cox s regression model for counting processes a large sample study the cox regression model for censored survival data specifies that covariates have a proportional effect on the hazard function of the life time distribution of an individual in this paper we discuss how this model can be extended to a model where covariate processes have a proportional effect on the intensity process of a multivariate counting process this permits a statistical regression analysis of the intensity of a recurrent event allowing for complicated censoring patterns and time dependent covariates furthermore this formulation gives rise to proofs with very simple structure using martingale techniques for the asymptotic properties of the estimators from such a model finally an example of a statistical analysis is included

misspecified proportional hazard models summary the properties of an estimator based on a proportional hazards model are investigated when the model is incorrect the estimator from the partial likelihood is shown to be consistent for a parameter that is defined implicitly the results are used to investigate the effects on estimation if the true model is accelerated failure time or if covariates are omitted from the proportional hazards model

semiparametric regression calibration for general hazard models in survival analysis with covariate measurement error surprising performance under linear hazard observational epidemiological studies often confront the problem of estimating exposure disease relationships when the exposure is not measured exactly regression calibration rc is a common approach to correct for bias in regression analysis with covariate measurement error in survival analysis with covariate measurement error it is well known that the rc estimator may be biased when the hazard is an exponential function of the covariates in the paper we investigate the rc estimator with general hazard functions including exponential and linear functions of the covariates when the hazard is a linear function of the covariates we show that a risk set regression calibration rrc is consistent and robust to a working model for the calibration function under exponential hazard models there is a trade off between bias and efficiency when comparing rc and rrc however one surprising finding is that the trade off between bias and efficiency in measurement error research is not seen under linear hazard when the unobserved covariate is from a uniform or normal distribution under this situation the rrc estimator is in general slightly better than the rc estimator in terms of both bias and efficiency the methods are applied to the nutritional biomarkers study of the women s health initiative

impact of element level static condensation on iterative solver performance this paper provides theoretical estimates that quantify and clarify the savings associated to the use of element level static condensation as a first step of an iterative solver these estimates are verified numerically the numerical evidence shows that static condensation at the element level is beneficial for higher order methods for lower order methods or when the number of iterations required for convergence is low the setup cost of the elimination as well as its implementation may offset the benefits obtained during the iteration process however as the iteration count e g above 50 or the polynomial order e g above cubics grows the benefits of element level static condensation are significant

a semilinear parabolic system arising in the theory of superconductivity arises in the theory of superconductivity of liquids proposed by abrahams and tsuneto in 11 system 1 has also been briefly treated as an example by chueh conley and smaller in 61 throughout this paper we shall assume that d is a bounded domain in r n 1 2 or 3 with smooth boundary x a denotes the laplacian all the results of the paper which hold for n 1 also hold when a is replaced by any second order symmetric elliptic differential expression with positive spectrum because of the very symmetric nature of l it is reasonable to expect solutions of 1 to have nice properties we show in this paper that this is in fact the case in section 2 we give a complete description of steady state solutions of 1 satisfying dirichlet or neumann boundary conditions we show that often all steady states are of the form u u t au for some constant a where u is the solution of a single differential equation it is possible using standard techniques for scalar nonlinear boundary value problems to obtain lots of information about the solutions of this single equation and so about solutions of 1 in 6 1 it is shown that all solutions of 1 are attracted towards the set u v u v 1 in section 3 we show that 1 has a lyapunov function and use this fact and the standard theory of w limit sets to prove that any solution of 1 tends as t 03 towards the family of all steady state solutions of 1 often in the case of scalar equations it can be shown that

on errors in variables for binary regression models summary we consider binary regression models when some of the predictors are measured with error for normal measurement errors structural maximum likelihood estimates are considered we show that if the measurement error is large the usual estimate of the probability of the event in question can be substantially in error especially for high risk groups in the situation of large measurement error we investigate a conditional maximum likelihood estimator and its properties

hierarchic models for laminated plates and shells the definition essential properties and formulation of hierarchic models for laminated plates and shells are presented the hierarchic models satisfy three essential requirements approximability asymptotic consistency and optimality of convergence rate aspects of implementation are discussed and the performance characteristics are illustrated by examples

proportional hazards model with covariates subject to measurement error when covariates of a proportional hazards model are subject to measurement error the maximum likelihood estimates of regression coefficients based on the partial likelihood are asymptotically biased prentice 1982 biometrika 69 331 342 presents an example of such bias and suggests a modified partial likelihood this paper applies the corrected score function method nakamura 1990 biometrika 77 127 137 to the proportional hazards model when measurement errors are additive and normally distributed the result allows a simple correction to the ordinary partial likelihood that yields asymptotically unbiased estimates the validity of the correction is confirmed via a limited simulation study

failure time regression with continuous covariates measured with error summary we consider failure time regression analysis with an auxiliary variable in the presence of a validation sample we extend the nonparametric inference procedure of zhou and pepe to handle a continuous auxiliary or proxy covariate we estimate the induced relative risk function with a kernel smoother and allow the selection probability of the validation set to depend on the observed covariates we present some asymptotic properties for the kernel estimator and provide some simulation results the method proposed is illustrated with a data set from an on going epidemiologic study

eigenvalues and eigenfunctions of the laplacian via inverse iteration with shift in this paper we present an iterative method inspired by the inverse iteration with shift technique of finite linear algebra designed to find the eigenvalues and eigenfunctions of the laplacian with homogeneous dirichlet boundary condition for arbitrary bounded domains r n this method which has a direct functional analysis approach does not approximate the eigenvalues of the laplacian as those of a finite linear operator it is based on the uniform convergence away from nodal surfaces and can produce a simple and fast algorithm for computing the eigenvalues with minimal computational requirements instead of using the ubiquitous rayleigh quotient of finite linear algebra also an alternative expression for the rayleigh quotient in the associated infinite dimensional sobolev space which avoids the integration of gradients is introduced and shown to be more efficient the method can also be used in order to produce the spectral decomposition of any given function u l 2

