missing data dial m for the question mark is common notation for the missing data that occur in most applied statistical analyses over the past century statisticians and other scientists not only have invented numerous methods for handling missing incomplete data but also have invented many forms of missing data including data augmentation hidden states latent variables potential outcome and auxiliary variables purposely constructing unobserved unobservable variables offers an extraordinarily flexible and powerful framework for both scientific modeling and computation and is one of the central statistical contributions to natural engineering and social sciences in parallel much research has been devoted to better understanding and modeling of real life missing data mechanisms that is the unintended data selection process that prevents us from observing our intended data this article is a very brief and personal tour of these developments and thus necessarily has much missing history and citations the tour consists of a number of m s starting with a historic story of the mysterious method of mckendrick for analyzing an epidemic study and its link to the em algorithm the most popular and powerful method of the twentieth century for fitting models involving missing data and latent variables the remaining m s touch on theoretical methodological and practical aspects of missingdata problems highlighted with some common applications in social computational biological medical and physical sciences

gmres implementations and residual smoothing techniques for solving ill posed linear systems there are verities of useful krylov subspace methods to solve nonsymmetric linear system of equations gmres is one of the best krylov solvers with several different variants to solve large sparse linear systems any gmres implementation has some advantages as the solution of ill posed problems are important in this paper some gmres variants are discussed and applied to solve these kinds of problems residual smoothing techniques are efficient ways to accelerate the convergence speed of some iterative methods like cg variants at the end of this paper some residual smoothing techniques are applied for different gmres methods to test the influence of these techniques on gmres implementations

composite linear models for incomplete multinomial data a composite linear model clm is a matrix model for incomplete multinomial data a clm provides a unified approach for maximum likelihood inference which is applicable to a wide variety of problems involving incomplete multinomial data by formulating a model as a clm one can simplify computation of maximum likelihood estimates and asymptotic standard errors as an example we use clm to test marginal homogeneity for ordered categories subject to both ignorable and non ignorable missing data mechanisms

two grid analysis of minimal residual smoothing as a multigrid acceleration technique we analyze the two level method accelerated by a minimal residual smoothing mrs technique the two grid analysis is sufficient for our purpose because our mrs acceleration scheme is only applied on the finest level of the multigrid method we prove that the mrs acceleration scheme is a semi iterative method with respect to the underlying two level iteration and that the mrs accelerated two level method is a polynomial acceleration of first order we explain why mrs may not effectively accelerate standard multigrid method for solving poisson like problems the iteration matrices for the mrs accelerated coarse grid correction operator and the mrs accelerated two level operator are obtained we give bounds for the residual reduction rates of the accelerated two level method numerical experiments are employed to support the analytical results

a note on convergence of quasi minimal residual smoothing in this paper we give convergence result on a quasi minimal residual qmr smoothing approach int j comput math 62 1996 249 appl numer math to appear for solving nonsymmetric linear systems

on the stable implementation of the generalized minimal error method the paper reviews several implementations of the generalized minimal error method gmerr method for solving nonsymmetric systems of linear equations that minimize the euclidean norm of the error in the related generalized krylov subspace we show the relation to the methods in the symmetric indefinite case a new variant of the gmerr method is proposed and the stable implementation based on the householder transformations is discussed numerical stability of the most frequent implementations is analyzed and the theoretical results are illustrated by numerical examples

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

on a variable smoothing procedure for krylov subspace methods abstract it is known that the convergence behavior of galerkin krylov subspace methods for solving linear systems can be very erratic a smoothing technique or a minimal residual seminorm variant of these galerkin methods can be proposed to eliminate this problem in this paper we examine a class of minimal residual seminorm methods and show that this class of methods can be obtained from a variable smoothing technique applied to galerkin methods

a quasi minimal residual variant of idrstab using the residual smoothing technique abstract quasi minimal residual qmr type methods have been developed for improving the oscillations in the residual norms of the bi conjugate gradient bi cg method and the hybrid bi cg methods the idrstab method which combines the induced dimension reduction idr s method with higher order stabilizing polynomials is often more effective than idr s and the hybrid bi cg methods but the residual norms oscillate in this paper we design a qmr variant of idrstab by using a residual smoothing technique in order to obtain smoother convergence behavior numerical experiments show the effectiveness of our qmr variant of idrstab

