indefinite overlapping schwarz methods for time dependent stokes problems abstract a class of indefinite overlapping schwarz methods is introduced and studied for time dependent stokes problems discretized with either mixed finite elements or mixed spectral elements the methods proposed are based on the solution of local time dependent stokes problems on overlapping subdomains and on the solution of a coarse time dependent stokes problem defined on the coarse subdomain mesh the resulting preconditioner is accelerated by an appropriate krylov space method producing a very efficient scalable and parallel solver

a preconditioner for the steady state navier stokes equations we present a new method for solving the sparse linear system of equations arising from the discretization of the linearized steady state navier stokes equations also known as the oseen equations the solver is an iterative method of krylov subspace type for which we devise a preconditioner through a heuristic argument based on the fundamental solution tensor for the oseen operator the preconditioner may also be conceived through a weaker heuristic argument involving differential operators computations indicate that convergence for the preconditioned discrete oseen problem is only mildly dependent on the viscosity inverse reynolds number and most importantly that the number of iterations does not grow as the mesh size is reduced indeed since the preconditioner is motivated through analysis of continuous operators the number of iterations decreases for smaller mesh size which accords with better approximation of these operators

a block preconditioner for non isothermal flow in porous media abstract in petroleum reservoir simulation the industry standard preconditioner the constrained pressure residual method cpr is a two stage process which involves solving a restricted pressure system with algebraic multigrid amg initially designed for isothermal models this approach is often used in the thermal case however it does not have a specific treatment of the additional energy conservation equation and temperature variable we seek to develop preconditioners which better capture thermal effects such as heat diffusion in order to study the effects of both pressure and temperature on fluid and heat flow we consider a model of non isothermal single phase flow through porous media for this model we develop a block preconditioner with an efficient schur complement approximation both the pressure block and the approximate schur complement are approximately inverted using an amg v cycle the resulting solver is scalable with respect to problem size and parallelization

preconditioned mixed spectral element methods for elasticity and stokes problems this paper continues our study of efficient preconditioners for the indefinite systems obtained by discretizing the linear elasticity and stokes problems with mixed spectral methods in three dimensions our study is extended here in two new directions to include i more general multidomain mixed spectral element methods and ii a new class of block triangular preconditioners the stiffness matrices resulting from stokes and mixed elasticity problems have the structure of saddle point problems with a penalty term which is associated with the poisson ratio for elasticity problems or with stabilization techniques for stokes problems the main result of this paper shows that the convergence rate of the proposed preconditioned iterative methods is independent of the penalty parameter nu the number of spectral elements n and mildly dependent on the spectral degree n via the inf sup constant the number of iterations required by the block triangular preconditioner is much smaller than the one required by the block diagonal preconditioner on the other hand the block triangular preconditioner is no longer symmetric and has a small extra cost these results and several numerical experiments presented in the final section show that these algorithms are a practical and efficient strategy for the iterative solution of the indefinite systems arising from mixed spectral element discretizations of elliptic systems

overlapping schwarz methods for mixed linear elasticity and stokes problems overlapping schwarz preconditioners are introduced and studied for saddle point problems with a penalty term such as stokes equations and mixed formulations of linear elasticity these preconditioners are based on the solution of local saddle point problems on overlapping subdomains and the solution of a coarse saddle point problem numerical experiments show that these are parallel and scalable preconditioners since the rate of convergence of the preconditioned operator is independent of the mesh size h the number of subdomains n and the penalty parameter

on the domain dependence of the inf sup and related constants via conformal mapping abstract in this paper we investigate the domain dependence of the inf sup stability constant in the family of two dimensional simply connected domains using its connection to the optimal constant figuring in friedrichs inequality for conjugate harmonic functions and the conformal mapping of the domain a lower estimation of the inf sup constant is also given in terms of the conformal mapping provided the boundary of the domain is smooth enough we illustrate the results with several examples

probability and fuzziness we discuss some relationships between probability theory and statistics on one hand and the theory of fuzzy sets on the other hand we develop various statistical techniques for the analysis of imprecise data and for inference based on imprecise data we study the estimation of fuzzy parameters testing problems with fuzzy probabilities and confidence estimation under uncertainty finally we indicate some possible applications to the assessment of credibility in expert systems

iterative techniques for time dependent stokes problems abstract in this paper we consider solving the coupled systems of discrete equations which arise from implicit time stepping procedures for the time dependent stokes equations using a mixed finite element spatial discretization at each time step a two by two block system corresponding to a perturbed stokes problem must be solved although there are a number of techniques for iteratively solving this type of block system to be effective they require a good preconditioner for the resulting pressure operator schur complement in contrast to the time independent stokes equations where the pressure operator is well conditioned the pressure operator for the perturbed system becomes more ill conditioned as the time step is reduced and or the reynolds number is increased in this paper we shall describe and analyze preconditioners for the resulting pressure systems these preconditioners give rise to iterative rates of convergence which are independent of both the mesh size h as well as the time step and reynolds number parameter k