eigenmodes of isospectral drums recently it was proved that there exist nonisometric planar regions that have identical laplace spectra that is one cannot hear the shape of a drum the simplest isospectral regions known are bounded by polygons with reentrant corners while the isospectrality can be proven mathematically analytical techniques are unable to produce the eigenvalues themselves furthermore standard numerical methods for computing the eigenvalues such as adaptive finite elements are highly inefficient physical experiments have been performed to measure the spectra but the accuracy and flexibility of this method are limited we describe an algorithm due to descloux and tolley comput methods appl mech engrg 39 1983 pp 37 53 that blends singular finite elements with domain decomposition and show that with a modification that doubles its accuracy this algorithm can be used to compute efficiently the eigenvalues for polygonal regions we present results accurate to 12 digits for the most famous pair of isospectral drums as well as results for another pair

estimation of dependence between paired correlated failure times in the presence of covariate measurement error summary in many biomedical studies covariates are subject to measurement error although it is well known that the regression coefficients estimators can be substantially biased if the measurement error is not accommodated there has been little study of the effect of covariate measurement error on the estimation of the dependence between bivariate failure times we show that the dependence parameter estimator in the clayton oakes model can be considerably biased if the measurement error in the covariate is not accommodated in contrast with the typical bias towards the null for marginal regression coefficients the dependence parameter can be biased in either direction we introduce a bias reduction technique for the bivariate survival function in copula models while assuming an additive measurement error model and replicated measurement for the covariates and we study the large and small sample properties of the dependence parameter estimator proposed

unsymmetric multi level hanging nodes and anisotropic polynomial degrees in h1 conforming higher order finite element methods abstract the implementation of higher order finite element schemes that can handle multi level hanging nodes is known to be a difficult task in fact most of the available literature on hanging nodes in finite element schemes restricts to one level hanging nodes resulting from symmetric bisections the intent of this paper is to provide all data structures and algorithms that are necessary for an implementation of a h 1 conforming higher order finite element method the corresponding finite element spaces are defined via tensor products of hierarchic as well as nodal polynomials for quadrilateral and hexahedron based meshes with unsymmetric multi level hanging nodes and arbitrary anisotropic polynomial degree distributions where special care is given to possible orientation problems the meshes may even result from non matching refinements given these data structures and algorithms an extension for serendipity spaces is described in detail along with some other techniques to improve computational efficiency numerical results from an implementation based on these data structures and algorithms serve as a validation and show the broad possibilities that highly flexible higher order finite element schemes have to offer to the best of our knowledge this paper offers the most comprehensive numerical results so far for various three dimensional benchmark problems using in particular finite element spaces defined for hierarchical as well as nodal polynomials on meshes with unsymmetric refinement ratios as well as multi level hanging nodes most notably the numerical results indicate that unsymmetric refinements are indeed favorable over symmetric refinements with respect to convergence rates however the actual optimal refinement ratio for a given problem seems to depend on the type and magnitude of singularities to be resolved as well as on the chosen full tensor product or serendipity finite element spaces in addition to these numerical results we find that systems of equations defined via finite element spaces using the nodal lagrange polynomials with gauss lobatto quadrature points as support points yield drastically improved condition numbers

proportional hazards regression with missing covariates abstract nonparametric maximum likelihood npml is used to estimate regression parameters in a proportional hazards regression model with missing covariates the npml estimator is shown to be consistent and asymptotically normally distributed under some conditions em type algorithms are applied to solve the maximization problem variance estimates of the regression parameters are obtained by a profile likelihood approach that uses em aided numerical differentiation simulation results indicate that the npml estimates of the regression parameters are more efficient than the approximate partial likelihood estimates and estimates from complete case analysis when missing covariates are missing completely at random and that the proposed method corrects for bias when the missing covariates are missing at random

survival analysis with heterogeneous covariate measurement error this article is motivated by a time to event analysis where the covariate of interest was measured at the wrong time we show that the problem can be formulated as a special case of survival analysis with heterogeneous covariate measurement error and develop a general analytic framework we study the asymptotic behavior of the naive partial likelihood estimates and analytically demonstrate that under the heterogeneous measurement error structure and the assumption that all components of the covariate vector and the measurement error vector combined are mutually independent these naive estimates will shrink toward 0 and that the degree of attenuation increases as the measurement error increases we also give counterexamples for reverse attenuation when the independence conditions are violated we use our analytical results to derive a simple bias correcting estimator that performs well in simulations for small and moderate amounts of measurement error our framework can be used to provide insight into th

experimental versus numerical eigenvalues of a bunimovich stadium billiard a comparison we compare the statistical properties of eigenvalue sequences for a gamma 1 bunimovich stadium billiard the eigenvalues have been obtained in two ways one set results from a measurement of the eigenfrequencies of a superconducting microwave resonator real system and the other set is calculated numerically ideal system we show influence of mechanical imperfections of the real system in the analysis of the spectral fluctuations and in the length spectra compared to the exact data of the ideal system we also discuss the influence of a family of marginally stable orbits the bouncing ball orbits in two microwave stadium billiards with different geometrical dimensions

reduced bezier element quadrature rules for quadratic and cubic splines in isogeometric analysis abstract we explore the use of various element based reduced quadrature strategies for bivariate and trivariate quadratic and cubic spline elements used in isogeometric analysis the rules studied encompass tensor product gauss and gauss lobatto rules and certain so called monomial rules that do no possess a tensor product structure the objective of the study is to determine quadrature strategies which enjoy the same accuracy and stability behavior as full gauss quadrature but with significantly fewer quadrature points several cases emerge that satisfy this objective and also demonstrate superior efficiency compared with standard c 0 continuous finite elements of the same order

