primal dual interior point methods for semidefinite programming convergence rates stability and numerical results primal dual interior point path following methods for semidefinite programming are considered several variants are discussed based on newton s method applied to three equations primal feasibility dual feasibility and some form of centering condition the focus is on three such algorithms called the xz xz zx and q methods for the xz zx and q algorithms the newton system is well defined and its jacobian is nonsingular at the solution under nondegeneracy assumptions the associated schur complement matrix has an unbounded condition number on the central path under the nondegeneracy assumptions and an additional rank assumption practical aspects are discussed including mehrotra predictor corrector variants and issues of numerical stability compared to the other methods considered the xz zx method is more robust with respect to its ability to step close to the boundary converges more rapidly and achieves higher accuracy

some convergence estimates for semidiscrete type schemes for time dependent nonselfadjoint parabolic equations l2 norm error estimates are shown for semidiscrete continuous in time galerkin finite element type approximations to solutions of general time dependent nonselfadjoint second order parabolic equations under dirichlet boundary conditions the semidiscrete solutions are defined in terms of given methods for the corresponding elliptic problem such as the standard galerkin method in which the boundary conditions are satisfied exactly but also methods for which this is not necessary the results are proved by energy arguments and include estimates for the homogeneous equation with both smooth and nonsmooth initial data

local polynomial variance function estimation the conditional variance function in a heteroscedastic nonparametric regression model is estimated by linear smoothing of squared residuals attention is focused on local polynomial smoothers both the mean and variance functions are assumed to be smooth but neither is assumed to be in a parametric family the biasing effect of preliminary estimation of the mean is studied and a degrees of freedom correction of bias is proposed the corrected method is shown to be adaptive in the sense that the variance function can be estimated with the same asymptotic mean and variance as if the mean function were known a proposal is made for using standard bandwidth selectors for estimating both the mean and variance functions the proposal is illustrated with data from the lidar method of measuring atmospheric pollutants and from turbulence model computations

lorelogram a regression approach to exploring dependence in longitudinal categorical responses abstract we propose flexible regression estimators of the marginal pairwise log odds ratio measure of association for longitudinal categorical responses the function that we estimate is the log odds ratio analog of the correlogram hence we name the function the lorelogram measuring the association of categorical responses on the log odds scale allows ease of interpretation and allows pairwise association to remain unconstrained by the marginal means a feature not shared by correlations with binary or multinomial responses estimation of the function is achieved through the use of standard parametric estimating equations or through an extension of generalized additive models that allows nonparametric estimation of dependence functions for fixed smoothing parameters we apply the methodology to binary longitudinal data where scientific interest focuses on the dependence structure

an interior point constrained trust region method for a special class of nonlinear semidefinite programming problems in this paper an interior point trust region algorithm for the solution of a class of nonlinear semidefinite programming sdp problems is described and analyzed such nonlinear and nonconvex programs arise e g in the design of optimal static or reduced order output feedback control laws and have the structure of abstract optimal control problems in a finite dimensional hilbert space the algorithm treats the abstract states and controls as independent variables in particular an algorithm for minimizing a nonlinear matrix objective functional subject to a nonlinear sdp condition a positive definiteness condition and a nonlinear matrix equation is considered the algorithm is designed to take advantage of the structure of the problem it is an extension of an interior point trust region method to nonlinear and nonconvex sdps with a special structure which applies sequential quadratic programming techniques to a sequence of barrier problems and uses trust regions to ensure robustness of the iteration some convergence results are given and finally several numerical examples demonstrate the applicability of the considered algorithm

convergence estimates for semidiscrete parabolic equation approximations semidiscrete methods for approximating the solutions of initial boundary value problems for parabolic equations are studied the construction of these semidiscrete methods is based upon the availability of several different galerkin type approximation methods for the associated elliptic steadystate problem the properties required of the spacial discretization methods are listed and estimates of the error made by the resulting semidiscrete approximations and of their time derivatives are given in particular estimates are given that require only weak smoothness assumptions on the initial data verifications of the required properties for various galerkin type methods are provided

