of a mixture model usually leads to intractable calculations since the posterior distribution takes into account all the partitions of the sample we present approximation methods which evaluate the posterior distribution and bayes estimators by gibbs sampling relying on the missing data structure of the mixture model the data augmentation method is shown to converge geometrically since a duality principle transfers properties from the discrete missing data chain to the parameters the fully conditional gibbs alternative is shown to be ergodic and geometric convergence is established in the normal case we also consider non informative approximations associated with improper priors assuming that the sample corresponds exactly to a k component mixture

spatial statistics and bayesian computation on wednesday may 6th 1992 professor b w silverman in the chair summary markov chain monte carlo mcmc algorithms such as the gibbs sampler have provided a bayesian inference machine in image analysis and in other areas of spatial statistics for several years founded on the pioneering ideas of ulf grenander more recently the observation that hyperparameters can be included as part of the updating schedule and the fact that almost any multivariate distribution is equivalently a markov random field has opened the way to the use of mcmc in general bayesian computation in this paper we trace the early development of mcmc in bayesian inference review some recent computational progress in statistical physics based on the introduction of auxiliary variables and discuss its current and future relevance in bayesian applications we briefly describe a simple mcmc implementation for the bayesian analysis of agricultural field experiments with which we have some practical experience

superconvergence analysis of linear fem based on polynomial preserving recovery for helmholtz equation with high wave number abstract although superconvergence properties of the polynomial preserving recovery ppr for finite element methods fem for coercive elliptic problems are well understood the superconvergence behavior of ppr for the helmholtz equation with high wave number still remains unclear we study the superconvergence property of the linear fem with ppr for the two dimensional helmholtz equation on triangulations satisfying the o h 1 approximate parallelogram property for some constant 0 we prove the supercloseness on the h 1 seminorm between the finite element solution and the linear interpolant of the exact solution under k k h 2 c 0 where k is the wave number and h is the mesh size then we obtain the superconvergence result based on the ppr technique which says that the ppr improves only the interpolation error but keeps the pollution error unchanged finally we provide numerical tests to verify the superconvergence property and to demonstrate that the ppr combining with the continuous interior penalty technique is much effective for improving both the interpolation error and the pollution error

the calculation of posterior distributions by data augmentation abstract the idea of data augmentation arises naturally in missing value problems as exemplified by the standard ways of filling in missing cells in balanced two way tables thus data augmentation refers to a scheme of augmenting the observed data so as to make it more easy to analyze this device is used to great advantage by the em algorithm dempster laird and rubin 1977 in solving maximum likelihood problems in situations when the likelihood cannot be approximated closely by the normal likelihood maximum likelihood estimates and the associated standard errors cannot be relied upon to make valid inferential statements from the bayesian point of view one must now calculate the posterior distribution of parameters of interest if data augmentation can be used in the calculation of the maximum likelihood estimate then in the same cases one ought to be able to use it in the computation of the posterior distribution it is the purpose of this article to explain how this can be done the basic idea