three dimensional induction logging problems part i an integral equation solution and model comparisons a 3 d frequency domain solution based on a volume integral equation approach has been implemented to simulate induction log responses in our treatment of the problem we assume that the electrical properties of the bedding as well as the borehole and invasion zones can exhibit transverse anisotropy the solution process uses a krylov subspace iteration to solve the scattering equation which is based on the modified iterative dissipative method internal consistency checks and comparisons with mode matching and finite difference solutions for vertical borehole models demonstrate the accuracy of the solution there are no known analytical solutions for induction log responses arising from deviated boreholes intersecting horizontal bed boundaries to simulate such responses requires the numerical solution of maxwell s equations in three dimensions along with independent tests to validate the solution approach and its accuracy in this paper we compare two independent 3 d frequency domain solutions for the

acceleration and stabilization properties of minimal residual smoothing technique in multigrid we analyze the standard multigrid method accelerated by a minimal residual smoothing mrs technique we show that mrs can accelerate the convergence of the slow residual components thus accelerates the overall multigrid convergence we prove that under certain hypotheses mrs stabilizes the divergence of certain slow residual components and thus stabilizes the divergent multigrid iteration the analysis is customarily conducted on the two level method

twisted alexander norms give lower bounds on the thurston norm we introduce twisted alexander norms of a compact connected orientable 3 manifold with first betti number greater than one generalizing norms of mcmullen and turaev we show that twisted alexander norms give lower bounds on the thurston norm of a 3 manifold using these we completely determine the thurston norm of many 3 manifolds which are not determined by norms of mcmullen and turaev

minimal residual smoothing in multi level iterative method a minimal residual smoothing mrs technique is employed to accelerate the convergence of the multi level iterative method by smoothing the residuals of the original iterative sequence the sequence with smoothed residuals is re introduced into the multi level iterative process the new sequence generated by this acceleration procedure converges much faster than both the sequence generated by the original multi level method and the sequence generated by mrs technique the cost of this acceleration scheme is independent of the original operator and in many cases is negligible the emphasis of this paper is on the practical implementation of mrs acceleration techniques in the multi level method the discussions are focused on the two level method because the acceleration scheme is only applied on the finest level of the multi level method numerical experiments using the proposed mrs acceleration scheme to accelerate both the two level and multi level methods are conducted to show the efficiency and the cost effectiveness of this acceleration scheme

regression estimation and post stratification in factor analysis regression estimation and poststratification are methods used in survey sampling to estimate a population mean when additional information is available for some auxiliary variables the extension of these methods to factor analysis is considered these methods may be used either to improve the efficiency of estimation or to adjust for the effects of nonrandom selection the estimation procedure may be formulated in a lisrel framework

adjusting for nonignorable drop out using semiparametric nonresponse models abstract consider a study whose design calls for the study subjects to be followed from enrollment time t 0 to time t t at which point a primary endpoint of interest y is to be measured the design of the study also calls for measurements on a vector v t of covariates to be made at one or more times t during the interval 0 t we are interested in making inferences about the marginal mean 0 of y when some subjects drop out of the study at random times q prior to the common fixed end of follow up time t the purpose of this article is to show how to make inferences about 0 when the continuous drop out time q is modeled semiparametrically and no restrictions are placed on the joint distribution of the outcome and other measured variables in particular we consider two models for the conditional hazard of drop out given v t y where v t denotes the history of the process v t through time t t 0 t in the first model we assume that q t v t y exp 0 y where 0 is a scalar paramet

regression analysis for categorical variables with outcome subject to nonignorable nonresponse abstract we develop a log linear model for categorical response subject to nonignorable nonresponse the paper differs from fay 1986 in its focus on estimation and hypothesis testing in a regression setting as opposed to imputation in a multivariate setting we present several new results concerning the existence of solutions on the boundary of the parameter space and the construction of confidence intervals for estimates we illustrate the method by estimating the proportion of voters preferring truman in a 1948 preelection poll mosteller hyman mccarthy marks and truman 1949 results may depend strongly on the model assumed for nonresponse goodness of fit tests are available for comparing alternative models

