bayesian analysis of dichotomous quantal response models abstract qualitative response models qrm s are analyzed from the bayesian point of view using diffuse and informative prior distributions exact finite sample bayesian and large sample bayesian and non bayesian estimation results are compared in addition the paper provides 1 plots and discussion of the properties of likelihood functions for qrm s 2 posterior distributions for logit models derivatives and elasticities 3 bayesian prediction procedures for qrm s 4 new estimates for the median and other fractiles of the logistic distribution 5 posterior odds ratios for model selection problems and 6 comparisons of two alternative monte carlo numerical integration procedures it is concluded that asymptotic approximations are not accurate for small to moderate sized samples even when only a single input variable is used and that operational bayesian methods are available for providing both exact small sample and large sample approximate inferences for drm s

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

universality at the edge of the spectrum for unitary orthogonal and symplectic ensembles of random matrices we prove universality at the edge of the spectrum for unitary 2 orthogonal 1 and symplectic 4 ensembles of random matrices in the scaling limit for a class of weights w x e v x where v is a polynomial v x 2mx2m 2m 0 the precise statement of our results is given in theorem 1 1 and corollaries 1 2 and 1 4 below for the same class of weights a proof of universality in the bulk of the spectrum is given in 12 for the unitary ensembles and in 9 for the orthogonal and symplectic ensembles our starting point in the unitary case is 12 and for the orthogonal and symplectic cases we rely on our recent work 9 which in turn depends on the earlier work of widom 46 and tracy and widom 42 as in 9 the uniform plancherel rotach type asymptotics for the orthogonal polynomials found in 12 plays a central role the formulae in 46 express the correlation kernels for 1 4 as a sum of a christoffel darboux cd term as in the case 2 together with a correction term in the bulk scaling limit 9 the correction term is of lower order and does not contribute to the limiting form of the correlation kernel by contrast in the edge scaling limit considered here the cd term and the correction term contribute to the same order this leads to additional technical difficulties over and above 49 2006 wiley periodicals inc

universality of the stochastic airy operator we introduce a new method for studying universality of random matrices let t n be the jacobi matrix associated to the dyson beta ensemble with uniformly convex polynomial potential we show that after scaling tn converges to the stochastic airy operator in particular the top edge of the dyson beta ensemble and the corresponding eigenvectors are universal as a byproduct these ideas lead to conjectured operator limits for the entire family of soft edge distributions c 2015 wiley periodicals inc

correlation functions cluster functions and spacing distributions for random matrices the usual formulas for the correlation functions in orthogonal and symplectic matrix models express them as quaternion determinants from this representation one can deduce formulas for spacing probabilities in terms of fredholm determinants of matrix valued kernels the derivations of the various formulas are somewhat involved in this article we present a direct approach which leads immediately to scalar kernels for the unitary ensembles and matrix kernels for the orthogonal and symplectic ensembles and the representations of the correlation functions cluster functions and spacing distributions in terms of them

some universal properties for restricted trace gaussian orthogonal unitary and symplectic ensembles consider fixed and bounded trace gaussian orthogonal unitary and symplectic ensembles closely related to gaussian ensembles without any constraint for three restricted trace gaussian ensembles we prove universal limits of correlation functions at zero and at the edge of the spectrum edge our argument also applies to restricted trace ensembles with monomial potentials in addition by using the universal result in the bulk for fixed trace gaussian unitary ensemble which has been obtained by gotze and gordin we also prove the universal limits of correlation functions everywhere in the bulk for bounded trace gaussian unitary ensemble

a novel multigrid based preconditioner for heterogeneous helmholtz problems an iterative solution method in the form of a preconditioner for a krylov subspace method is presented for the helmholtz equation the preconditioner is based on a helmholtz type differential operator with a complex term a multigrid iteration is used for approximately inverting the preconditioner the choice of multigrid components for the corresponding preconditioning matrix with a complex diagonal is validated with fourier analysis multigrid analysis results are verified by numerical experiments high wavenumber helmholtz problems in heterogeneous media are solved indicating the performance of the preconditioner