graphical model checking with correlated response data correlated response data arise often in biomedical studies the generalized estimation equation gee approach is widely used in regression analysis for such data however there are few methods available to check the adequacy of regression models in gee in this paper a graphical method is proposed based on cook and weisberg s marginal model plot a bootstrap method is applied to obtain the reference band to assess statistical uncertainties in comparing two marginal mean functions we also propose using the generalized additive model gam in a similar fashion the proposed two methods are easy to implement by taking advantage of existing smoothing and gam softwares for independent data the usefulness of the methodology is demonstrated through application to a correlated binary data set drawn from a clinical trial the lung health study

cosine methods for second order hyperbolic equations with time dependent coefficients we analyze efficient high order accurate methods for the approximation of the solutions of linear second order hyperbolic equations with time dependent coefficients the methods are based on galerkin type discretizations in space and on a class of fourth order accurate two step cosine time stepping schemes preconditioned iterative techniques are used to solve linear systems with the same operator at each time step the schemes are supplemented by single step high order starting procedures and need no evaluations of derivatives of operators l2 optimal error estimates are proved throughout

higher order single step fully discrete approximations for second order hyperbolic equations with time dependent coefficients error estimates are proved for finite element approximations to the solution of an initial boundary value problem for a second order hyperbolic partial differential equation with time dependent coefficients optimal order rates of convergence in l 2 are shown for single step fully discrete schemes the order nu of the single step schemes is greater than or equal to one and less than or equal to four a class of iterative methods for approximately solving the fully discrete equations is analyzed and optimal order rates of convergence are also obtained these methods avoid the repeated work of new matrix factorizations at each time level

error estimates for semidiscrete galerkin type approximations to semilinear evolution equations with nonsmooth initial data error estimates for the semidiserete galerkin method for abstract semilinear evolution equations with non smooth initial data are given in concrete cases almost optimal order of convergence for linear finite elements results

galerkin runge kutta discretizations for parabolic equations with time dependent coefficients a new class of fully discrete galerkin runge kutta methods is constructed and analyzed for linear parabolic initial boundary value problems with time dependent coefficients unlike any classical counterpart this class offers arbitrarily high order convergence while significantly avoiding what has been called order reduction in support of this claim error estimates are proved and computational results are presented additionally since the time stepping equations involve coefficient matrices changing at each time step a preconditioned iterative technique is used to solve the linear systems only approximately nevertheless the resulting algorithm is shown to preserve the original convergence rate while using only the order of work required by the base scheme applied to a linear parabolic problem with time independent coefficients furthermore it is noted that special runge kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method

numerical approximations for a miscible displacement process in porous media a reservoir simulation problem describing a miscible displacement process is stated and analyzed and a semidiscrete numerical scheme based on a galerkin method with mesh refinements is presented for its solution the underlying physical problem involves two miscible incompressible fluids of equal viscosity flowing in a porous medium injection and production wells are rigorously modelled which leads to a formulation containing nearly singular source and sink terms the singularities arising in the solution are described in some detail and it is shown how knowledge of their forms motivate the choice of an appropriate mesh refinement for the numerical scheme optimal order quadratic convergence is shown for the resulting semidiscrete approximation using piecewise linear elements

vector generalized additive models vector smoothing is used to extend the class of generalized additive models in a very natural way to include a class of multivariate regression models the resulting models are called vector generalized additive models the class of models for which the methodology gives generalized additive extensions includes the multiple logistic regression model for nominal responses the continuation ratio model and the proportional and non proportional odds models for ordinal responses and the bivariate probit and bivariate logistic models for correlated binary responses they may also be applied to generalized estimating equations

varying coefficient models we explore a class of regression and generalized regression models in which the coefficients are allowed to vary as smooth functions of other variables general algorithms are presented for estimating the models flexibly and some examples are given this class of models ties together generalized additive models and dynamic generalized linear models into one common framework when applied to the proportional hazards model for survival data this approach provides a new way of modelling departures from the proportional hazards assumption

convergence of runge kutta approximations for parabolic problems with neumann boundary conditions we analyze a class of algebraically stable runge kutta standard galerkin methods for inhomogeneous linear parabolic equations with time dependent coefficients under neumann boundary conditions and derive an l 2 error bound of o h r k min nu p 3 over2 provided frac k h is bounded