constructing a cut off point for a quantitative diagnostic test this paper is concerned with the problem of constructing a suitable cut off point for a quantitative diagnostic test from a sample of diseased and non diseased individuals when the specificity and or sensitivity required for the resulting diagnostic classification rule are specified statistical procedures are discussed which yield an explicit cut off point together with the necessary confidence limits on the true specificity and sensitivity of the test operating at this point by contrast to confidence limits referring to an unknown optimal choice of the cut off point sample size formulae are derived showing a relative efficiency of up to 1 5 for a new procedure introduced in this paper over the usual procedure based on tolerance limits

the mean field theory in em procedures for markov random fields in many signal processing and pattern recognition applications the hidden data are modeled as markov processes and the main difficulty of using the maximisation em algorithm for these applications is the calculation of the conditional expectations of the hidden markov processes it is shown how the mean field theory from statistical mechanics can be used to calculate the conditional expectations for these problems efficiently the efficacy of the mean field theory approach is demonstrated on parameter estimation for one dimensional mixture data and two dimensional unsupervised stochastic model based image segmentation experimental results indicate that in the 1 d case the mean field theory approach provides results comparable to those obtained by baum s 1987 algorithm which is known to be optimal in the 2 d case where baum s algorithm can no longer be used the mean field theory provides good parameter estimates and image segmentation for both synthetic and real world images

high order direct stokes solvers with or without temporal splitting numerical investigations of their comparative properties a recently proposed direct stokes solver which decouples the velocity and pressure operators without calling for a temporal scheme is numerically analyzed in comparison first with the splitting scheme proposed by g karniadakis m israeli and s orszag in j comput phys 97 1991 pp 414 443 and with the unique grid bbb p n bbb p n 2 uzawa approach for the space accuracy and computational costs the chebyshev collocation approximation is used to analyze the spectra of the continuous temporal evolution operators and their discrete time versions from the first to the fourth order an explicit boundary condition is also involved in the proposed stokes solver and it is numerically shown that the trace of bf delta bf u on the boundary must be evaluated through its bf nabla times bf nabla times contribution only otherwise the ellipticity is lost before proceeding to the time discretization the explicit evaluation of the rotational boundary term does not prevent the first and second order in time schemes from being unconditionally stable while the schemes built at the next two higher orders are limited by the usual explicit cal o n 4 stability criterion on the time step the proposed bf u p decoupling gives this limitation a less restrictive coefficient and supplies the expected temporal orders on the whole explored range of time step sizes the effective accuracy obtained for the navier stokes 2d solutions has been measured for the re 1000 lid driven cavity problem and has been found equivalent to what is supplied by the much more expensive uzawa decoupling method

splitting techniques with staggered grids for the navier stokes equations in the 2d case abstract a pseudo spectral approximation for the navier stokes equations in the 2d case is presented using a new splitting technique based on the uzawa algorithm the system is decoupled into helmholtz equations for the velocity and an equation with the pseudo laplacian for the pressure staggered grids with gauss and gauss lobatto nodes are employed preconditioning with finite differences is considered by extrapolation a stable second order method in time for the velocity and at least a first order method for the pressure can be achieved

a fourier spectral element algorithm for thermal convection in rotating axisymmetric containers we present a fourier spectral element approach for modeling thermal convection in a rotating axisymmetric container following the theory detailed in bernardi et al c bernardi m dauge y maday spectral methods for axisymmetric domains gauthier villars paris 1999 a fourier expansion of the field variables is performed in the periodic direction and the resulting collection of meridional problems is discretized by means of a parallel spectral element method a gauss lobatto jacobi 0 1 quadrature which incorporates the cylindrical radius in its weight is introduced to avoid a potential degeneracy of the discrete set of equations due to those nodes located on the axis of symmetry a second order timestepping scheme is presented which treats the coriolis and viscous forces implicitly numerical comparisons with analytical and published numerical solutions in spherical and cylindrical geometries are presented which highlight the accuracy of the model and demonstrate its spectral convergence properties

balancing neumann neumann methods for incompressible stokes equations this paper describes the unerlying mathematical model and the balancing neumann neumann methods are introduced and studied for incompressible stokes equations discretized with mixed finite or spectral elements with discontinuous pressures after decomposing the original domain of the problem into nonoverlapping subdomains the interior unknowns which are the interior velocity component and all except the constant pressure component of each subdomain problem are implicitly eliminated the resulting saddle point schur complement is solved with a krylov space method with a balancing neumann neumann preconditioner based on the solution of a coarse stokes problem with a few degrees of freedom per subdomain and on the solution of local stokes problems with natural neumann velocity and essential boundary conditions on the subdomains this preconditioner is of hybrid form in which the coarse problem is treated multiplicatively while the local problems are treated additively the condition number of the preconditioned operator is independent of the number of subdomains and is bounded from above by the product of the square of the logarithm of the local number of unknowns in each subdomain and the inverse of the inf sup constants of the discrete problem and of the coarse subproblem numerical results show that the method is quite fast they are also fully consistent with the theory