an additive schwarz method for the h b version of the finite element method in three dimensions in this paper we study the additive schwarz method for the h p version of the finite element method in three dimensions the main idea is to treat separately the h version linear components and the p version high order components by a vertex based method it can also be viewed as a three level method with the level being the linear finite element approximation on the coarse mesh the linear finite element approximation on the fine mesh and the high order finite element approximation on the fine mesh respectively the resulting algorithm can be implemented in parallel on the subdomain level for the h version components and on the element level for the p version components the condition number is of order max limits i 1 ln hi pi hi 2 where hi stands for the diameter of the subdomain omega i hi is the diameter of the elements in omega i and pi is the maximum of the polynomial degrees used in omega i

a polylogarithmic bound for an iterative substructuring method for spectral elements in three dimensions iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems a p version finite element method based on continuous piecewise q p functions is considered for second order elliptic problems in three dimensions this special method can also be viewed as a conforming spectral element method an iterative method is designed for which the condition number of the relevant operator grows only in proportion to 1 log p 2 this bound is independent of jumps in the coefficient of the elliptic problem across the interfaces between the subregions numerical results are also reported which support the theory

toward a universal h p adaptive finite element strategy part 1 constrained approximation and data structure abstract this is the first of a series of papers devoted to the development of an adaptive finite element method to solve various boundary value problems arising in solid and fluid mechanics the paper deals with the general concepts of h p finite element methods constrained approximation and data structure and related issues several numerical examples illustrate the main ideas of the method

non overlapping domain decomposition methods for adaptive hp approximations of the stokes problem with discontinuous pressure fields abstract this paper proposes a framework for non overlapping domain decomposition methods applied to viscous incompressible flows governed by stokes equations modelled by the hp version of the finite element method preconditioning methods developed earlier for elliptic problems are extended to this case suitable condition number bounds are obtained for both the hierarchical hp preconditioners of the type proposed in 9 and neumann neumann preconditioners proposed in 10 systematic procedures for constructing compatible hp approximations for incompressible flow are also described

hazard ratio estimation for biomarker calibrated dietary exposures uncertainty concerning the measurement error properties of self reported diet has important implications for the reliability of nutritional epidemiology reports biomarkers based on the urinary recovery of expended nutrients can provide an objective measure of short term nutrient consumption for certain nutrients and when applied to a subset of a study cohort can be used to calibrate corresponding self report nutrient consumption assessments a non standard measurement error model that makes provision for systematic error and subject specific error along with the usual independent random error is needed for the self report data three estimation procedures for hazard ratio cox model parameters are extended for application to this more complex measurement error structure these procedures are risk set regression calibration conditional score and nonparametric corrected score an estimator for the cumulative baseline hazard function is also provided the performance of each method is assessed in a simulation study the methods are then applied to an example from the women s health initiative dietary modification trial

fully automatic hp adaptivity in three dimensions the paper presents a three dimensional implementation of the fully automatic hp adaptive mesh refinement strategy proposed in l demkowicz w rachowicz ph devloo a fully automatic hp adaptivity j sci comput 17 1 3 2002 127 155 the methodology is based on a minimization of the projection based interpolation error with a fixed number of degrees of freedom a coarse grid is first uniformly refined in both h and p to yield a corresponding fine grid a two grid solver is used to solve the problem on the fine grid or iterate towards the fine grid solution the fine grid solution possibly partially converged serves then as a reference solution to be interpolated on the original coarse mesh and the next optimal coarse grid to be determined the difference between the coarse and fine grid solutions provides an excellent error estimate for the coarse grid error the procedure is repeated with the coarse and fine meshes generated during the adaptive process simultaneously until a prescribed error tolerance for the coarse grid error is met the solution obtained on the last fine grid is considered to be the final product delivered by the adaptive procedure

a class of difference schemes with flexible local approximation solutions of many physical problems have salient local features that are qualitatively known a priori for example singularities at point sources edge and corners boundary layers derivative jumps at material interfaces strong dipole field components near polarized spherical particles cusps of electronic wavefunctions at the nuclei electrostatic double layers around colloidal particles etc the known methods capable of providing flexible local approximation of such features include the generalized finite element partition of unity method special variational difference schemes in broken sobolev spaces and a few other specialized techniques in the proposed new class of flexible local approximation methods flame a desirable set of local approximating functions such as cylindrical or spherical harmonics plane waves harmonic polynomials etc defines a finite difference scheme on a chosen grid stencil one motivation is to minimize the notorious staircase effect at curved and slanted interface boundaries however the new approach has much broader applications as illustrative examples the paper presents arbitrarily high order 3 point schemes for the 1d schrodinger equation and a 1d singular equation schemes for electrostatic interactions of colloidal particles electromagnetic wave propagation and scattering plasmon resonances moreover many classical finite difference schemes including the collatz mehrstellen schemes are direct particular cases of flame