estimating equations for parameters in means and covariances of multivariate discrete and continuous responses generalized estimating equations are introduced in an ad hoc fashion for the covariance matrix of a multivariate response these equations are to be solved jointly with score equations from a generalized linear model for mean parameters a class of quadratic exponential models is used to develop joint estimating equations for mean and covariance parameters in a more systematic fashion and proposals for the use of such equations are developed comments on the relative merits of the ad hoc and model based approaches to estimation are given and a regression illustration with a bivariate response is provided

semidiscrete and single step fully discrete approximations for second order hyperbolic equations with time dependent coefficients l2 norm error estimates are proved for finite element approximations to the solutions of initial boundary value problems for second order hyperbolic partial differential equations with time dependent coefficients optimal order rates of convergence are shown for semidiscrete and single step fully discrete schemes using specially constructed initial data the initial data are designed so that the data used for the fully discrete equation is reasonable to compute and so that the optimal order estimates can be proved

on the nesterov todd direction in semidefinite programming we study different choices of search direction for primal dual interior point methods for semidefinite programming problems one particular choice we consider comes from a specialization of a class of algorithms developed by nesterov and todd for certain convex programming problems we discuss how the search directions for the nesterov todd nt method can be computed efficiently and demonstrate how they can be viewed as newton directions this last observation also leads to convenient computation of accelerated steps using the mehrotra predictor corrector approach in the nt framework we also provide an analytical and numerical comparison of several methods using different search directions and suggest that the method using the nt direction is more robust than alternative methods

goodness of fit tests for gee modeling with binary responses analysis of data with repeated measures is often accomplished through the use of generalized estimating equations gee methodology although methods exist for assessing the adequacy of the fitted models for uncorrelated data with likelihood methods it is not appropriate to use these methods for models fitted with gee methodology we propose model based and robust empirically corrected goodness of fit tests for gee modeling with binary responses based on partitioning the space of covariates into distinct regions and forming score statistics that are asymptotically distributed as chi square random variables with the appropriate degrees of freedom the null distribution and the statistical power of the proposed goodness of fit tests were assessed using simulated data the proposed goodness of fit tests are illustrated by two examples using data from clinical studies

two numerical methods for optimizing matrix stability consider the affine matrix family a x a0 m 1 xkak mapping a design vector x r m into the space of n n real matrices we are interested in the question of how to choose x to optimize the stability of the matrix a x a typical motivation is that one wishes to control the stability of the dynamical system a x z a classic example is stabilization by output feedback we take two approaches let x denote the spectral abscissa the largest real part of the eigenvalues of a matrix x as is well known this quantity bounds the asymptotic decay rate of the trajectories of the associated dynamical system z xz our first approach to optimizing stability is to directly minimize the function a x the spectral abscissa x is a continuous but non smooth in fact non lipschitz function of the matrix argument x and finding a global minimizer of a x is hard we introduce a novel random gradient bundle method for approximating local minimizers motivated by recent work on nonsmooth analysis of the function x our second approach is to minimize a related function a x where is a robustness parameter in 0 1 one motivation for the definition of the robust spectral abscissa x is that it bounds transient peaks as well as asymptotic decay of trajectories of xz the function x is lipschitz but non convex for 0 1

some new error estimates of a semidiscrete finite volume element method for a parabolic integro differential equation with nonsmooth initial data a semidiscrete finite volume element fve approximation to a parabolic integro differential equation pide is analyzed in a two dimensional convex polygonal domain an optimal order l 2 error estimate for smooth initial data and nearly the same optimal order l 2 error estimate for nonsmooth initial data are obtained more precisely for homogeneous equations an elementary energy technique and a duality argument are used to derive an error estimate of order o left t 1 h 2 ln h right in the l 2 norm for positive time when the given initial function is only in l 2

graphics for assessing the adequacy of regression models abstract graphical paradigms for assessing the adequacy of nearly any regression model are discussed the fundamental idea is to examine the fit of a model using a sequence of marginal model plots on each plot nonparametric estimates of fit derived from the model are compared to nonparametric fits based on the observed data several examples are given

fully discrete galerkin approximations of parabolic boundary value problems with nonsmooth boundary data in this paper we study the convergence properties of a fully discrete galerkin approximation with a backwark euler time discretization scheme an approach based on semigroup theory is used to deal with the nonsmooth dirichlet boundary data which cannot be handled by standard techniques this approach gives rise to optimal rates of convergence inl p o t l 2 norms for boundary conditions inl p o t l 2 1 p