the analysis of categorical data from complex sample surveys chi squared tests for goodness of fit and independence in two way tables abstract the effect of stratification and clustering on the asymptotic distributions of standard pearson chi squared test statistics for goodness of fit simple hypothesis and independence in a two way contingency table denoted as x 2 and xi 2 respectively is investigated it is shown that both x 2 and xi 2 are asymptotically distributed as weighted sums of independent 1 2 random variables the weights are then related to the familiar design effects deffs used by survey samplers a simple correction to x 2 which requires only the knowledge of variance estimates or deffs for individual cells in the goodness of fit problem is proposed and empirical results on the performance of corrected x 2 provided empirical work on xi 2 indicated that the distortion of nominal significance level is substantially smaller with xi 2 than with x 2 some results under simple models for clustering are also given

segmentation of brain mr images through a hidden markov random field model and the expectation maximization algorithm the finite mixture fm model is the most commonly used model for statistical segmentation of brain magnetic resonance mr images because of its simple mathematical form and the piecewise constant nature of ideal brain mr images however being a histogram based model the fm has an intrinsic limitation no spatial information is taken into account this causes the fm model to work only on well defined images with low levels of noise unfortunately this is often not the the case due to artifacts such as partial volume effect and bias field distortion under these conditions fm model based methods produce unreliable results here the authors propose a novel hidden markov random field hmrf model which is a stochastic process generated by a mrf whose state sequence cannot be observed directly but which can be indirectly estimated through observations mathematically it can be shown that the fm model is a degenerate version of the hmrf model the advantage of the hmrf model derives from the way in which the spatial information is encoded through the mutual influences of neighboring sites although mrf modeling has been employed in mr image segmentation by other researchers most reported methods are limited to using mrf as a general prior in an fm model based approach to fit the hmrf model an em algorithm is used the authors show that by incorporating both the hmrf model and the em algorithm into a hmrf em framework an accurate and robust segmentation can be achieved more importantly the hmrf em framework can easily be combined with other techniques as an example the authors show how the bias field correction algorithm of guillemaud and brady 1997 can be incorporated into this framework to achieve a three dimensional fully automated approach for brain mr image segmentation

on the enforcement of the zero mean pressure condition in the spectral element approximation of the stokes problem this paper considers the spectral element approximation of the stokes problem and the conditioning of the resulting discrete problem the well posedness of the variational formulation of the stokes problem and in particular the uniqueness of the pressure has been demonstrated when the subspace of square integrable functions having vanishing mean is chosen for the pressure space in the discrete setting a conforming subspace is chosen for the discrete pressure space and the analysis of the discrete problem is based on this choice however this space is not used in practical implementations of finite or spectral element methods for the stokes problem in this paper it is shown how the zero volume condition on the pressure may be accommodated within the trial space by modifying the continuity equation in a consistent manner the dependence of the condition number of the uzawa operator and the accuracy of the spectral element approximation on the weighting factor is investigated

a model for evaluating sensitivity and specificity for correlated diagnostic tests in efficacy studies with an imperfect reference test abstract the purpose of a diagnostic efficacy study is to evaluate and compare the sensitivities and specificities of several diagnostic tests usually the diagnostic tests are correlated conditional on disease status and the reference test is subject to error in the chlamydia trachomatis study five screening tests for detecting chlamydia in endocervical specimens were compared the five tests are correlated and the reference test the cell culture test has less than 100 sensitivity the conventional method ignores both the correlations between the tests and the misclassification of the reference test and thus cannot provide a valid analysis we propose a model to evaluate and compare the sensitivities and specificities of correlated diagnostic tests when there is either an imperfect reference test or even no reference test the model also can estimate the effects of covariates it is a generalized linear mixed model with two unobserved variables one continuous and one dichotomous we use a hybrid

analysis of iterative methods for the steady and unsteady stokes problem application to spectral element discretizations a new and detailed analysis of the basic uzawa algorithm for decoupling of the pressure and the velocity in the steady and unsteady stokes operator is presented the paper focuses on the following new aspects explicit construction of the uzawa pressure operator spectrum for a semiperiodic model problem general relationship of the convergence rate of the uzawa procedure to classical inf sup discretization analysis and application of the method to high order variational discretization

a family of nonparametric statistics for comparing diagnostic markers with paired or unpaired data summary in this paper we study a broad class of nonparametric statistics for comparing two diagnostic markers one can compare the sensitivities of these diagnostic markers over restricted ranges of specificity by selecting an appropriate statistic from this class as special cases one can compare the entire area under the receiver operator curve hanley mcneil 1982 or one can compare the sensitivities at a fixed common specificity usually we would recommend a comparison based on an average of sensitivities over a restricted high level of specificities test procedures and confidence intervals are based on asymptotic normality these procedures are applicable for paired data in which both diagnostic markers are performed on each subject and for unpaired data the procedures may be used to compare two real functions of multiple diagnostic markers as well as to compare individual markers