a direct solver for elliptic pdes in three dimensions based on hierarchical merging of poincare steklov operators a numerical method for variable coefficient elliptic pdes on three dimensional domains is described the method is designed for problems with smooth solutions and is based on a multidomain spectral collocation discretization scheme the resulting system of linear equations can very efficiently be solved using a nested dissection style direct as opposed to iterative solver this makes the scheme particularly well suited to solving problems for which iterative solvers struggle in particular for problems with oscillatory solutions a principal feature of the scheme is that once the solution operator has been constructed the actual solve is extremely fast an upper bound on the asymptotic cost of the build stage of o n 4 3 is proved for the case where the pde is held fixed as n increases the solve stage has close to linear complexity the scheme requires a relatively large amount of storage per degree of freedom but since it is a high order scheme a small number of degrees of freedom is sufficient to achieve high accuracy the method is presented for the case where there is no body load present but it can with little difficulty be generalized to the non homogeneous case numerical experiments demonstrate that the scheme is capable of solving helmholtz type equations on a domain of size 20 20 20 wavelengths to three correct digits on a modest personal workstation with n 2 10 6

generalized christoffel darboux formula for classical skew orthogonal polynomials we show that skew orthogonal functions defined with respect to jacobi weight wa b x 1 x a 1 x b a b 1 including the limiting cases of laguerre wa x xa e x a 1 and gaussian weight satisfy three term recursion relation in the quaternion space from this we derive generalized christoffel darboux gcd formulae for kernel functions arising in the study of the corresponding orthogonal and symplectic ensembles of random 2n 2n matrices using the gcd formulae we calculate the level densities and prove that in the bulk of the spectrum under appropriate scaling the eigenvalue correlations are universal we also provide evidence to show that there exists a mapping between skew orthogonal functions arising in the study of orthogonal and symplectic ensembles of random matrices

random matrix central limit theorems for nonintersecting random walks we consider nonintersecting random walks satisfying the condition that the increments have a finite moment generating function we prove that in a certain limiting regime where the number of walks and the number of time steps grow to infinity several limiting distributions of the walks at the mid time behave as the eigenvalues of random hermitian matrices as the dimension of the matrices grows to infinity

efficient structured multifrontal factorization for general large sparse matrices rank structures provide an opportunity to develop new efficient numerical methods for practical problems when the off diagonal blocks of certain dense intermediate matrices have small numerical

universality in several matrix models via approximate transport maps we construct approximate transport maps for perturbative several matrix models as a consequence we deduce that local statistics have the same asymptotic as in the case of independent gue or goe matrices i e they are given by the sine kernel in the bulk and the tracy widom distribution at the edge and we show averaged energy universality i e universality for averages of m points correlation functions around some energy level e in the bulk as a corollary these results yield universality for self adjoint polynomials in several independent gue or goe matrices which are close to the identity

illustration of bayesian inference in normal data models using gibbs sampling abstract the use of the gibbs sampler as a method for calculating bayesian marginal posterior and predictive densities is reviewed and illustrated with a range of normal data models including variance components unordered and ordered means hierarchical growth curves and missing data in a crossover trial in all cases the approach is straightforward to specify distributionally and to implement computationally with output readily adapted for required inference summaries

efficient preconditioning of hp fem matrix sequences with slowly varying coefficients an application to topology optimization abstract we previously introduced a preconditioner that has proven effective for h p fem discretizations of various challenging elliptic and hyperbolic problems the construction is inspired by standard nested dissection and relies on the assumption that the schur complements can be approximated to high precision by hierarchically semi separable matrices the preconditioner is built as an approximate l d m t factorization through a divide and conquer approach this implies an enhanced flexibility which allows to handle unstructured geometric meshes anisotropies and discontinuities we build on our previous numerical experiments and develop a preconditioner update strategy that allows us to handle matrix sequences arising from problems with slowly varying coefficients we investigate the performance of the preconditioner along with the update strategy in context of topology optimization of an acoustic cavity

strong asymptotics of orthogonal polynomials with respect to exponential weights we consider asymptotics of orthogonal polynomials with respect to weights w x dx e q x dx on the real line where q x 2m k 0 qkx k q2m 0 denotes a polynomial of even order with positive leading coefficient the orthogonal polynomial problem is formulated as a riemann hilbert problem following 22 23 we employ the steepest descent type method introduced in 18 and further developed in 17 19 in order to obtain uniform plancherel rotach type asymptotics in the entire complex plane as well as asymptotic formulae for the zeros the leading coefficients and the recurrence coefficients of the orthogonal polynomials c 1999 john wiley sons inc