on the computation of a very large number of eigenvalues for selfadjoint elliptic operators by means of multigrid methods recent results in the study of quantum manifestations in classical chaos raise the problem of computing a very large number of eigenvalues of selfadjoint elliptic operators the standard numerical methods for large eigenvalue problems cover the range of applications where a few of the leading eigenvalues are needed they are not appropriate and generally fail to solve problems involving a number of eigenvalues exceeding a few hundreds further the accurate computation of a large number of eigenvalues leads to much larger problem dimension in comparison with the usual case dealing with only a few eigenvalues a new method is presented which combines multigrid techniques with the lanczos process the resulting scheme requires o mn arithmetic operations and o n storage requirement where n is the number of unknowns and m the number of needed eigenvalues the discretization of the considered differential operators is realized by means of p finite elements and is applicable on general geometries numerical experiments validate the proposed approach and demonstrate that it allows to tackle problems considered to be beyond the range of standard iterative methods at least on current workstations the ability to compute more than 9000 eigenvalues of an operator of dimension exceeding 8 million on a pc shows the potential of this method practical applications are found e g in the numerical simulation of quantum billiards

computing eigenvalues of very large symmetric matrices an implementation of a lanczos algorithm with no reorthogonalization many engineering and scientific applications require the computation of eigenvalues and eigenvectors of very large symmetric or hermitian matrices we describe a lanczos procedure which allows us to compute either few or many eigenvalues of such matrices in any intervals specified by the user this procedure can even be used to compute all of the eigenvalues the desired eigenvalues are computed as eigenvalues of an associated symmetric tridiagonal matrix tm whose order depends upon the distribution of the eigenvalues in the given matrix a and upon which portions of the spectrum of a are desired the storage requirements depend linearly upon the order of a if the the storage required to generate the products ax is also linear in the order the amount of computation required depends directly upon the distribution of the desired eigenvalues and upon the cost of computing ax numerical results for a very large matrix of order 4900 demonstrate that this procedure can be used on very large matrices

on condition numbers in hp fem with gauss lobatto based shape functions sharp bounds on the condition number of stiffness matrices arising in hp spectral discretizations for two dimensional problems elliptic problems are given two types of shape functions that are based on lagrange interpolation polynomials in the gauss lobatto points are considered these shape functions result in condition numbers o p and o p ln p for the condensed stiffness matrices where p is the polynomial degree employed locally refined meshes are analyzed for the discretization of dirichlet problems on meshes that are refined geometrically toward singularities the conditioning of the stiffness matrix is shown to be independent of the number of layers of geometric refinement

covariate measurement errors and parameter estimation in a failure time regression model summary estimation in cox s failure time regression model is considered when the regression vector is subject to measurement error a hazard function model is induced for the failure rate given the measured covariate and a partial likelihood function is derived for the relative risk parameters this partial likelihood function may involve the baseline hazard function as well as the regression parameter but useful inference techniques arise for testing whether the regression parameter equals zero and for more general inferences in important special cases explicit consideration is given to testing equality of survival curves when group membership is subject to misclassification and to relative risk estimation with normally distributed covariates approaches to regression estimation using the overall likelihood function and a marginal likelihood function based on failure time ranks are also indicated illustration of the possible effect of covariate errors on relative risk estimation is provided

a collocated c0 finite element method reduced quadrature perspective cost comparison with standard finite elements and explicit structural dynamics summary we demonstrate the potential of collocation methods for efficient higher order analysis on standard nodal finite element meshes we focus on a collocation method that is variationally consistent and geometrically flexible converges optimally embraces concepts of reduced quadrature and leads to symmetric stiffness and diagonal consistent mass matrices at the same time it minimizes the evaluation cost per quadrature point thus reducing formation and assembly effort significantly with respect to standard galerkin finite element methods we provide a detailed review of all components of the technology in the context of elastodynamics that is weighted residual formulation nodal basis functions on gauss lobatto quadrature points and symmetrization by averaging with the ultra weak formulation we quantify potential gains by comparing the computational efficiency of collocated and standard finite elements in terms of basic operation counts and timings our results show that collocation is significantly less expensive for problems dominated by the formation and assembly effort such as higher order elastostatic analysis furthermore we illustrate the potential of collocation for efficient higher order explicit dynamics throughout this work we advocate a straightforward implementation based on simple modifications of standard finite element codes we also point out the close connection to spectral element methods where many of the key ideas are already established copyright 2014 john wiley sons ltd

the robust inference for the cox proportional hazards model abstract we derive the asymptotic distribution of the maximum partial likelihood estimator for the vector of regression coefficients under a possibly misspecified cox proportional hazards model as in the parametric setting this estimator converges to a well defined constant vector in addition the random vector n 1 2 is asymptotically normal with mean 0 and with a covariance matrix that can be consistently estimated the newly proposed robust covariance matrix estimator is similar to the so called sandwich variance estimators that have been extensively studied for parametric cases for many misspecified cox models the asymptotic limit or part of it can be interpreted meaningfully in those circumstances valid statistical inferences about the corresponding covariate effects can be drawn based on the aforementioned asymptotic theory of and the related results for the score statistics extensive studies demonstrate that the proposed robust tests and interval estimation procedures

bifurcation analysis for phase transitions in superconducting rings with nonuniform thickness we perform an analytic study of the problem of transitions between normal and superconducting phases for a sample which encloses a magnetic flux as in the classic little parks experiment for a sample of uniform thickness the order parameter is uniform but even infinitesimal deviations from uniform thickness give rise to a mixed state which avoids enclosing the magnetic field the stability domain of this mixed state is a line segment in the magnetic field temperature plane delimited by two critical points the phase diagram contains several bifurcation lines which are systematically analyzed

an isoparametric spectral element method for solution of the navier stokes equations in complex geometry abstract an isoparametric spectral element technique is presented for solution of the two dimensional navier stokes equations in arbitrary curvy geometries the spatial discretization is described and the method is illustrated by solution of poisson s equation in a cylindrical annulus a fractional step navier stokes solver based on a discretely consistent poisson equation for the pressure is then presented and exponential convergence in space is demonstrated for the case of separated flow between eccentric rotating cylinders the technique is made efficient by the use of explicit collocation rather than galerkin operators for the convective terms and use of static condensation to solve the elliptic equations resulting from the stokes problem the algorithm is amenable to a high degree of parallelism in both multiprocessor and vector pipeline environments