testing for conditional multiple marginal independence summary survey respondents are often prompted to pick any number of responses from a set of possible responses categorical variables that summarize this kind of data are called pick any c variables counts from surveys that contain a pick any c variable along with a group variable r levels and stratification variable q levels can be marginally summarized into an r c q contingency table a question that may naturally arise from this setup is to determine if the group and pick any c variable are marginally independent given the stratification variable a test for conditional multiple marginal independence cmmi can be used to answer this question since subjects may pick any number out of c possible responses the cochran 1954 biometrics10 417 451 and mantel and haenszel 1959 journal of the national cancer institute22 719 748 tests cannot be used directly because they assume that units in the contingency table are independent of each other therefore new testing methods are developed cochran s test statistic is extended to r 2 q tables and a modified version of this statistic is proposed to test cmmi its sampling distribution can be approximated through bootstrapping other cmmi testing methods discussed are bootstrap p value combination methods and bonferroni adjustments simulation findings suggest that the proposed bootstrap procedures and the bonferroni adjustments consistently hold the correct size and provide power against various alternatives

a high order splitting method for time dependent domains abstract we present a temporal splitting scheme for the semi discrete convection diffusion equation and the semi discrete incompressible navier stokes equations in time dependent geometries the proposed splitting scheme can be considered as an extension of the oif method proposed in maday et al y maday a t patera e m ronquist an operator integration factor splitting method for time dependent problems application to incompressible fluid flow j sci comput 5 4 1990 263 292 in the sense that it can be interpreted as a semi lagrangian method for time dependent domains the semi discrete equations are derived from an arbitrary lagrangian eulerian ale formulation of the governing equations and are discretized in space using high order spectral elements the proposed splitting scheme has been tested numerically on model problems with known analytical solutions and first second and third order convergence in time has been obtained we also show that it is not necessary for the interior mesh velocity to be obtained through the use of an elliptic solver numerical tests show that it is sufficient that the mesh velocity is regular within each spectral element and only c 0 continuous across element boundaries this is consistent with the theoretical results presented in formaggia and nobile l formaggia f nobile a stability analysis for the arbitrary lagrangian eulerian formulation with finite elements east west j numer math 7 2 1999 105 131 in addition the mesh velocity should be regular in the time direction

homogeneous and heterogeneous domain decomposition methods for plate bending problems we consider the approximation of fourth order problems derived e g by the kirchoff plate model and the heterogeneous coupling between a fourth order problem and a reduced second order one describing e g a plate membrane model this paper is devoted to the analysis of an iteration by subdomain method the so called dirichlet neumann method which is used to solve both homogeneous and heterogeneous couplings numerical results obtained by the spectral element method are shown

beta mapping and beta regression for changes of ordinal rating measurements on likert scales a comparison of the change scores among multiple treatment groups patient reported outcome and observer evaluative studies in clinical trials and post hoc analyses often use instruments that measure responses on ordinal rating or likert scales we propose a flexible distributional approach by modeling the change scores from the baseline to the end of the study using independent beta distributions the two shape parameters of the fitted beta distributions are estimated by matching moments covariates and the interaction terms are included in multivariate beta regression analyses under generalized linear mixed models these methods are illustrated on the treatment satisfaction data in an overactive bladder drug study with four treatment arms monte carlo simulations were conducted to compare the type 1 errors and statistical powers using a beta likelihood ratio test of the proposed method against its fully nonparametric or parametric alternatives

preconditioned conjugate residual methods for mixed spectral discretizations of elasticity and stokes problems abstract mixed spectral methods applied to elasticity and stokes problems in three dimensions produce large symmetric indefinite systems of equations these systems are sparse and more ill conditioned than those produced by h version finite element methods these stiffness matrices have the structure of saddle point problems with a penalty term in this paper preconditioned iterative methods of conjugate residual type are introduced for these systems the convergence rates of the resulting algorithms are proven to be independent of the penalty parameter and mildly dependent on the spectral degree n several numerical experiments on three dimensional model problems confirm the results obtained

domain decomposition preconditioners for p and hp finite element approximation of stokes equations abstract domain decomposition preconditioning techniques are developed in the context of hp finite element approximation of the stokes problem two basic types of preconditioner are considered a block diagonal scheme based on decoupling the velocity and pressure components and a scheme based on an indefinite system similar to the original stokes system for each type of scheme theoretical estimates are obtained for the location of the eigenvalues of the preconditioned operators in terms of the polynomial degree the mesh sizes on the coarse and fine grids and the inf sup constant for the method theoretical estimates show that the growth of the bounds is modest as the mesh is refined and the polynomial order is increased the preconditioners are shown to be applicable to various iterative schemes for the stokes systems the theoretical bounds are compared with actual quantities obtained in practical computations for several representative problems