transport maps for beta matrix models and universality we construct approximate transport maps for non critical beta matrix models that is maps so that the push forward of a non critical beta matrix model with a given potential is a non critical beta matrix model with another potential up to a small error in the total variation distance one of the main features of our construction is that these maps enjoy regularity estimates that are uniform in the dimension in addition we find a very useful asymptotic expansion for such maps which allows us to deduce that local statistics have the same asymptotic behavior for both models

correlation kernels for discrete symplectic and orthogonal ensembles in 49 h widom derived formulae expressing correlation functions of orthogonal and symplectic ensembles of random matrices in terms of orthogonal polynomials we obtain similar results for discrete ensembles with rational discrete logarithmic derivative and compute explicitly correlation kernels associated to the classical meixner and charlier orthogonal polynomials

discrete painleve equations and their appearance in quantum gravity we discuss an algorithmic approach for both deriving discrete analogues of painleve equations as well as using such equations to characterize similarity reductions of spatially discrete integrable evolution equations as a concrete example we show that a discrete analogue of painleve i can be used to characterize similarity solutions of the kac moerbeke equation it turns out that these similarity solutions also satisfy a special case of painleve iv equation in addition we discuss a methodology for obtaining the relevant continuous limits not only at the level of equations but also at the level of solutions as an example we use the wkb method in the presence of two turning points of the third order to parametrize at the continuous limit the solution of painleve i in terms of the solution of discrete painleve i finally we show that these results are useful for investigating the partition function of the matrix model in 2d quantum gravity associated with the measure exp t1z2 t2z4 t3z6

power analyses for longitudinal study designs with missing data existing methods for power analysis for longitudinal study designs are limited in that they do not adequately address random missing data patterns although the pattern of missing data can be assessed during data analysis it is unknown during the design phase of a study the random nature of the missing data pattern adds another layer of complexity in addressing missing data for power analysis in this paper we model the occurrence of missing data with a two state first order markov process and integrate the modelling information into the power function to account for random missing data patterns the markov model is easily specified to accommodate different anticipated missing data processes we develop this approach for the two most popular longitudinal models the generalized estimating equations gee and the linear mixed effects model under the missing completely at random mcar assumption for gee we also limit our consideration to the working independence correlation model the proposed methodology is illustrated with numerous examples that are motivated by real study designs

improving multifrontal methods by means of block low rank representations matrices coming from elliptic partial differential equations pdes have been shown to have a low rank property well defined off diagonal blocks of their schur complements can be approximated by low rank products given a suitable ordering of the matrix which gives to the blocks a geometrical meaning such approximations can be computed using an svd or a rank revealing qr factorization the resulting representation offers a substantial reduction of the memory requirement and gives efficient ways to perform many of the basic dense algebra operations several strategies have been proposed to exploit this property we propose a low rank format called block low rank blr and explain how it can be used to reduce the memory footprint and the complexity of direct solvers for sparse matrices based on the multifrontal method we present experimental results that show how the blr format delivers gains that are comparable to those obtained with hierarchical formats such as hierarchical matrices h matrices and hierarchically semi separable hss matrices but provides much greater flexibility and ease of use which are essential in the context of a general purpose algebraic solver

small sample properties of probit model estimators abstract when maximum likelihood estimates of the coefficients in a nonlinear model such as the probit model are obtained there are a number of asymptotically equivalent covariance matrix estimators that can be used these covariance matrix estimators are typically associated with different computer algorithms for example with the newton raphson algorithm the inverse of the negative of the hessian matrix from the log likelihood function is used with the method of scoring the inverse of the information matrix is used and with a procedure proposed by berndt hall hall and hausman 1974 the inverse of the outer product of the first derivatives of the log likelihood function is used although these three estimators are asymptotically equivalent their performance can vary in finite samples the main objective of this article is to use a monte carlo experiment to investigate the finite sample properties of the three covariance matrix estimators in the context of maximum likelihood estimation of the pr

edge universality of beta ensembles we prove the edge universality of the beta ensembles for any beta ge 1 provided that the limiting spectrum is supported on a single interval and the external potential is fancyscript c 4 and regular we also prove that the edge universality holds for generalized wigner matrices for all symmetry classes moreover our results allow us to extend bulk universality for beta ensembles from analytic potentials to potentials in class fancyscript c 4