every missingness not at random model has a missingness at random counterpart with equal fit over the last decade a variety of models to analyse incomplete multivariate and longitudinal data have been proposed many of which allowing for the missingness to be not at random in the sense that the unobserved measurements influence the process governing missingness in addition to influences coming from observed measurements and or covariates the fundamental problems that are implied by such models to which we refer as sensitivity to unverifiable modelling assumptions has in turn sparked off various strands of research in what is now termed sensitivity analysis the nature of sensitivity originates from the fact that a missingness not at random mnar model is not fully verifiable from the data rendering the empirical distinction between mnar and missingness at random mar where only covariates and observed outcomes influence missingness difficult or even impossible unless we are willing to accept the posited mnar model in an unquestioning way we show that the empirical distinction between mar and mnar is not possible in the sense that each mnar model fit to a set of observed data can be reproduced exactly by an mar counterpart of course such a pair of models will produce different predictions of the unobserved outcomes given the observed outcomes theoretical considerations are supplemented with an illustration that is based on the slovenian public opinion survey which has been analysed before in the context of sensitivity analysis

maximum likelihood estimation and likelihood ratio test for square tables with missing data missing data complicate the analysis of paired categorical data this paper considers with the use of the em algorithm the maximum likelihood estimation and likelihood ratio test for incomplete square tables with missing data an example involving the susceptibility of micro organisms to antimicrobial drugs illustrates the procedure

on the geometric approach to multivariate selection multivariate selection can be represented as a linear transformation in a geometric framework this approach has led to considerable simplification in the study of the effects of selection on factor analysis in this note this approach is extended to describe the effects of selection on regression analysis and to adjust for the effects of selection using the inverse of the linear transformation

causal models for patterns of nonresponse abstract the problem of missing data for categorical variables is examined from the perspective of modeling the mechanisms of nonresponse log linear causal models as formulated by goodman are studied for the relationship of the survey variables to response under some conditions several such models are estimable from the observed data for nested patterns of nonresponse a specific causal model represents exactly the assumption of ignorable response most causal models however imply nonignorable response mechanisms and yield alternative estimates for the distribution of the survey variables

qmr smoothing for lanczos type product methods based on three term rrecurrences for the solution of large sparse non hermitian linear systems lanczos type product methods that are based on the lanczos three term recurrence are derived in a systematic way these methods either square the lanczos process or combine it with a local minimization of the residual for them a quasi minimal residual qmr smoothing is proposed that can also be implemented by short term recurrences the practical performance of these methods and the qmr smoothing is demonstrated in a number of numerical examples

multi level minimal residual smoothing a family of general purpose mutigrid acceleration techniques abstract we employ multi level minimal residual smoothing mrs as a pre optimization technique to accelerate standard multigrid convergence the mrs method is used to improve the current multigrid iterate by smoothing its corresponding residual before the latter is projected to the coarse grid we develop different schemes for implementing mrs technique on the finest grid and on the coarse grids and several versions of the inexact mrs technique numerical experiments are conducted to show the efficiency of the multi level and inexact mrs techniques

multiple imputation for an incomplete covariate that is a ratio we are concerned with multiple imputation of the ratio of two variables which is to be used as a covariate in a regression analysis if the numerator and denominator are not missing simultaneously it seems sensible to make use of the observed variable in the imputation model one such strategy is to impute missing values for the numerator and denominator or the log transformed numerator and denominator and then calculate the ratio of interest we call this passive imputation alternatively missing ratio values might be imputed directly with or without the numerator and or the denominator in the imputation model we call this active imputation in two motivating datasets one involving body mass index as a covariate and the other involving the ratio of total to high density lipoprotein cholesterol we assess the sensitivity of results to the choice of imputation model and as an alternative explore fully bayesian joint models for the outcome and incomplete ratio fully bayesian approaches using winbugs were unusable in both datasets because of computational problems in our first dataset multiple imputation results are similar regardless of the imputation model in the second results are sensitive to the choice of imputation model sensitivity depends strongly on the coefficient of variation of the ratio s denominator a simulation study demonstrates that passive imputation without transformation is risky because it can lead to downward bias when the coefficient of variation of the ratio s denominator is larger than about 0 1 active imputation or passive imputation after log transformation is preferable 2013 the authors statistics in medicine published by john wiley sons ltd