a legendre spectral element model for sloshing and acoustic analysis in nearly incompressible fluids a new spectral finite element formulation is presented for modeling the sloshing and the acoustic waves in nearly incompressible fluids the formulation makes use of the legendre polynomials in deriving the finite element interpolation shape functions in the lagrangian frame of reference the formulated element uses gauss lobatto legendre quadrature scheme for integrating the volumetric stiffness and the mass matrices while the conventional gauss legendre quadrature scheme is used on the rotational stiffness matrix to completely eliminate the zero energy modes which are normally associated with the lagrangian fe formulation the numerical performance of the spectral element formulated here is examined by doing the inf sup test on a standard rectangular rigid tank partially filled with liquid the eigenvalues obtained from the formulated spectral element are compared with the conventional equally spaced node locations of the h type lagrangian finite element and the predicted results show that these spectral elements are more accurate and give superior convergence the efficiency and robustness of the formulated elements are demonstrated by solving few standard problems involving free vibration and dynamic response analysis with undistorted and distorted spectral elements and the obtained results are compared with available results in the published literature

survival analysis with error prone time varying covariates a risk set calibration approach occupational environmental and nutritional epidemiologists are often interested in estimating the prospective effect of time varying exposure variables such as cumulative exposure or cumulative updated average exposure in relation to chronic disease endpoints such as cancer incidence and mortality from exposure validation studies it is apparent that many of the variables of interest are measured with moderate to substantial error although the ordinary regression calibration approach is approximately valid and efficient for measurement error correction of relative risk estimates from the cox model with time independent point exposures when the disease is rare it is not adaptable for use with time varying exposures by re calibrating the measurement error model within each risk set a risk set regression calibration method is proposed for this setting an algorithm for a bias corrected point estimate of the relative risk using an rrc approach is presented followed by the derivation of an estimate of its variance resulting in a sandwich estimator emphasis is on methods applicable to the main study external validation study design which arises in important applications simulation studies under several assumptions about the error model were carried out which demonstrated the validity and efficiency of the method in finite samples the method was applied to a study of diet and cancer from harvard s health professionals follow up study hpfs

a preconditioner based on domain decomposition for h p finite element approximation on quasi uniform meshes the problem of solving the algebraic systems arising from the discretization of a symmetric elliptic boundary value problem using h p finite element methods in two dimensions is addressed a preconditioning technique similar to those of bramble pasciak and schatz math comp 47 1986 pp 103 135 based on domain decomposition also known as substructuring is developed the method is applicable to problems on general domains involving differential operators with quite general coefficients the algorithm reduces to that of bramble pasciak and schatz if linear elements are used it is shown that the condition number of the preconditioned system which determines the rate of convergence of the iterative method grows at most as 1 log 2 p 1 log 2 hp h where p is the polynomial degree h is the size of the elements and h is the size of the subdomains

on using the cox proportional hazards model with missing covariates summary we propose two methods for handling missing covariates in using the cox proportional hazards model the maximum partial likelihood estimator based only on study subjects having complete covariates does not utilise all available information also it is biased when the probability of missingness depends on the failure or censoring time our suggestion is to impute the conditional expectation of the statistic involving missing covariates given the available information the proposed method provides a consistent regression parameter estimator when the probability of missingness depends on the failure or censoring time as well as on the observed covariates also the proposed estimator is more efficient than the estimator suggested previously by lin ying 1993 when data are missing completely at random

regression calibration in failure time regression in this paper we study a regression calibration method for failure time regression analysis when data on some covariates are missing or mismeasured the method estimates the missing data based on the data structure estimated from a validation data set a random subsample of the study cohort in which covariates are always observed ordinary cox 1972 journal of the royal statistical society series b 34 187 220 regression is then applied to estimate the regression coefficients using the observed covariates in the validation data set and the estimated covariates in the nonvalidation data set the method can be easily implemented we present the asymptotic theory of the proposed estimator finite sample performance is examined and compared with an estimated partial likelihood estimator and other related methods via simulation studies where the proposed method performs well even though it is technically inconsistent finally we illustrate the method with a mouse leukemia data set

a spectral element methodology tuned to parallel implementations abstract we present in this paper a modification of the conforming spectral element method for the approximation of the poisson equation this new procedure allows the implementation on a parallel machine in an easier and faster way the resulting method is nonconforming but the numerical analysis shows that the error between the exact solution and the discrete solution is of the same asymptotic order as the error of the original conforming method

fully discrete hp finite elements fast quadrature a fully discrete hp finite element method fem is presented it combines the features of the standard hp fem conforming galerkin formulation variable order quadrature schemes geometric meshes static condensation and of the spectral element method special shape functions and spectral quadrature techniques the speed up relative to standard hp elements is analyzed in detail both theoretically and computationally

assessment of finite and spectral element shape functions for efficient iterative simulations of interior acoustics several shape function families currently used in the spectral element and the finite element method fem are reviewed widely known shape functions such as the conventional lagrange functions various p fem shapes and recent developments in the area of spectral methods are compared especially shape functions derived from bernstein polynomials are discussed due to their ease of formulation for almost any geometric entity their applicability and efficiency for acoustic simulations are assessed in particular the efficiency is measured in terms of performance of current krylov solvers