schwarz preconditioners for the spectral element discretization of the steady stokes and navier stokes equations the q n q n 2 spectral element discretization of the stokes equation gives rise to an ill conditioned indefinite symmetric linear system for the velocity and pressure degrees of freedom we propose a domain decomposition method which involves the solution of a low order global and several local problems related to the vertices edges and interiors of the subdomains the original system is reduced to a symmetric equation for the velocity which can be solved with the conjugate gradient method we prove that the condition number of the iteration operator is bounded from above by c 1 log n 3 beta n where c is a positive constant independent of the degree n and the number of subdomains and beta n is the inf sup condition of the pair q n q n 2 we also consider the stationary navier stokes equations in each newton step a non symmetric indefinite problem is solved using a schwarz preconditioner by using an especially designed low order global space and the solution of local problems analogous to those decribed above for the stokes equation we are able to present a complete theory for the method we prove that the number of iterations of the gmres method at each newton step is bounded from above by c 1 log n 3 beta n the constant c does not depend on the number of subdomains or n and it does not deteriorate as the newton iteration proceeds

a regression modelling framework for receiver operating characteristic curves in medical diagnostic testing summary receiver operating characteristic curves roc s are used to evaluate diagnostic tests when test results are not binary they describe the inherent capacity of the test for distinguishing between truly diseased and nondiseased subjects although methodology for estimating and for comparing roc s is well developed to date no general framework exists for evaluating covariate effects on roc s we formulate a general regression model which allows the effects of covariates on test accuracy to be succinctly summarised such covariates might include for example characteristics of the patient or test environment test type or severity of disease the regression models are shown to arise naturally from some classic models for continuous or ordinal test data regression parameters are fitted using an estimating equation approach the method is illustrated on data from a study of multiformat photographic images used for scintigraphy

splitting techniques for the unsteady stokes equations a pseudospectral or collocation approximation of the unsteady stokes equations is presented using the uzawa algorithm the spectral system is decoupled into helmholtz equations for the velocity components and an equation with the pseudo laplacian for the pressure both the velocity components and the pressure are approximated by polynomials of the same degree n compatible velocity and pressure approximations are constructed to ensure that the discrete problem is well posed the resulting discrete velocity field is identically divergence free only one grid no staggered grids with the standard chebyshev gauss lobatto nodes is used here we further compare our treatment with an inhomogeneous neumann condition for the pressure this scheme was proposed in g karniadakis m israeli and s a orszag j comp phys 97 1991 pp 414 443 the comparison shows that the neumann condition yields somewhat better approximations for the pressure in the time discretization a high order backward differentiation scheme for the intermediate velocity is combined with a high order extrapolant for the pressure it is numerically shown that a stable third order method in time can be achieved in the case of periodic boundary conditions this result is confirmed by fourier analysis

all at once solution of time dependent stokes control the solution of time dependent pde constrained optimization problems subject to unsteady flow equations presents a challenge to both algorithms and computing power in this paper we present an all at once approach where we solve for all time steps of the discretized unsteady stokes problem at once the most desirable feature of this approach is that for all steps of an iterative scheme we only need approximate solutions of the discretized stokes operator this leads to an efficient scheme which exhibits mesh independent behaviour

spatially regularized estimation for the analysis of dynamic contrast enhanced magnetic resonance imaging data competing compartment models of different complexities have been used for the quantitative analysis of dynamic contrast enhanced magnetic resonance imaging data we present a spatial elastic net approach that allows to estimate the number of compartments for each voxel such that the model complexity is not fixed a priori a multi compartment approach is considered which is translated into a restricted least square model selection problem this is done by using a set of basis functions for a given set of candidate rate constants the form of the basis functions is derived from a kinetic model and thus describes the contribution of a specific compartment using a spatial elastic net estimator we chose a sparse set of basis functions per voxel and hence rate constants of compartments the spatial penalty takes into account the voxel structure of an image and performs better than a penalty treating voxels independently the proposed estimation method is evaluated for simulated images and applied to an in vivo dataset copyright 2013 john wiley sons ltd

three validation metrics for automated probabilistic image segmentation of brain tumours the validity of brain tumour segmentation is an important issue in image processing because it has a direct impact on surgical planning we examined the segmentation accuracy based on three two sample validation metrics against the estimated composite latent gold standard which was derived from several experts manual segmentations by an em algorithm the distribution functions of the tumour and control pixel data were parametrically assumed to be a mixture of two beta distributions with different shape parameters we estimated the corresponding receiver operating characteristic curve dice similarity coefficient and mutual information over all possible decision thresholds based on each validation metric an optimal threshold was then computed via maximization we illustrated these methods on mr imaging data from nine brain tumour cases of three different tumour types each consisting of a large number of pixels the automated segmentation yielded satisfactory accuracy with varied optimal thresholds the performances of these validation metrics were also investigated via monte carlo simulation extensions of incorporating spatial correlation structures using a markov random field model were considered

an interpretation for the roc curve and inference using glm procedures the accuracy of a medical diagnostic test is often summarized in a receiver operating characteristic roc curve this paper puts forth an interpretation for each point on the roc curve as being a conditional probability of a test result from a random diseased subject exceeding that from a random nondiseased subject this interpretation gives rise to new methods for making inference about roc curves it is shown that inference can be achieved with binary regression techniques applied to indicator variables constructed from pairs of test results one component of the pair being from a diseased subject and the other from a nondiseased subject within the generalized linear model glm binary regression framework roc curves can be estimated and we highlight a new semiparametric estimator covariate effects can also be evaluated with the glm models the methodology is applied to a pancreatic cancer dataset where we use the regression framework to compare two different serum biomarkers asymptotic distribution theory is developed to facilitate inference and to provide insight into factors influencing variability of estimated model parameters