interval estimation for log linear models with one variable subject to non ignorable non response log linear models for multiway contingency tables where one variable is subject to non ignorable non response will often yield boundary solutions with the probability of non respondents being classified in some cells of the table estimated as 0 the paper considers the effect of this non standard behaviour on two methods of interval estimation based on the distribution of the maximum likelihood estimator the first method relies on the estimator being approximately normally distributed with variance equal to the inverse of the information matrix it is shown that the information matrix is singular for boundary solutions but intervals can be calculated after a simple transformation for the second method based on the bootstrap asymptotic results suggest that the coverage properties may be poor for boundary solutions both methods are compared with profile likelihood intervals in a simulation study based on data from the british general election panel study the results of this study indicate that all three methods perform poorly for a parameter of the non response model whereas they all perform well for a parameter of the margin model irrespective of whether or not there is a boundary solution

itemwise conditionally independent nonresponse modelling for incomplete multivariate data summary we introduce a nonresponse mechanism for multivariate missing data in which each study variable and its nonresponse indicator are conditionally independent given the remaining variables and their nonresponse indicators this is a nonignorable missingness mechanism in that nonresponse for any item can depend on values of other items that are themselves missing we show that under this itemwise conditionally independent nonresponse assumption one can define and identify nonparametric saturated classes of joint multivariate models for the study variables and their missingness indicators we also show how to perform sensitivity analysis with respect to violations of the conditional independence assumptions encoded by this missingness mechanism we illustrate the proposed modelling approach with data analyses

vml a class of virtual multi level iterative methods for solving partial differential equations we introduce virtual multi level iterative methods vml which attempt to remove the low frequency errors by conducting some special smoothing residual norm minimization procedure with respect to the coarse grids however there is no coarse grid formed explicitly no inter grid transfer operator is needed and even the smoothing procedure can be done almost locally these properties are attractive to parallel computers vml with different relaxation schemes and different smoothing techniques constitute a class of vml iterative methods they may be used to accelerate general single level iterative methods or be used with the standard real multi grid method to alleviate the inherent lack of parallelism numerical experiments with some relaxation and smoothing techniques are used to show how the vml iterative methods work

models for categorical data with nonignorable nonresponse abstract when categorical outcomes are subject to nonignorable nonresponse log linear models may be used to adjust for the nonresponse the models are fitted to the data in an augmented frequency table in which one index corresponds to whether or not the subject is a respondent the likelihood function is maximized over pseudo observed cell frequencies with respect to this log linear model using an em algorithm each e step of the em algorithm determines the pseudo observed cell frequencies and the m step yields the maximum likelihood estimators mle s of these pseudo observed cell frequencies this approach may produce boundary estimates for the expected cell frequencies of the nonrespondents in these cases the estimators of the log linear model parameters are not uniquely determined and may be unstable following the approach of clogg et al we propose a bayesian method that uses smoothing constants to adjust the pseudo observed cell frequencies so that the solution is not on the boundary the role

closed form estimates for missing counts in two way contingency tables one method for analyzing contingency tables with missing observations is to model the missing data mechanism using log linear models previous methods for obtaining estimates of missing counts and parameters have required an iterative algorithm in many cases however one can obtain estimates by use of a simple algebraic formula we illustrate the method with data on smoking and birth weight