sparsity optimized high order finite element functions for h div on simplices this paper deals with conforming high order finite element discretizations of the vector valued function space h div in 2 and 3 dimensions a new set of hierarchic basis functions on simplices with the following two main properties is introduced when working with an affine simplicial triangulation first the divergence of the basis functions is l 2 orthogonal and secondly the l 2 inner product of the interior basis functions is sparse with respect to the polynomial degree p the construction relies on a tensor product based construction with properly weighted jacobi polynomials as well as an explicit splitting of the higher order basis functions into solenoidal and non solenoidal ones the basis is suited for fast assembling strategies the general proof of the sparsity result is done by the assistance of computer algebra software tools several numerical experiments document the proved sparsity patterns and practically achieved condition numbers for the parameter dependent div div problem even on curved elements a tremendous improvement in condition numbers is observed the precomputed mass and stiffness matrix entries in general form are available online

asymptotic error expansion and richardson extrapolation for linear fine elements the elliptic ritz projection with linear finite elements is shown to admit asymptotic error expansions on certain uniform meshes this justifies the application of richardson extrapolation for increasing the accuracy

estimating the parameters in the cox model when covariate variables are measured with error the cox proportional hazards model is commonly used to model survival data as a function of covariates because of the measuring mechanism or the nature of the environment covariates are often measured with error and are not directly observable a naive approach is to use the observed values of the covariates in the cox model which usually produces biased estimates of the true association of interest an alternative strategy is to take into account the error in measurement which may be carried out for the cox model in a number of ways we examine several such approaches and compare and contrast them through several simulation studies we introduce a likelihood based approach which we refer to as the semiparametric method and show that this method is an appealing alternative the methods are applied to analyze the relationship between survival and cd4 count in patients with aids

a p version finite element approach for two and three dimensional problems of the j2 flow theory with non linear isotropic hardening in this paper an implementation of a two and three dimensional p version approach to the j 2 flow theory with non linear isotropic hardening for small displacements and small strains is presented based on higher order quadrilateral and hexahedral element formulations a newton raphson iteration scheme combined with a radial return algorithm is applied to find approximate solutions for the underlying physically non linear model problem curved boundaries are taken care of with the blending function method allowing an accurate representation of geometry with only a few p elements numerical examples demonstrate that the p version supplies efficient and accurate approximations to this class of physically non linear problems

a risk set calibration method for failure time regression by using a covariate reliability sample regression parameter estimation in the cox failure time model is considered when regression variables are subject to measurement error assuming that repeat regression vector measurements adhere to a classical measurement model we can consider an ordinary regression calibration approach in which the unobserved covariates are replaced by an estimate of their conditional expectation given available covariate measurements however since the rate of withdrawal from the risk set across the time axis due to failure or censoring will typically depend on covariates we may improve the regression parameter estimator by recalibrating within each risk set the asymptotic and small sample properties of such a risk set regression calibration estimator are studied a simple estimator based on a least squares calibration in each risk set appears able to eliminate much of the bias that attends the ordinary regression calibration estimator under extreme measurement error circumstances corresponding asymptotic distribution theory is developed small sample properties are studied using computer simulations and an illustration is provided

on the use of three dimensional h and p version finite elements in solving vibration response problems abstract the potential for reliably solving vibration response problems by using h and p versions of the finite element fe method is investigated each fe model is based on full three dimensional 3 d displacement based continuum theory special attention is given to the ability to handle thin and or nearly incompressible elastic and viscoelastic materials steady state time harmonic problems in the low and mid frequency ranges are treated convergence studies are performed on plate like model problems with simply supported boundary conditions using a series elasto dynamic navier solution extended to the viscoelasto dynamic case by means of the elasto viscoelastic correspondence principle the importance of eliminating the discretisation error when using numerical solutions to estimate the frequency dependent viscoelastic material parameters from experiments is stressed the superior efficiency of p enrichments compared to h refinements for resolving regular elastostatic solutions is observed in elasto viscoelasto dynamics as well requirements on p enrichments to avoid shear and volumetric locking are given the frequently used polynomial interpolation functions employed are shown to produce a mass contribution to the dynamic stiffness whose condition number deteriorates with increasing polynomial order the practical implications and limitations of this observation are outlined the recommended approach is corroborated using the measured frequency response on a laminate consisting of two aluminium plates and a constrained viscoelastic polymer damping treatment

rational krylov sequence methods for eigenvalue computation abstract algorithms to solve large sparse eigenvalue problems are considered a new class of algorithms which is based on rational functions of the matrix is described the lanczos method the arnoldi method the spectral transformation lanczos method and rayleigh quotient iteration all are special cases but there are also new algorithms which correspond to rational functions with several poles in the simplest case a basis of a rational krylov subspace is found in which the matrix eigenvalue problem is formulated as a linear matrix pencil with a pair of hessenberg matrices

how to implement the spectral transformation the general linear eigenvalue equations h x m z 0 where h and m are real symmetric matrices with m positive semidefinite must be transformed if the lanczos algorithm is to be used to compute eigenpairs x z when the matrices are large and sparse but not diagonal some factorization must be performed as part of the transformation step if we are interested in only a few eigenvalues x near a specified shift then the spectral transformation of ericsson and ruhe 1 proved itself much superior to traditional methods of reduction the purpose of this note is to show that a small variant of the spectral transformation is preferable in all respects perhaps the lack of symmetry in our formulation deterred previous investigators from choosing it it arises in the use of inverse iteration a second goal is to introduce a systematic modification of the computed ritz vectors which improves the accuracy when m is ill conditioned or singular we confine our attention to the simple lanczos algorithm although the first two sections apply directly to the block algorithms as well 1 overview this contribution is an addendum to the paper by ericsson and ruhe 1 and also 7 the value of the spectral transformation is reiterated in a later section here we outline our implementation of this transformation the equation to be solved for an eigenvalue a and eigenvector z is 1 h xm z 0 h and m are real symmetric n x n matrices and m is positive semidefinite a practical instance of 1 occurs in dynamic analysis of structures where h and m are the stiffness and mass matrices respectively we assume that a linear combination of h and m is positive definite it then follows that all eigenvalues x are real in addition one has a real scalar a distinct from any eigenvalue and we seek a few eigenvalues a close to a together with their eigenvectors z ericsson and ruhe replace 1 by a standard eigenvalue equation 2 c h am ct v y 0 where c is the choleski factor of m m ctc and y cz if m is singular then so is c but fortunately the eigenvector z can be recovered from y via z h am icty of course there is no intention to invert h a m explicitly the received may 14 1984 revised december 20 1985 1980 mathematics subject classification primary 65f15 this research was supported in part by the afosr contract f49620 84 c 0090 the third author was also supported in part by the swedish natural science research council the paper was written while this author was visiting the center for pure and applied mathematics university of california berkeley california 94720 1987 american mathematical society 0025 5718 87 1 00 25 per page