bddc preconditioners for spectral element discretizations of almost incompressible elasticity in three dimensions balancing domain decomposition by constraints bddc algorithms are constructed and analyzed for the system of almost incompressible elasticity discretized with gauss lobatto legendre spectral elements in three dimensions initially mixed spectral elements are employed to discretize the almost incompressible elasticity system but a positive definite reformulation is obtained by eliminating all pressure degrees of freedom interior to each subdomain into which the spectral elements have been grouped appropriate sets of primal constraints can be associated with the subdomain vertices edges and faces so that the resulting bddc methods have a fast convergence rate independent of the almost incompressibility of the material in particular the condition number of the bddc preconditioned operator is shown to depend only weakly on the polynomial degree n the ratio h h of subdomain and element diameters and the inverse of the inf sup constants of the subdomains and the underlying mixed formulation while being scalable i e independent of the number of subdomains and robust i e independent of the poisson ratio and young s modulus of the material considered these results also apply to the related dual primal finite element tearing and interconnect feti dp algorithms defined by the same set of primal constraints numerical experiments carried out on parallel computing systems confirm these results

role of the lbb condition in weak spectral projection methods 405 this paper investigates the relevance of the ladyshenskaya babuska brezzi con dition in spectral projection methods we consider the stability and convergence properties for a first order nonincremental projection method and a second order incremental projection method both based on a spectral galerkin legendre spatial discretization we show that the convergence of both projection methods is controlled by the ability of the spectral framework to approximate correctly the steady stokes problem c 2001 elsevier science

balancing neumann neumann preconditioners for mixed approximations of heterogeneous problems in linear elasticity balancing neumann neumann methods are extented to mixed formulations of the linear elasticity system with discontinuous coefficients discretized with mixed finite or spectral elements with discontinuous pressures these domain decomposition methods implicitly eliminate the degrees of freedom associated with the interior of each subdomain and solve iteratively the resulting saddle point schur complement using a hybrid preconditioner based on a coarse mixed elasticity problem and local mixed elasticity problems with natural and essential boundary conditions a polylogarithmic bound in the local number of degrees of freedom is proven for the condition number of the preconditioned operator in the constant coefficient case parallel and serial numerical experiments confirm the theoretical results indicate that they still hold for systems with discontinuous coefficients and show that our algorithm is scalable parallel and robust with respect to material heterogeneities the results on heterogeneous general problems are also supported in part by our theory

positive definite balancing neumann neumann preconditioners for nearly incompressible elasticity in this paper a positive definite balancing neumann neumann bnn solver for the linear elasticity system is constructed and analyzed the solver implicitly eliminates the interior degrees of freedom in each subdomain and solves iteratively the resulting schur complement involving only interface displacements using a bnn preconditioner based on the solution of a coarse elasticity problem and local elasticity problems with natural and essential boundary conditions while the schur complement becomes increasingly ill conditioned as the materials becomes almost incompressible the bnn preconditioned operator remains well conditioned the main theoretical result of the paper shows that the proposed bnn method is scalable and quasi optimal in the constant coefficient case this bound holds for material parameters arbitrarily close to the incompressible limit while this result is due to an underlying mixed formulation of the problem both the interface problem and the preconditioner are positive definite numerical results in two and three dimensions confirm these good convergence properties and the robustness of the methods with respect to the almost incompressibility of the material

preconditioning of the p version of the finite element method the p version finite element method for linear second order elliptic equations in an arbitrary sufficiently smooth incl polygonal bounded domain is studied in the framework of the domain decomposition dd method two types of square reference elements are used with coordinate functions given by the products of the integrated legendre polynomials estimates for the condition numbers and some useful inequalities are given we consider preconditioning of the problems arising on subdomains and of the schur complement as well as the derivation and analysis of the dd preconditioner for the entire system this is done for a class of curvilinear finite elements we obtain several dd preconditioners for which the generalized condition numbers vary from o log p 3 to o 1 this paper is based on 19 21 27 we have omitted most of the proofs in order to shorten it and have described instead what could be done as well as outlined some additional ideas the full proofs omitted can in most cases be found in 19 20 27

efficient solution of large scale saddle point systems arising in riccati based boundary feedback stabilization of incompressible stokes flow we investigate numerical methods for solving large scale saddle point systems which arise during the feedback control of flow problems we focus on the instationary stokes equations that describe instationary incompressible flows for moderate viscosities after a mixed finite element discretization we get a differential algebraic system of differential index two j weickert navier stokes equations as a differential algebraic system preprint sfb393 96 08 department of mathematics chemnitz university of technology chemnitz germany 1996 to reduce this index we follow the analytic ideas of j p raymond siam j control optim 45 2006 pp 790 828 coupled with the projection idea of m heinkenschloss d c sorensen and k sun siam j sci comput 30 2008 pp 1038 1063 avoiding this explicit projection leads to solving a series of large scale saddle point systems in this paper we construct iterative methods to solve such saddle point systems by deriving efficient preconditioners b