hankel determinant and orthogonal polynomials for a gaussian weight with a discontinuity at the edge we compute asymptotics for hankel determinants and orthogonal polynomials with respect to a discontinuous gaussian weight in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support their behavior is described in terms of the ablowitz segur family of solutions to the painleve ii equation our results complement the ones in 33 as consequences of our results we conjecture asymptotics for an airy kernel fredholm determinant and total integral identities for painleve ii transcendents and we also prove a new result on the poles of the ablowitz segur solutions to the painleve ii equation we also highlight applications of our results in random matrix theory

domain decomposition based ℌ lu preconditioners hierarchical matrices provide a data sparse way to approximate fully populated matrices the two basic steps in the construction of an h matrix are a the hierarchical construction of a matrix block partition and b the blockwise approxi mation of matrix data by low rank matrices in this paper we develop a new approach to construct the necessary partition based on domain decomposition compared to standard geometric bisection based h matrices this new approach yields h lu fac torizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements these rigorously proven and numerically verified improvements result from an h matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an lu factorization we provide numerical results in which a domain decom position based h lu factorization is used as a preconditioner in the iterative solution of the discrete three dimensional convection diffusion equation

mathcal h h matrix approximability of the inverses of fem matrices we study the question of approximability for the inverse of the fem stiffness matrix for scalar second order elliptic boundary value problems by blockwise low rank matrices such as those given by the mathcal h h matrix format introduced by hackbusch computing 62 2 89 108 1999 we show that exponential convergence in the local block rank r r can be achieved we also show that exponentially accurate lu lu decompositions in the mathcal h h matrix format are possible for the stiffness matrices arising in the fem our analysis avoids any coupling of the block rank r r to the mesh width h h we also cover fairly general boundary conditions of mixed dirichlet neumann robin boundary conditions

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

rank 1 real wishart spiked model in this paper we consider n dimensional real wishart matrices y in the class input amssym w bbb r sigma m in which all but one eigenvalue of is 1 let the nontrivial eigenvalue of be 1 then as n m with m n 2 finite and nonzero the eigenvalue distribution of y will converge into the marchenko pastur distribution inside a bulk region when increases from 0 one starts to see a stray eigenvalue of y outside of the support of the marchenko pastur density as this stray eigenvalue leaves the bulk region a phase transition will occur in the largest eigenvalue distribution of the wishart matrix in this paper we will compute the asymptotics of the largest eigenvalue distribution when the phase transition occurs we will first establish the results that are valid for all n and m and will use them to carry out the asymptotic analysis in particular we have derived a contour integral formula for the harish chandra itzykson zuber integral when x and y are real symmetric and y is a rank 1 matrix this allows us to write down a fredholm determinant formula for the largest eigenvalue distribution and analyze it using orthogonal polynomial techniques as a result we obtain an integral formula for the largest eigenvalue distribution in the large n limit characterized by painleve transcendents the approach used in this paper is very different from a recent paper by bloemenal and virag in which the largest eigenvalue distribution was obtained using a stochastic operator method in particular the painleve formula for the largest eigenvalue distribution obtained in this paper is new 2012 wiley periodicals inc

toda versus pfaff lattice and related polynomials the pfaff lattice was introduced by us in the context of a lie algebra splitting of gl infinity into sp infinity and an algebra of lower triangular matrices the pfaff lattice is equivalent to a set of bilinear identities for the wave functions which yield the existence of a sequence of tau functions the latter satisfy their own set of bilinear identities which moreover characterize them in the semi infinite case the tau functions are pfaffians in the same way that for the toda lattice the tau functions are hankel determinants interesting examples occur in the theory of random matrices where one considers symmetric and symplectic matrix integrals for the pfaff lattice and hermitian matrix integrals for the toda lattice there is a striking parallel between the pfaff lattice and the toda lattice and even more striking there is a map from one to the other mapping skew orthogonal to orthogonal polynomials in particular we exhibit two maps dual to each other mapping hermitian matrix integrals to either symmetric matrix integrals or symplectic matrix integrals

universality of the double scaling limit in random matrix models we study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support we establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit the limiting kernels are constructed out of functions associated with the second painleve equation this extends a result of bleher and its for the special case of a critical quartic potential the two main tools we use are equilibrium measures and riemann hilbert problems in our treatment of equilibrium measures we allow a negative density near the critical point which enables us to treat all cases simultaneously the asymptotic analysis of the riemann hilbert problem is done with the deift zhou steepest descent analysis for the construction of a local parametrix at the critical point we introduce a modification of the approach of baik deift and johansson so that we are able to satisfy the required jump properties exactly c 2005 wiley periodicals inc