methods for conducting sensitivity analysis of trials with potentially nonignorable competing causes of censoring we consider inference for the treatment arm mean difference of an outcome that would have been measured at the end of a randomized follow up study if during the course of the study patients had not initiated a nonrandomized therapy or dropped out we argue that the treatment arm mean difference is not identified unless unverifiable assumptions are made we describe identifying assumptions that are tantamount to postulating relationships between the components of a pattern mixture model but that can also be interpreted as imposing restrictions on the cause specific censoring probabilities of a selection model we then argue that although sufficient for identification these assumptions are insufficient for inference due to the curse of dimensionality we propose reducing dimensionality by specifying semiparametric cause specific selection models these models are useful for conducting a sensitivity analysis to examine how inference for the treatment arm mean difference changes as one varies the magnitude of the cause specific selection bias over a plausible range we provide methodology for conducting such sensitivity analysis and illustrate our methods with an analysis of data from the aids clinical trial group actg study 002

every missing not at random model has got a missing at random counterpart with equal fit over the last decade a variety of models to analyze incomplete multivariate and longitudinal data have been proposed many of which allowing for the missingness to be not at random mnar in the sense that the unobserved measurements influence the process governing missingness in addition to influences coming from observed measurements and or covariates the fundamental problems implied by such models to which we refer as sensitivity to unverifiable modeling assumptions has in turn sparked off various strands of research in what is now termed sensitivity analysis the nature of sensitivity originates from the fact that an mnar model is not fully verifiable from the data rendering the empirical distinction between mnar and random missingness mar where only covariates and observed outcomes influence missingness hard or even impossible unless one is prepared to accept the posited mnar model in an unquestioning way in this paper we show that the empirical distinction between mar and mnar is not possible in the sense that each mnar model fit to a set of observed data can be reproduced exactly by an mar counterpart of course such a pair of models will produce different predictions of the unobserved outcomes given the observed ones theoretical considerations are supplemented with an illustration based on the slovenian public opinion survey analyzed before in the context of sensitivity analysis missing data can be seen as latent variables such a view allows extension of our results to other forms of coarsening such as grouping and censoring in addition the technology applies to random effects models where a parametric form for the random effects can be replaced by certain other parametric and non parametric form without distorting the model s fit latent classes latent variables etc

models for nonresponse in sample surveys abstract the literature on the analysis of incomplete data using models is reviewed in the context of nonresponse in sample surveys the modeling approach provides a large body of methods for handling unit and item nonresponse some of which cannot be derived from the randomization theory of inference for surveys key concepts from the literature on incomplete data such as factorizations of the likelihood for special data patterns the em algorithm for general data patterns and ignorability of the response mechanism are discussed within the survey context model based procedures are related to common methods for handling nonresponse in surveys such as weighting or imputation of means within subclasses of the population

flexible inner outer krylov subspace methods flexible krylov methods refers to a class of methods which accept preconditioning that can change from one step to the next given a krylov subspace method such as cg gmres qmr etc for the solution of a linear system ax b instead of having a fixed preconditioner m and the right preconditioned equation am 1 y b mx y one may have a different matrix say mk at each step in this paper the case where the preconditioner itself is a krylov subspace method is studied there are several papers in the literature where such a situation is presented and numerical examples given a general theory is provided encompassing many of these cases including truncated methods the overall space where the solution is approximated is no longer a krylov subspace but a subspace of a larger krylov space we show how this subspace keeps growing as the outer iteration progresses thus providing a convergence theory for these inner outer methods numerical tests illustrate some important implementation aspects that make the discussed inner outer methods very appealing in practical circumstances

semiparametric regression for repeated outcomes with nonignorable nonresponse abstract we consider inference about the parameter indexing the conditional mean of a vector of correlated outcomes given a vector of explanatory variables when some of the outcomes are missing in a subsample of the study and the probability of response depends on both observed and unobserved data values that is nonresponse is nonignorable we propose a class of augmented inverse probability of response weighted estimators that are consistent and asymptotically normal can for estimating when the response probabilities can be parametrically modeled and a can estimator exists the proposed estimators do not require full specification of a parametric likelihood and their computation does not require numerical integration our estimators can be viewed as an extension of generalized estimating equation estimators that allows for nonignorable nonresponse we show that our class essentially consists of all can estimators of we also show that the asymptotic variance of the optimal estimator in our