bifurcation analysis of the eckhaus instability the eckhaus instability 1 is the only purely two dimensional instability occurring in convection 2 taylor couette flow 3 liquid crystals 4 and other pattern forming systems despite its widespread occurrence the eckhaus instability has not been analyzed in bifurcation theoretic terms it can be studied analytically by averaging over the depth 5 leading to the classic one dimensional ginzburg landau equation frac partial a partial t mu a frac partial 2 a partial x 2 left a right 2 a 1

a pseudo partial likelihood method for semiparametric survival regression with covariate errors this article presents an estimator for the regression coefficient vector in the cox proportional hazards model with covariate error the estimator is obtained by maximizing a likelihood type function similar to the cox partial likelihood the likelihood function involves the cumulative baseline hazard function for which a simple estimator is substituted the method is capable of handling general covariate error structures it is not restricted to the independent additive error model it can be applied to studies with either an external or internal validation sample and also to studies with replicate measurements of the surrogate covariate the estimator is shown to be consistent and asymptotically normal and an estimate of the asymptotic covariance matrix is derived some extensions to general transformation survival models are indicated simulation studies are presented for a setup with a single error prone binary covariate and a setup with a single error prone normally distributed covariate these si

finite element preconditioning for legendre spectral collocation approximations to elliptic equations and systems spectral collocation approximations based on legendre gauss lobatto lgl points for helmholtz equations as well as for the linear elasticity system in rectangular domains are studied the collocati

the multi level hp method for three dimensional problems dynamically changing high order mesh refinement with arbitrary hanging nodes abstract one main challenge of the h p version of the finite element method is the high implementational complexity of the method resulting from the added need of handling hanging nodes appropriately the multi level h p formulation recently introduced for two dimensional applications aims at alleviating these difficulties without compromising the approximation quality this is achieved by changing from the conventional refine by replacement approach to a refine by superposition idea the current work shows that the multi level h p approach can be extended naturally to three dimensional refinement without increasing the complexity of the rule set ensuring linear independence and compatibility of the shape functions in this way a three dimensional h p refinement scheme is formulated in which hanging nodes are avoided by definition this ease of complexity allows for a highly flexible discretization kernel featuring arbitrary irregular meshes and a continuous refinement and coarsening throughout the simulation runtime different numerical examples demonstrate that even in the presence of singularities this novel refinement scheme yields exponential convergence with respect to both the number of unknowns and the computational time it is further shown that the refinement scheme is able to capture complex solution features that demand for three dimensional refinement patterns the dynamic discretization properties of the approach are demonstrated by continuously refining and coarsening the mesh during the simulation runtime to keep the refinement zone local to a moving singularity finally it is shown that the high approximation power of the multi level h p scheme also carries over to curved geometries common in engineering practice without a significant detrimental effect on the conditioning of the stiffness matrix

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

global bifurcation structure of a one dimensional ginzburg landau model we consider an equation of a simplified ginzburg landau model of superconductivity in a one dimensional ring the equation for a complex order parameter has two real parameters and related to the magnitude of an applied magnetic field and the ginzburg landau parameter respectively the purpose of this paper is to reveal a global bifurcation structure for the equation in the parameter space in particular we show that there exist modulating amplitude solutions which bifurcate from constant amplitude solutions and how the bifurcation branches of such solutions continue or disappear as varies we also determine the minimizer of the energy functional

semiparametric modeling of longitudinal measurements and time to event data a two stage regression calibration approach summary in this article we investigate regression calibration methods to jointly model longitudinal and survival data using a semiparametric longitudinal model and a proportional hazards model in the longi tudinal model a biomarker is assumed to followa semiparametric mixed model where covariate effects are modeled parametrically and subject specific time profiles are modeled nonparametrially using a population smoothing spline and subject specific random stochastic processes the cox model is assumed for survival data by including both the current measure and the rate of change of the underlying longitudinal trajectories as covariates as motivated by a prostate cancer study application we develop a two stage semiparametric regression calibration rc method two variations of the rc method are considered risk set regression calibration and a computationally simpler ordinary regression calibration simulation results showthat the two stage rc approach performs well in practice and effectively corrects the bias from the naive method we apply the proposed methods to the analysis of a dataset for evaluating the effects of the longitudinal biomarker psa on the recurrence of prostate cancer

new shape functions for triangular p fem using integrated jacobi polynomials in this paper the second order boundary value problem x y u f is discretized by the finite element method using piecewise polynomial functions of degree p on a triangular mesh on the reference element we define integrated jacobi polynomials as interior ansatz functions if is a constant function on each triangle and each triangle has straight edges we prove that the element stiffness matrix has not more than nonzero matrix entries an application for preconditioning is given numerical examples show the advantages of the proposed basis