large gap asymptotics for airy kernel determinants with discontinuities we obtain large gap asymptotics for airy kernel fredholm determinants with any number m of discontinuities these m point determinants are generating functions for the airy point process and encode probabilistic information about eigenvalues near soft edges in random matrix ensembles our main result is that the m point determinants can be expressed asymptotically as the product of m 1 point determinants multiplied by an explicit constant pre factor which can be interpreted in terms of the covariance of the counting function of the process

on the proof of universality for orthogonal and symplectic ensembles in random matrix theory we give a streamlined proof of a quantitative version of a result from p deift and d gioev universality in random matrix theory for orthogonal and symplectic ensembles imrp int math res pap in press which is crucial for the proof of universality in the bulk p deift and d gioev universality in random matrix theory for orthogonal and symplectic ensembles imrp int math res pap in press and also at the edge p deift and d gioev universality at the edge of the spectrum for unitary orthogonal and symplectic ensembles of random matrices comm pure appl math in press for orthogonal and symplectic ensembles of random matrices as a byproduct this result gives asymptotic information on a certain ratio of the 1 2 4 partition functions for log gases

asymptotics of the airy kernel determinant the authors use riemann hilbert methods to compute the constant that arises in the asymptotic behavior of the airy kernel determinant of random matrix theory

the largest eigenvalue of real symmetric hermitian and hermitian self dual random matrix models with rank one external source part i we consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric 1 hermitian 2 and hermitian self dual 4 random matrix models with rank 1 external source they are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues assuming the one band condition and certain regularities of the potential function we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value when the nonzero eigenvalue of the external source matrix is less than or equal to the critical value the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper in this paper we also give a definition of the external source model for all 0

on the relation between orthogonal symplectic and unitary matrix ensembles for the unitary ensembles of n n hermitian matrices associated with a weight function w there is a kernel expressible in terms of the polynomials orthogonal with respect to the weight function which plays an important role for the orthogonal and symplectic ensembles of hermitian matrices there are 2 2 matrix kernels usually constructed using skew orthogonal polynomials which play an analogous role these matrix kernels are determined by their upper left hand entries we derive formulas expressing these entries in terms of the scalar kernel for the corresponding unitary ensembles we also show that whenever w w is a rational function the entries are equal to the scalar kernel plus some extra terms whose number equals the order of w w general formulas are obtained for these extra terms we do not use skew orthogonal polynomials in the derivations

orthogonal and symplectic matrix models universality and other properties we study orthogonal and symplectic matrix models with polynomial potentials and multi interval supports of the equilibrium measures for these models we find the bounds similar to those for the hermitian matrix models for the rate of convergence of linear eigenvalue statistics and for the variance of linear eigenvalue statistics and find the logarithms of partition functions up to the order o 1 we prove also the universality of local eigenvalue statistics in the bulk

bayesian analysis of binary and polychotomous response data abstract a vast literature in statistics biometrics and econometrics is concerned with the analysis of binary and polychotomous response data the classical approach fits a categorical response regression model using maximum likelihood and inferences about the model are based on the associated asymptotic theory the accuracy of classical confidence statements is questionable for small sample sizes in this article exact bayesian methods for modeling categorical response data are developed using the idea of data augmentation the general approach can be summarized as follows the probit regression model for binary outcomes is seen to have an underlying normal regression structure on latent continuous data values of the latent data can be simulated from suitable truncated normal distributions if the latent data are known then the posterior distribution of the parameters can be computed using standard results for normal linear models draws from this posterior are used to sample new latent data and t

classical skew orthogonal polynomials and random matrices skew orthogonal polynomials arise in the calculation of the n point distribution function for the eigenvalues of ensembles of random matrices with orthogonal or symplectic symmetry in particular the distribution functions are completely determined by a certain sum involving the skew orthogonal polynomials in the case that the eigenvalue probability density function involves a classical weight function explicit formulas for the skew orthogonal polynomials are given in terms of related orthogonal polynomials and the structure is used to give a closed form expression for the sum this theory treates all classical cases on an equal footing giving formulas applicable at once to the hermite laguerre and jacobi cases