model based inference for categorical survey data subject to non ignorable non response we consider non response models for a single categorical response with categorical covariates whose values are always observed we present bayesian methods for ignorable models and a particular non ignorable model and we argue that standard methods of model comparison are inappropriate for comparing ignorable and non ignorable models uncertainty about ignorability of non response is incorporated by introducing parameters describing the extent of non ignorability into a pattern mixture specification and integrating over the prior uncertainty associated with these parameters our approach is illustrated using polling data from the 1992 british general election panel survey we suggest sample size adjustments for surveys when non ignorable non response is expected

the fisher information matrix for log linear models arguing conditionally on observed explanatory variable summary for poisson or multinomial contingency table data the conditional distribution is product multinomial when conditioning on observed values of explanatory variables birch 1963 showed that under the restriction formed by keeping the marginal totals of one margin fixed at their observed values the poisson multinomial and product multinomial likelihoods are proportional and give the same estimates for common parameters in the log linear model here the inverses of the fisher information matrices are shown to be identical over common parameters so that the asymptotic covariance matrices of the estimates correspond

least squares theory based on general distributional assumptions with an application to the incomplete observations problem the linear regression modely x e is reanalyzed taking the modest position that x is an approximation of the best predictor ofy we derive the asymptotic distribution ofb andr 2 under mild assumptions the method of derivation yields an easy answer to the estimation of from a data set which contains incomplete observations where the incompleteness is random

maximum likelihood estimation and model selection in contingency tables with missing data abstract in many studies the values of one or more variables are missing for subsets of the original sample this article focuses on the problem of obtaining maximum likelihood estimates mle for the parameters of log linear models under this type of incomplete data the appropriate systems of equations are presented and the expectation maximization em algorithm dempster laird and rubin 1977 is suggested as one of the possible methods for solving them the algorithm has certain advantages but other alternatives may be computationally more effective tests of fit for log linear models in the presence of incomplete data are considered the data from the protective services project for older persons blenkner bloom and nielsen 1971 blenkner bloom and weber 1974 are used to illustrate the procedures discussed in the article

twisted alexander polynomials and symplectic structures let n be a closed oriented 3 manifold a folklore conjecture states that s 1 n admits a symplectic structure only if n admits a fibration over the circle the purpose of this paper is to provide evidence to this conjecture studying suitable twisted alexander polynomials of n and showing that their behavior is the same as of those of fibered 3 manifolds in particular we will obtain new obstructions to the existence of symplectic structures and to the existence of symplectic forms representing certain cohomology classes of s 1 n as an application of these results we will show that s 1 n p does not admit a symplectic structure where n p is the 0 surgery along the pretzel knot p 5 3 5 answering a question of peter kronheimer

a quasi minimal residual variant of the bi cgstab algorithm for nonsymmetric systems motivated by a recent method of freund siam j sci comput 14 1993 pp 470 482 who introduced a quasi minimal residual qmr version of the conjugate gradients squared cgs algorithm a qmr variant of the biconjugate gradient stabilized bi cgstab algorithm of van der vorst that is called qmrcgstab is proposed for solving nonsymmetric linear systems the motivation for both qmr variants is to obtain smoother convergence behavior of the underlying method the authors illustrate this by numerical experiments that also show that for problems on which bi cgstab performs better than cgs the same advantage carries over to qmrcgstab

algebraic multigrid for complex symmetric matrices and applications this paper is concerned with the numerical study of an algebraic multigrid preconditioner for complex symmetric system matrices we use several different krylov subspace methods as an outer iteration namely the qmr method proposed by freund and nachtigal the bicgcr method of clemens and the csym method of bunse gerstner and stoever in addition we compare the results with the standard jacobi preconditioner for complex symmetric problems we test our approach on the numerical simulation of high voltage insulators

a transpose free quasi minimal residual algorithm for non hermitian linear systems the biconjugate gradient method bcg for solving general non hermitian linear systems ax b and its transpose free variant the conjugate gradients squared algorithm cgs both typically exhib

smoothing iterative block methods for linear systems with multiple right hand sides in the present paper we present smoothing procedures for iterative block methods for solving nonsymmetric linear systems of equations with multiple right hand sides these procedures generalize those known when solving one right hand linear systems we give some properties of these new methods and then using these procedures we show connections between some known iterative block methods finally we give some numerical examples