galerkin approximations of abstract parabolic boundary value problems with rough boundary data l p theory etude des approximations de galerkin d un probleme aux valeurs limites parabolique abstrait avec des donnees aux limites grossieres determination des vitesses de convergence optimales dans les normes lp

higher order single step fully discrete approximations for nonlinear second order hyperbolic equations abstract error estimates are proved for finite element approximations to the solution of an initial boundary value problem for nonlinear second order hyperbolic partial differential equations optimal order rates of convergence in l 2 are shown for single step fully discrete schemes which are linearized by extrapolations the order of accuracy in time is 2 3 or 4 a class of iterative methods for approximately solving the fully discrete equations is analyzed and optimal order rates of convergence are also obtained

additive extensions to generalized estimating equation methods the methods underlying vector generalized additive models are extended to provide additive extensions to the generalized estimating equations approaches to multivariate regression problems of liang and zeger and the subsequent literature the methods are illustrated by two examples using correlated binary data concerning the presence or absence of several plant species at the same geographical site and chest pain in male workers from a workforce study with minor modification the extensions are shown to apply to longitudinal data

nonparametric smoothing estimates of time varying coefficient models with longitudinal data this paper considers nonparametric estimation in a varying coefficient model with repeated measurements y ij x ij t ij for i 1 n and j 1 n i where x ij x ij x ijk t and y ij x ij t ij denote the jth outcome covariate and time design points respectively of the ith subject the model considered here is y ij x ij t t ij e i t ij where t 0 t k t t for k 0 are smooth nonparametric functions of interest and e i t is a zero mean stochastic process the measurements are assumed to be independent for different subjects but can be correlated at different time points within each subject two nonparametric estimators of t namely a smoothing spline and a locally weighted polynomial are derived for such repeatedly measured data a crossvalidation criterion is proposed for the selection of the corresponding smoothing parameters asymptotic properties such as consistency rates of convergence and asymptotic mean squared errors are established for kernel estimators a special case of the local polynomials these asymptotic results give useful insights into the reliability of our general estimation methods an example of predicting the growth of children born to hiv infected mothers based on gender hiv status and maternal vitamin a levels shows that this model and the corresponding nonparametric estimators are useful in epidemiological studies

error estimates for semidiscrete finite element approximations of linear and semilinear parabolic equations under minimal regularity assumptions semidiscrete finite element error estimates for linear parabolic equations are derived under minimal regularity with the help of l2 projectors then analogous minimal regularity semidiscrete error estimates for semilinear parabolic equations are derived

goodness of fit for gee an example with mental health service utilization suppose we use generalized estimating equations to estimate a marginal regression model for repeated binary observations there are no established summary statistics available for assessing the adequacy of the fitted model in this paper we propose a goodness of fit test statistic which has an approximate chi squared distribution when we have specified the model correctly the proposed statistic can be viewed as an extension of the hosmer and lemeshow goodness of fit statistic for ordinary logistic regression to marginal regression models for repeated binary responses we illustrate the methods using data from a study of mental health service utilization by children the repeated responses are a set of binary measures of service use we fit a marginal logistic regression model to the data using generalized estimating equations and we apply the proposed goodness of fit statistic to assess the adequacy of the fitted model

locally weighted regression an approach to regression analysis by local fitting abstract locally weighted regression or loess is a way of estimating a regression surface through a multivariate smoothing procedure fitting a function of the independent variables locally and in a moving fashion analogous to how a moving average is computed for a time series with local fitting we can estimate a much wider class of regression surfaces than with the usual classes of parametric functions such as polynomials the goal of this article is to show through applications how loess can be used for three purposes data exploration diagnostic checking of parametric models and providing a nonparametric regression surface along the way the following methodology is introduced a a multivariate smoothing procedure that is an extension of univariate locally weighted regression b statistical procedures that are analogous to those used in the least squares fitting of parametric functions c several graphical methods that are useful tools for understanding loess estimates and checking the a

runge kutta nystrom methods for hyperbolic problems with time dependent coefficients abstract galerkin fully discrete approximations for hyperbolic equations with time dependent coefficients are analyzed the schemes are based on implicit runge kutta nystrom methods and are coupled with preconditioned iterative methods for the efficient albeit approximate solution of the resulting linear systems arbitrarily high order both conditionally and unconditionally stable methods are considered in addition it is shown that a specific class of methods exhibit parallel features that can be exploited to reduce the final execution time to that of a low order method