quantification of mr brain images by mixture density and partial volume modeling the problem of automatic quantification of brain tissue by utilizing single valued single echo magnetic resonance imaging mri brain scans is addressed it is shown that this problem can be solved without classification or segmentation a method that may be particularly useful in quantifying white matter lesions where the range of values associated with the lesions and the white matter may heavily overlap the general technique utilizes a statistical model of the noise and partial volume effect together with a finite mixture density description of the tissues the quantification is then formulated as a minimization problem of high order with up to six separate densities as part of the mixture this problem is solved by tree annealing with and without partial volume utilized the results compared and the sensitivity of the tree annealing algorithm to various parameters is exhibited the actual quantification is performed by two methods a classification based method called bayes quantification and parameter estimation results from each method are presented for synthetic and actual data

a non parametric permutation method for assessing agreement for distance matrix observations distance matrix data are occurring ever more frequently in medical research particularly in fields such as genetics dna research and image analysis we propose a non parametric permutation method for assessing agreement when the data under study are distance matrices we apply agglomerative hierarchical clustering and accompanying dendrograms to visualize the internal structure of the matrix observations the accompanying test is based on random permutations of the elements within individual matrix observations and the corresponding matrix mean of these permutations we compare the within matrix element sum of squares wmess for the observed mean against the wmess for the permutation means the methodology is exemplified using simulations and real data from magnetic resonance imaging copyright 2013 john wiley sons ltd

a variational bound nip element method for geometrically stiff problems application to thermal composites and porous media geometrically stiff problems problems which exhibit disparate length scales geometric in origin arise frequently in fluid mechanics and heat transfer perhaps most prevalently in the analysis of multicomponent media in this paper we present a hybrid analytico computational approach to the treatment of geometrically stiff problems for which the quantity of interest can be expressed as the extremum of a functional more precisely we consider in depth the effective conductivity of random thermal composites and briefly the permeability of random porous media the approach proceeds by an inner outer decomposition in which analytical approximations in inner nip regions are folded into modified outer problems defined over geometrically more homogeneous domains the technique considerably mitigates the numerical difficulties associated with geometric stiffness by reduced degrees of freedom improved conditioning less stringent mesh generation requirements and better parallel load balancing while simultaneously providing rigorous upper and lower bounds for the quantity of interest furthermore the method enjoys certain convergence properties the upper lower bound gap can be rendered systematically smaller by decreasing the extent of the inner nip regions we first describe a series of bound procedures and associated modified outer weak forms we next discuss finite element implementation of the outer problem on message passing multicomputers here the intel ipsc 860 hypercube and finally we present several illustrative examples

advances in statistical methodology for the evaluation of diagnostic and laboratory tests the roc plot is a useful tool in the evaluation of the performance of medical tests for separating two populations for a two state decision rule based on such a test the roc plot is the graph of all observed 1 specificity sensitivity pairs each point on this empirical plot can be represented by a 2 x 2 contingency table the non parametric statistics of mann whitney and kolmogorov smirnov can be immediately identified on this plot local non parametric confidence interval procedures related to the theoretical roc curve are briefly reviewed for continuous data two new simultaneous confidence regions associated with the roc curve are presented one based on kolmogorov smirnov confidence bands for distribution functions and the other based on bootstrapping two different tests on the same patients can be compared on the roc scale for continuous data one important problem concerns the comparison of two roc plots as would arise from two correlated diagnostic tests on each patient using a sup norm this metric can detect differences that the roc area cannot the distribution of a statistic based on this norm is studied using the bootstrap a biomedical example illustrates the methodologies

spectral element methods for large scale parallel navier stokes calculations abstract we analyze the computational complexity of a recently developed two level iteration scheme for spectral element solution of the time dependent incompressible navier stokes equations in complex domains we present several algorithmic advances which significantly enhance the scalability of this approach including implementation of an advanced combine operation for degrees of freedom on subdomain edges parallel solution of the fine grained coarse grid problem and implementation of local low order finite element preconditioners for the find grid problem timings on the 512 node intel delta machine show that the combined improvements lead to a fourfold reduction in navier stokes solution time for the particular case of a three dimensional boundary layer calculation consisting of one million degrees of freedom

spectral approximation of the periodic nonperiodic navier stokes equations in order to approximate the navier stokes equations with periodic boundary conditions in two directions and a no slip boundary condition in the third direction by spectral methods we justify by theoretical arguments an appropriate choice of discrete spaces for the velocity and the pressure the compatibility between these two spaces is checked via an infsup condition we analyze a spectral and a collocation pseudo spectral method for the stokes problem and a collocation pseudo spectral method for the navier stokes equations we derive error bounds of spectral type i e which behave likem wherem depends on the number of degrees of freedom of the method and represents the regularity of the data