degree of l 2 alexander torsion for 3 manifolds for an irreducible orientable compact 3 manifold n with empty or incompressible toral boundary the full l 2 alexander torsion tau 2 n phi t associated to any real first cohomology class phi of n is represented by a function of a positive real variable t the paper shows that tau 2 n phi is continuous everywhere positive and asymptotically monomial in both ends moreover the degree of tau 2 n phi equals the thurston norm of phi the result confirms a conjecture of j dubois s friedl and w luck and addresses a question of w li and w zhang associated to any admissible homomorphism gamma pi 1 n rightarrow g the l 2 alexander torsion tau 2 n gamma phi is shown to be continuous and everywhere positive provided that g is residually finite and n gamma is weakly acyclic in this case a generalized degree can be assigned to tau 2 n gamma phi moreover the generalized degree is bounded by the thurston norm of phi

discrete qmr and bcg in the numerical solution of linear systems of odes in this paper we consider the use of qmr and bcg in the numerical solution of linear systems of odes reformulated as second kind integral equations the implementation of the methods is performed in practice by applying them on suitably discretized equations an important issue in this contest is to compare the behaviour of the discrete and continuous iteration here we consider the case of linear systems of odes the discretization is performed using a continuous runge kutta method a numerical experiment concludes the paper

changing the norm in conjugate gradient type algorithms a conjugate gradient type method for solving ax b or more precisely an orthogonal error method operatorname oe b c a is determined by a formal inner product matrix b and a left preconditioning matrix c relations are pointed out between quantities such as iterates residuals direction vectors and recurrence coefficients of orthogonal error methods with different but in a particular way related matrices b namely for operatorname oe b ca k c a k 0 1 ldots and for operatorname oe a h c h k b c a k 0 1 ldots the relations for the first sequence of methods have to do with rutishauser s lr algorithm those for the second one are based on a generalization of the schonauer weiss smoothing algorithm relevant for practice are the cases k 0 1 for b ca and b ca h

hybrid procedures for solving linear equations in this paper we introduce the notion of hybrid procedures for solving a system of linear equations a hybrid procedure consists in a combination of two arbitrary approximate solutions with coefficients summing up to one thus the combination only depends on one parameter whose value is chosen in order to minimize the euclidean norm of the residual vector obtained by the hybrid procedure properties of such procedures are studied in detail the two approximate solutions which are combined in a hybrid procedure are usually obtained by two iterative methods several strategies for combining these two methods together or with the previous iterate of the hybrid procedure itself are discussed and their properties are analyzed numerical experiments illustrate the various procedures

breakdowns in the implementation of the lanczos method for solving linear systems abstract the lanczos method for solving systems of linear equations is based on formal orthogonal polynomials its implementation is realized via some recurrence relationships between polynomials of a family of orthogonal polynomials or between those of two adjacent families of orthogonal polynomials a division by zero can occur in such recurrence relations thus causing a breakdown in the algorithm which has to be stopped in this paper two types of breakdowns are discussed the true breakdowns which are due to the nonexistence of some polynomials and the ghost breakdowns which are due to the recurrence relationship used among all the recurrence relationships which can be used and all the algorithms for implementing the lanczos method which came out from them the only reliable algorithm is lanczos orthodir which can only suffer from true breakdowns it is shown how to avoid true breakdowns in this algorithm other algorithms are also discussed and the case of near breakdown is treated the same treatment applies to other methods related to lanczos