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

cox regression with accurate covariates unascertainable a nonparametric correction approach abstract many survival studies involve covariates that are not accurately ascertainable cd4 lymphocyte count in hiv aids research is a typical example for which the gold standard of the measurement is not available this article proposes a consistent estimation procedure for cox regression under the additive measurement error model distinct from existing methods the proposed estimation for regression coefficients does not require any additional assumptions we establish the normalized partial score function as a functional of empirical processes which facilitates the construction of an estimating function with the same limit using replicated mismeasured covariates the resulting regression coefficient estimators are shown to be consistent and asymptotically normal a consistent sandwich variance estimate is presented we also suggest estimators for the baseline cumulative hazard function numerical studies demonstrate that the procedure performs well under practical sample sizes application to an aid

auxiliary covariate data in failure time regression summary we consider the problem of missing covariate data in the context of censored failure time relative risk regression auxiliary covariate data which are considered informative about the missing data but which are not explicitly part of the relative risk regression model may be available full covariate information is available for a validation set an estimated partial likelihood method is proposed for estimating relative risk parameters this method is an extension of the estimated likelihood regression analysis method for uncensored data pepe 1992 pepe fleming 1991 a key feature of the method is that it is nonparametric with respect to the association between the missing and observed including auxiliary covariate components asymptotic distribution theory is derived for the proposed estimated partial likelihood estimator in the case where the auxiliary or mismeasured covariates are categorical asymptotic efficiencies are calculated for exponential failure times using an exponential relative risk model the estimated partial likelihood estimator compares favourably with a fully parametric maximum likelihood analysis comparisons are also made with a standard partial likelihood analysis which ignores the incomplete observations important efficiency gains can be made with the estimated partial likelihood method small sample properties are investigated through simulation studies

bifurcation and stability of periodic traveling waves for a reaction diffusion system v plane having a stable critical point at 1 o and unstable critical points at 0 o and cos sin moreover if we take 0 rc then flows starting with 3 i rc make large excursions before decaying to 1 0 whereas those with 0 decay monotonically to l o this excitability is a feature in common with the nonlinear vector field in the hodgkin huxley equations and also with the simpler fitzhugh nagumo equations we begin by describing some results of numerical simulations we made of

parallel domain decomposition solver for adaptive hp finite element methods in this paper the development and implementation of highly parallelizable domain decomposition solvers for adaptive hp finite element methods is discussed two level orthogonalization is used to obtain a reduced system which is preconditioned by a coarse grid operator the condition number of the preconditioned system for poisson problems in two space dimensions is proved to be bounded by c 1 hp h 2 1 log p 2 and cp 1 hp h 2 1 log p 2 for different choices of coarse grid operators where h is the subdomain size p is the maximum spectral order h is the size of the smallest element in the subdomain and c is a constant independent of the mesh parameters the work here extends the work of bramble et al math comp 47 1986 pp 103 134 on the h version and babuska et al siam j numer anal 29 1991 pp 624 661 on the p version of the finite element method a preliminary version of this solver was first announced by oden patra and feng in domain decomposition solver for adaptive hp finite elements vii conference on domain decomposition state college pa october 1993 numerical experiments show fast convergence of the solver and good control of the condition number on a variety of discretizations

cox regression with dependent error in covariates many survival studies have error contaminated covariates due to the lack of a gold standard of measurement furthermore the error distribution can depend on the true covariates but the structure may be difficult to characterize heteroscedasticity is a common manifestation we suggest a novel dependent measurement error model with minimal assumptions on the dependence structure and propose a new functional modeling method for cox regression when an instrumental variable is available this proposal accommodates much more general error contamination than existing approaches including nonparametric correction methods of huang and wang 2000 journal of the american statistical association 95 1209 1219 2006 statistica sinica 16 861 881 the estimated regression coefficients are consistent and asymptotically normal and a consistent variance estimate is provided for inference simulations demonstrate that the procedure performs well even under substantial error contamination illustration with a clinical study is provided

fast computation of high frequency dirichlet eigenmodes via spectral flow of the interior neumann to dirichlet map we present a new algorithm for numerical computation of large eigenvalues and associated eigenfunctions of the dirichlet laplacian in a smooth star shaped domain in d d 2 conventional boundary based methods require a root search in eigenfrequency k hence take o n3 effort per eigenpair found where n o kd 1 is the number of unknowns required to discretize the boundary our method is o n faster achieved by linearizing with respect to k the spectrum of a weighted interior neumann to dirichlet ntd operator for the helmholtz equation approximations to the square roots kj of all o n eigenvalues lying in k k where o 1 are found with o n3 effort we prove an error estimate with c independent of k we present a higher order variant with eigenvalue error scaling empirically as o 5 and eigenfunction error as o 3 the former improving upon the scaling method of vergini and saraceno for planar domains d 2 with an assumption of absence of spectral concentration we also prove rigorous error bounds that are close to those numerically observed for d 2 we compute robustly the spectrum of the ntd operator via potential theory nystrom discretization and the cayley transform at high frequencies 400 wavelengths across with eigenfrequency relative error 10 10 we show that the method is 103 times faster than standard ones based upon a root search 2014 wiley periodicals inc

non dissipative space time hp discontinuous galerkin method for the time dependent maxwell equations a finite element method for the solution of the time dependent maxwell equations in mixed form is presented the method allows for local hp refinement in space and in time to this end a space time galerkin approach is employed in contrast to the space time dg method introduced in 1 test and trial spaces do not coincide this allows for obtaining a non dissipative method to obtain an efficient implementation a hierarchical tensor product basis in space and time is proposed this allows to evaluate the local residual with a complexity of o p 4 and o p 5 for affine and non affine elements respectively