the chebyshev spectral element approximation with exact quadratures a new chebyshev spectral element method has been developed in this paper in which exact quadratures are used to overcome a shortfall of the gauss chebyshev quadrature in variational spectral element projections the method is validated with the stokes and the cauchy riemann problems it is shown that an enhancement of the approximation convergence rate is attained and numerical accuracy is much better than that from the gauss lobatto legendre spectral element method the paper addressed and assessed some of the issues concerning the numerical accuracy on the chebyshev spectral element method a new chebyshev spectral element method is developed by using exact quadratures in computing spectral elements the method is validated with the stokes and the cauchy riemann problems numerical results show that an enhancement of the approximation convergence rate is attained numerical accuracy is much better than that from other spectral element methods

on the first order rao scott correction of the umesh loughin scherer statistic summary decady and thomas 2000 biometrics56 893 896 propose a first order corrected umesh loughin scherer statistic to test for association in an r c contingency table with multiple column responses agresti and liu 1999 biometrics55 936 943 point out that such statistics are not invariant to the arbitrary designation of a zero or one to a positive response this paper shows that in addition the proposed testing procedure does not hold the correct size when there are strong pairwise associations between responses

iterative substructuring methods for spectral elements problems in three dimensions based on numerical quadrature abstract iterative substructuring methods form an important family of domain decomposition algorithms for elliptic finite element problems two preconditioners for p version finite element methods based on continuous piecewise qp functions are considered for second order elliptic problems in three dimensions these special methods can also be viewed as spectral element methods the first iterative method is designed for the galerkin formulation of the problem the second applies to linear systems for a discrete model derived by using gauss lobatto legendre quadrature for both methods it is established that the condition number of the relevant operator grows only in proportion to 1 log p 2 these bounds are independent of the number of elements into which the given region has been divided their diameters as well as the jumps in the coefficients of the elliptic equation between elements results of numerical computations are also given which provide upper bounds on the condition numbers as functions of p and which confirms the correctness of our theory

a hierarchical duality approach to bounds for the outputs of partial differential equations abstract we present a technique for generating lower and upper bounds to outputs which are linear functionals of the solutions to finiteelement discretizations of symmetric or nonsymmetric coercive linear partial differential equations the method is based upon the construction of an augmented lagrangian which integrates i a quadratic energy reformulation of the desired output as the objective to be minimized with ii the finite element equilibrium equations and conforming hybridized intersubdomain continuity conditions as the constraints to be satisfied the bounds are then derived by appealing to the associated dual unconstrained max min problem evaluated for optimally chosen candidate lagrange multipliers generated by a less expensive approximation such as a low dimensional finite element discretization as in many a posteriori error estimation techniques the bound calculation requires only the solution of subdomain local symmetric neumann problems on the refined truth mesh the technique is presented and illustrated for the case of the one dimensional convection diffusion equation

uniform preconditioners for the time dependent stokes problem implicit time stepping procedures for the time dependent stokes problem lead to stationary singular perturbation problems at each time step these singular perturbation problems are systems of saddle point type which formally approach a mixed formulation of the poisson equation as the time step tends to zero preconditioners for discrete analogous of these systems are discussed the preconditioners uses standard positive definite elliptic preconditioners as building blocks and lead to condition numbers which are bounded uniformly with respect to the time step and the spatial discretization the construction of the discrete preconditioners is related to the mapping properties of the corresponding continuous system

smooth non parametric receiver operating characteristic roc curves for continuous diagnostic tests receiver operating characteristic roc curves have seen increasing use to assess the accuracy of diagnostic tests that yield continuous test results we can calculate a fully non parametric roc curve from the empirical false positive rate fpr and true positive rate tpr at each possible decision threshold but it may be quite jagged we can obtain a smooth roc curve by directly fitting a parametric model for example binormal or bilogistic to the actual test results but substantial lack of fit may result if the distributional assumptions are not valid a recently proposed alternative algorithm labroc4 is insensitive to such departures but is not fully flexible in that the form of the resulting roc curve is restricted to be binormal we propose a smooth non parametric roc curve derived from kernel density estimates of the two test result distributions we obtain pointwise standard errors for the estimated tpr at each fpr and use them to construct pointwise confidence intervals for the roc curve using a logit transformation we also obtain standard errors for the estimated tpr and fpr at given thresholds and use them to construct confidence rectangles in fpr tpr space that correspond to a specific threshold we adapt existing methods for the unsmoothed non parametric roc curve to obtain the area under the roc curve and its standard error we also give partial areas and corresponding standard errors we have created a fortran algorithm roc rol available upon request we compare roc curves and areas obtained by applying our methods and its competitors to two data sets one of which is fit well by parametric methods and labroc4 the other of which is not 1997 john wiley sons ltd

a laguerre legendre spectral method for the stokes problem in a semi infinite channel a mixed spectral method is proposed and analyzed for the stokes problem in a semi infinite channel the method is based on a generalized galerkin approximation with laguerre functions in the x direction and legendre polynomials in the y direction the well posedness of this method is established by deriving a lower bound on the inf sup constant numerical results indicate that the derived lower bound is sharp rigorous error analysis is also carried out