error estimates for the semidiscrete finite element approximation of a damped wave equation with a smoothing property error estimates are given for the semidiscrete finite element approximation of an inhomogeneous wave equation damped by a constant times the positive square root of the elliptic operator in the wave equation times the time derivative of the solution the damping term smooths the solution the smoothing property is used to obtain higher order error estimates for positive time even when the initial data are non smooth

maximum norm analysis of completely discrete finite element methods for parabolic problems we consider full discretizations of linear parabolic initial value problems in any space dimension finite elements are used for the discretization in space the resulting semidiscrete problem is integrated in time by means of a strongly a theta acceptable rational method stability estimates are derived and error bounds are established in the maximum norm for both smooth and rough initial data

differentiability and convergence rates of approximating semigroups for retarded functional differential equations in this paper the averaging approximations i e semidiscrete finite difference scheme for linear retarded functional differential equations are considered in the context of the semigroups on the hilbert space r n times l 2 it is shown that the approximating semigroups are all differentiable uniformly with respect to the index n determining the mesh size this fact makes it possible to prove the existence of a uniform exponential decay or growth rate for the approximating semigroups which can be made arbitrarily close to the decay respectively growth rate of the original semigroup more importantly the uniform differentiability makes it possible to establish convergence rates that improve with time and to prove the convergence of the approximating semigroups in the uniform operator topology the dependence of the convergence rate on the initial conditions and system parameters is also established

finite element solution of diffusion problems with irregular data diffusion problems occuring in practice often involve irregularities in the initial or boundary data resulting in a local break down of the solution s regularity this may drastically reduce the accuracy of discretization schemes over the whole interval of integration unless certain precautions are taken the diagonal pade schemes of order 2 combined with a standard finite element discretization usually require an unnatural step size restriction in order to achieve even locally optimal accuracy it is shown here that this restriction can be avoided by means of a sample damping procedure which preserves the order of the discretization and in the case 1 does not increase the costs

models for longitudinal data a generalized estimating equation approach this article discusses extensions of generalized linear models for the analysis of longitudinal data two approaches are considered subject specific ss models in which heterogeneity in regression parameters is explicitly modelled and population averaged pa models in which the aggregate response for the population is the focus we use a generalized estimating equation approach to fit both classes of models for discrete and continuous outcomes when the subject specific parameters are assumed to follow a gaussian distribution simple relationships between the pa and ss parameters are available the methods are illustrated with an analysis of data on mother s smoking and children s respiratory disease

nonparametric function estimation for clustered data when the predictor is measured without with error abstract we consider local polynomial kernel regression with a single covariate for clustered data using estimating equations we assume that at most m observations are available on each cluster in the case of random regressors with no measurement error in the predictor we show that it is generally the best strategy to ignore entirely the correlation structure within each cluster and instead pretend that all observations are independent in the further special case of longitudinal data on individuals with fixed common observation times we show that equivalent to the pooled data approach is the strategy of fitting separate nonparametric regressions at each observation time and constructing an optimal weighted average we also consider what happens when the predictor is measured with error using the simex approach to correct for measurement error we construct an asymptotic theory for both the pooled and the weighted average estimators surprisingly for the same amount of smoothing the weighted a

longitudinal data analysis using generalized linear models summary this paper proposes an extension of generalized linear models to the analysis of longitudinal data we introduce a class of estimating equations that give consistent estimates of the regression parameters and of their variance under mild assumptions about the time dependence the estimating equations are derived without specifying the joint distribution of a subject s observations yet they reduce to the score equations for multivariate gaussian outcomes asymptotic theory is presented for the general class of estimators specific cases in which we assume independence m dependence and exchangeable correlation structures from each subject are discussed efficiency of the proposed estimators in two simple situations is considered the approach is closely related to quasi likelih ood some key ironh estimating equation generalized linear model longitudinal data quasi likelihood repeated measures

from finite differences to finite elements a short history of numerical analysis of partial differential equations abstract this is an account of the history of numerical analysis of partial differential equations starting with the 1928 paper of courant friedrichs and lewy and proceeding with the development of first finite difference and then finite element methods the emphasis is on mathematical aspects such as stability and convergence analysis