universality of the stochastic bessel operator we establish universality at the hard edge for general beta ensembles assuming that the background potential v is a polynomial such that x mapsto v x 2 is strongly convex beta ge 1 and the dimension difference parameter a ge 0 the method rests on the corresponding tridiagonal matrix models showing that their appropriate continuum scaling limit is given by the stochastic bessel operator as conjectured in edelman and sutton j stat phys 127 1121 1165 2007 and rigorously established in ramirez and rider commun math phys 288 887 906 2009 the latter characterizes the hard edge in the case of linear potential and all beta the classical beta laguerre ensembles

non intersecting squared bessel paths and multiple orthogonal polynomials for modified bessel weights we study a model of n non intersecting squared bessel processes in the confluent case all paths start at time t 0 at the same positive value x a remain positive and are conditioned to end at time t t at x 0 in the limit n after appropriate rescaling the paths fill out a region in the tx plane that we describe explicitly in particular the paths initially stay away from the hard edge at x 0 but at a certain critical time t the smallest paths hit the hard edge and from then on are stuck to it for t t we obtain the usual scaling limits from random matrix theory namely the sine airy and bessel kernels a key fact is that the positions of the paths at any time t constitute a multiple orthogonal polynomial ensemble corresponding to a system of two modified bessel type weights as a consequence there is a 3 3 matrix valued riemann hilbert problem characterizing this model that we analyze in the large n limit using the deift zhou steepest descent method there are some novel ingredients in the riemann hilbert analysis that are of independent interest

edge universality for orthogonal ensembles of random matrices we prove edge universality of local eigenvalue statistics for orthogonal invariant matrix models with real analytic potentials and one interval limiting spectrum our starting point is the result of shcherbina commun math phys 285 957 974 2009 on the representation of the reproducing matrix kernels of orthogonal ensembles in terms of scalar reproducing kernel of corresponding unitary ensemble

a fast block low rank dense solver with applications to finite element matrices this article presents a fast solver for the dense frontal matrices that arise from the multifrontal sparse elimination process of 3d elliptic pdes the solver relies on the fact that these matrices can be efficiently represented as a hierarchically off diagonal low rank hodlr matrix to construct the low rank approximation of the off diagonal blocks we propose a new pseudo skeleton scheme the boundary distance low rank approximation that picks rows and columns based on the location of their corresponding vertices in the sparse matrix graph we compare this new low rank approximation method to the adaptive cross approximation aca algorithm and show that it achieves better speedup specially for unstructured meshes using the hodlr direct solver as a preconditioner with a low tolerance to the gmres iterative scheme we can reach machine accuracy much faster than a conventional lu solver numerical benchmarks are provided for frontal matrices arising from 3d finite element problems corresponding to a wide range of applications

generalized covariance adjusted canonical correlation analysis with application to psychiatry the lack of control over covariates in practice motivates the need for their adjustment when measuring the degree of association between two sets of variables for which canonical correlation is traditionally used in most studies however there is also a lack of control over the attributes of responses for the sets of variables of interest in particular a portion of the response variable may be continuous and the other discrete for such settings the traditional partial canonical correlation approach is restrictive since a covariate adjustment for a set of continuous variables is assumed by ignoring the assumption of continuous variates and proceeding with a partial canonical correlation analysis in the presence of continuous and discrete variates results in canonical correlation estimates that are not consistent in this paper we generalize the traditional partial canonical correlation approach to covariate adjustment by allowing the response variables to contain continuous as well as discrete variates the methodology is illustrated with a psychiatric application for examining which sleep variables relate to which depressive symptoms as measured by commonly used constructs that presents with both continuous and discrete outcomes

universality at the edge of the spectrum in wigner random matrices we prove universality at the edge for rescaled correlation functions of wigner random matrices in the limit n as a corollary we show that after proper rescaling the 1th 2nd 3rd etc eigenvalues of wigner random hermitian resp real symmetric matrix weakly converge to the distributions established by tracy and widom in g u e g o e cases

fast structured lu factorization for nonsymmetric matrices in this paper an approximate lu factorization algorithm is developed for nonsymmetric matrices based on the hierarchically semiseparable matrix techniques it utilizes a technique involving orthogonal transformations and approximations to avoid the explicit computation of the schur complement in each factorization step a modified compression method is further developed for the case when some diagonal blocks have small singular values the complexity of the methods proposed in this paper is analyzed and shown to be o n 2k o n 2 k where n n is the dimension of matrix and k k is the maximum off diagonal numerical rank depending on the accuracy and efficiency requirements in the approximation this factorization can be used either as a direct solver or a preconditioner numerical results from applications are included to show the efficiency of our method