lanczos type variants of the cocr method for complex nonsymmetric linear systems motivated by the celebrated extending applications of the well established complex biconjugate gradient cbicg method to deal with large three dimensional electromagnetic scattering problems by pocock and walker m d pocock s p walker the complex bi conjugate gradient solver applied to large electromagnetic scattering problems computational costs and cost scalings ieee trans antennas propagat 45 1997 140 146 three lanczos type variants of the recent conjugate a orthogonal conjugate residual cocr method of sogabe and zhang t sogabe s l zhang a cocr method for solving complex symmetric linear systems j comput appl math 199 2007 297 303 are explored for the solution of complex nonsymmetric linear systems the first two can be respectively considered as mathematically equivalent but numerically improved popularizing versions of the bicr and crs methods for complex systems presented in sogabe s ph d dissertation and the last one is somewhat new and is a stabilized and more smoothly converging variant of the first two in some circumstances the presented algorithms are with the hope of obtaining smoother and hopefully faster convergence behavior in comparison with the cbicg method as well as its two corresponding variants this motivation is demonstrated by numerical experiments performed on some selective matrices borrowed from the university of florida sparse matrix collection by davis

fast iterative solution of stabilized navier stokes systems discretization and linearization of the steady state navier stokes equations give rise to a nonsymmetric indefinite linear system of equations in this paper we address the block triangular preconditioning iterative methods for solving this type of systems which result from stabilized mixed finite element discretization of the linearized navier stokes equations the convergence results are established and the results of the numerical experiments are presented

satellites and concordance of knots in 3 manifolds given a 3 manifold y and a free homotopy class in s1 y we investigate the set of topological concordance classes of knots in y 0 1 representing the given homotopy class the concordance group of knots in the 3 sphere acts on this set we show in many cases that the action is not transitive using two techniques our first technique uses reidemeister torsion invariants and the second uses linking numbers in covering spaces in particular we show using covering links that for the trivial homotopy class and for any 3 manifold that is not the 3 sphere the set of orbits is infinite on the other hand for the case that y s1 s2 we apply topological surgery theory to show that all knots with winding number one are concordant

imputation of missing values when the probability of response depends on the variable being imputed abstract a method is developed for imputing missing values when the probability of response depends upon the variable being imputed the missing data problem is viewed as one of parameter estimation in a regression model with stochastic censoring of the dependent variable the prediction approach to imputation is used to solve this estimation problem wages and salaries are imputed to non respondents in the current population survey and the results are compared to the nonrespondents irs wage and salary data the stochastic censoring approach gives improved results relative to a prediction approach that ignores the response mechanism

residual smoothing techniques for iterative methods an iterative method for solving a linear system ax b produces iterates x k with associated residual norms that in general need not decrease smoothly to zero residual smoothing techniques are considered that generate a second sequence y k via a simple relation y k 1 eta k y k 1 eta k x k the authors first review and comment on a technique of this form introduced by schonauer and weiss that results in y k with monotone decreasing residual norms this is referred to as minimal residual smoothing certain relationships between the residuals and residual norms of the biconjugate gradient bcg and quasi minimal residual qmr methods are then noted from which it follows that qmr can be obtained from bcg by a technique of this form this technique is extended to generally applicable quasi minimal residual smoothing the practical performance of these techniques is illustrated in a number of numerical experiments

on structural equation modeling with data that are not missing completely at random a general latent variable model is given which includes the specification of a missing data mechanism this framework allows for an elucidating discussion of existing general multivariate theory bearing on maximum likelihood estimation with missing data here missing completely at random is not a prerequisite for unbiased estimation in large samples as when using the traditional listwise or pairwise present data approaches the theory is connected with old and new results in the area of selection and factorial invariance it is pointed out that in many applications maximum likelihood estimation with missing data may be carried out by existing structural equation modeling software such as lisrel and liscomp several sets of artifical data are generated within the general model framework the proposed estimator is compared to the two traditional ones and found superior

an approach to categorical data with nonignorable nonresponse log linear models have been used to adjust for nonresponse when categorical outcomes are subject to nonignorable nonresponse the log models are fitted to the data in an augmented frequency table in which one index corresponds to whether the subject is a respondent park and brown 1994 journal of the american statistical association 89 44 52 proposed a pseudo bayesian method that has the effect of smoothing the unobserved cell frequencies their approach assigns prior observations only to the unobserved cells their method was shown to perform better than the maximum likelihood method which can produce unstable boundary estimates generalizing their approach we propose a new approach that assigns prior observations to both observed and unobserved cells through a simulation study we compare the proposed approach with park and brown s and the maximum likelihood approach