generalized linear models with random effects a gibbs sampling approach abstract generalized linear models have unified the approach to regression for a wide variety of discrete continuous and censored response variables that can be assumed to be independent across experimental units in applications such as longitudinal studies genetic studies of families and survey sampling observations may be obtained in clusters responses from the same cluster cannot be assumed to be independent with linear models correlation has been effectively modeled by assuming there are cluster specific random effects that derive from an underlying mixing distribution extensions of generalized linear models to include random effects has thus far been hampered by the need for numerical integration to evaluate likelihoods in this article we cast the generalized linear random effects model in a bayesian framework and use a monte carlo method the gibbs sampler to overcome the current computational limitations the resulting algorithm is flexible to easily accommodate changes in the number

on universality for orthogonal ensembles of random matrices we prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum our starting point is the tracy widom formula for the matrix reproducing kernel the key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a product of a positive toeplitz matrix and a two diagonal skew symmetric toeplitz matrix

universality for orthogonal and symplectic laguerre type ensembles we give a proof of the universality conjecture for orthogonal 1 and symplectic 4 random matrix ensembles of laguerre type in the bulk of the spectrum as well as at the hard and soft spectral edges our results are stated precisely in the introduction theorems 1 1 1 4 1 6 and corollaries 1 2 1 5 1 7 they concern the appropriately rescaled kernels k n correlation and cluster functions gap probabilities and the distributions of the largest and smallest eigenvalues corresponding results for unitary 2 laguerre type ensembles have been proved by the fourth author in ref 23 the varying weight case at the hard spectral edge was analyzed in ref 13 for 2 in this paper we do not consider varying weights our proof follows closely the work of the first two authors who showed in refs 7 8 analogous results for hermite type ensembles as in refs 7 8 we use the version of the orthogonal polynomial method presented in refs 22 25 to analyze the local eigenvalue statistics the necessary asymptotic information on the laguerre type orthogonal polynomials is taken from ref 23

universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles this paper is devoted to the rigorous proof of the universality conjecture of random matrix theory according to which the limiting eigenvalue statistics ofn n random matrices within spectral intervals ofo n 1 is determined by the type of matrix real symmetric hermitian or quaternion real and by the density of states we prove this conjecture for a certain class of the hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function the potential in the quantum field theory satisfying some regularity conditions

acoustic inverse scattering via helmholtz operator factorization and optimization we present a joint acoustic seismic inverse scattering and finite frequency reflection tomography program formulated as a coupled set of optimization problems in terms of inhomogeneous helmholtz equations we use a higher order finite difference scheme for these helmholtz equations to guarantee sufficient accuracy we adapt a structured approximate direct solver for the relevant systems of algebraic equations which addresses storage requirements through compression to yield a complexity for computing the gradients or images in the optimization problems that consists of two parts viz the cost for all the matrix factorizations which is roughly o rn for example o rnlogn when d 2 times the number of frequencies and the cost for all solutions by substitution which is roughly o n for example o nlog rlogn when d 2 times the number of frequencies times the number of sources events where n n d if n is the number of grid samples in any direction and r is a parameter depending on the preset accuracy and the problem at hand with this complexity the multi frequency approach to inverse scattering and finite frequency tomography becomes computationally feasible with large data sets in dimensions d 2 and 3

a bayesian approach to outlier detection and residual analysis summary an approach to detecting outliers in a linear model is developed an outlier is defined to be an observation with a large random error generated by the linear model under consideration outliers are detected by examining the posterior distribution of the random errors an augmented residual plot is also suggested as a graphical aid in finding outliers we propose a precise definition of an outlier in a linear model which appears to lead to simple ways of exploring data for the possibility of outliers the definition is such that if the parameters of the model are known then it is also known which observations are outliers alternatively if the parameters are unknown the posterior distribution can be used to calculate the posterior probability that any observation is an outlier in a linear model with normally distributed random errors ei with mean zero and variance a 2we declare the ith observation to be an outlier if iei i ko for some choice of k the value of k can be chosen so that the prior probability of an outlier is small and thus outliers are observations which are more extreme than is usually expected realizations of normally distributed errors of more than about three standard deviations from the mean are certainly surprising and worth further investigation such outlying observations can occur under the assumed model however and this should be taken into account when deciding what to do with outliers and in choosing k note that ei is the actual realization of the random error not the usual estimated residual i the problem of outliers is studied and thoroughly reviewed by barnett lewis 1984 hawkins 1980 beckman cook 1983 and pettit smith 1985 the usual bayesian approach to outlier detection uses the definition given by freeman 1980 freeman defines an outlier to be any observation that has not been generated by the mechanism that generated the majority of observations in the data set freeman s definition therefore requires that a model for the generation of outliers be specified and is implemented by for example box tiao 1968 guttman dutter freeman 1978 and abraham box 1978 our method differs in that we define outliers as arising from the model under consideration rather than arising from a separate expanded model our approach is similar to that described by zellner moulton 1985 and is an extension of the philosophy

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

hierarchical interpolative factorization for elliptic operators differential equations this paper introduces the hierarchical interpolative factorization for integral equations hif ie associated with elliptic problems in two and three dimensions this factorization takes the form of an approximate generalized lu decomposition that permits the efficient application of the discretized operator and its inverse hif ie is based on the recursive skeletonization algorithm but incorporates a novel combination of two key features 1 a matrix factorization framework for sparsifying structured dense matrices and 2 a recursive dimensional reduction strategy to decrease the cost thus higher dimensional problems are effectively mapped to one dimension and we conjecture that constructing applying and inverting the factorization all have linear or quasilinear complexity numerical experiments support this claim and further demonstrate the performance of our algorithm as a generalized fast multipole method direct solver and preconditioner hif ie is compatible with geometric adaptivity and can handle both boundary and volume problems matlab codes are freely available 2015 wiley periodicals inc

higher order analogues of the tracy widom distribution and the painleve ii hierarchy we study fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points the kernels are natural higher order analogues of the airy kernel and

low rank improvements of two level grid preconditioned matrices abstract as an alternative to basic two level and multilevel iteration preconditioners for elliptic partial differential equations it is shown that low rank approximations based on approximate eigenvectors to the largest eigenvalues of the inverse two level schur complement matrix can give arbitrarily accurate preconditioners that hold uniformly with respect to mesh sizes the methods are particularly efficient for problems with multiple right hand sides

effective matrix free preconditioning for the augmented immersed interface method we present effective and efficient matrix free preconditioning techniques for the augmented immersed interface method aiim aiim has been developed recently and is shown to be very effective for interface problems and problems on irregular domains gmres is often used to solve for the augmented variable s associated with a schur complement a in aiim that is defined along the interface or the irregular boundary the efficiency of aiim relies on how quickly the system for a can be solved for some applications there are substantial difficulties involved such as the slow convergence of gmres particularly for free boundary and moving interface problems and the inconvenience in finding a preconditioner due to the situation that only the products of a and vectors are available here we propose matrix free structured preconditioning techniques for aiim via adaptive randomized sampling using only the products of a and vectors to construct a hierarchically semiseparable matrix approximation to a several improvements over existing schemes are shown so as to enhance the efficiency and also avoid potential instability the significance of the preconditioners includes 1 they do not require the entries of a or the multiplication of a t with vectors 2 constructing the preconditioners needs only o log n matrix vector products and o n storage where n is the size of a 3 applying the preconditioners needs only o n flops 4 they are very flexible and do not require any a priori knowledge of the structure of a the preconditioners are observed to significantly accelerate the convergence of gmres with heuristical justifications of the effectiveness comprehensive tests on several important applications are provided such as navier stokes equations on irregular domains with traction boundary conditions interface problems in incompressible flows mixed boundary problems and free boundary problems the preconditioning techniques are also useful for several other problems and methods

universality in invariant random matrix models existence near the soft edge we consider two non gaussian ensembles of large hermitian random matrices with strong level confinement and show that near the soft edge of the spectrum both scaled density of states and eigenvalue correlations follow so called airy laws inherent in gaussian unitary ensemble this suggests that the invariant one matrix models should display universal eigenvalue correlations in the soft edge scaling limit

sparse regular random graphs spectral density and eigenvectors we examine the empirical distribution of the eigenvalues and the eigenvectors of adjacency matrices of sparse regular random graphs we find that when the degree sequence of the graph slowly increases to infinity with the number of vertices the empirical spectral distribution converges to the semicircle law moreover we prove concentration estimates on the number of eigenvalues over progressively smaller intervals we also show that with high probability all the eigenvectors are delocalized

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