modeling conditional distributions for functional responses with application to traffic monitoring via gps enabled mobile phones motivated by problems involving a traffic monitoring system in which trajectory data are obtained from global positioning system enabled mobile phones we propose a novel approach to functional regression modeling where instead of the usual mean regression the entire distribution of functional responses is modeled conditionally on predictors an approach that sensibly balances flexibility and stability is obtained by assuming that the response functions are drawn from a gaussian process the mean and covariance function of which depend on predictors the dependence of the mean function and covariance function of the response on the predictors is modeled additively we demonstrate the proposed methods by constructing predicted curves and corresponding prediction regions for traffic velocity trajectories for a future time period using current traffic velocity fields as predictor functions the proposed functional regression and conditional distribution approach is of general interest for functional respon

fast and high accuracy multigrid solution of the three dimensional poisson equation we employ a fourth order compact finite difference scheme fos with the multigrid algorithm to solve the three dimensional poisson equation we test the influence of different orderings of the grid space and different grid transfer operators on the convergence and efficiency of our high accuracy algorithm fourier smoothing analysis is conducted to show that fos has a smaller smoothing factor than the traditional second order central difference scheme cds a new method of fourier smoothing analysis is proposed for the partially decoupled red black gauss seidel relaxation with fos numerical results are given to compare the computed accuracy and the computational efficiency of fos with multigrid against cds with multigrid

estimation of functional derivatives situations of a functional predictor paired with a scalar response are increasingly encountered in data analysis predictors are often appropriately modeled as square integrable smooth random functions imposing minimal assumptions on the nature of the functional relationship we aim to estimate the directional derivatives and gradients of the response with respect to the predictor functions in statistical applications and data analysis functional derivatives provide a quantitative measure of the often intricate relationship between changes in predictor trajectories and those in scalar responses this approach provides a natural extension of classical gradient fields in vector space and provides directions of steepest descent we suggest a kernel based method for the nonparametric estimation of functional derivatives that utilizes the decomposition of the random predictor functions into their eigenfunctions these eigenfunctions define a canonical set of directions into which the gradient field is expanded the proposed method is shown to lead to asymptotically consistent estimates of functional derivatives and is illustrated in an application to growth curves

a 3d non newtonian fluid structure interaction model for blood flow in arteries the mathematical modelling and numerical simulation of the human cardiovascular system is playing nowadays an important role in the comprehension of the genesis and development of cardiovascular diseases in this paper we deal with two problems of 3d modelling and simulation in this field which are very often neglected in the literature on the one hand blood flow in arteries is characterized by travelling pressure waves due to the interaction of blood with the vessel wall on the other hand blood exhibits non newtonian properties like shear thinning viscoelasticity and thixotropy the present work is concerned with the coupling of a generalized newtonian fluid accounting for the shear thinning behaviour of blood with an elastic structure describing the vessel wall to capture the pulse wave due to the interaction between blood and the vessel wall we provide an energy estimate for the coupling and compare the numerical results with those obtained with an equivalent fluid structure interaction model using a newtonian fluid

a note on convergence of the multigrid v cycle several recent papers have discussed the convergence of the multigrid v cycle in particular there are several results for the symmetric case where the numbers of smoothings before the fine to coarse transfer and after the coarse to fine transfer are the same in most instances the smoother h i e 1a has been limited to the case where e is positive definite and the eigenvalues h of h satisfy 0

functional adaptive model estimation in this article we are interested in modeling the relationship between a scalar y and a functional predictor x t we introduce a highly flexible approach called functional adaptive model estimation fame which extends generalized linear models glms generalized additive models gams and projection pursuit regression ppr to handle functional predictors the fame approach can model any of the standard exponential family of response distributions that are assumed for glm or gam while maintaining the flexibility of ppr for example standard linear or logistic regression with functional predictors as well as far more complicated models can easily be applied using this approach we use a functional principal components decomposition of the predictor functions to aid visualization of the relationship between x t and y we also show how the fame procedure can be extended to deal with multiple functional and standard finite dimensional predictors possibly with missing data we illustrate the fame

nonparametric estimation of the mode of a distribution of random curves summary motivated by the need to develop meaningful empirical approximations to a typical data value we introduce methods for density and mode estimation when data are in the form of random curves our approach is based on finite dimensional approximations via generalized fourier expansions on an empirically chosen basis the mode estimation problem is reduced to a problem of kernel type multivariate estimation from vector data and is solved using a new recursive algorithm for finding the empirical mode the algorithm may be used as an aid to the identification of clusters in a set of data curves bootstrap methods are employed to select the bandwidth

non planar curvature and branching of arteries and non planar type flow in this study magnetic resonance imaging techniques have been used to examine the geometry of arterial curvature and branching in casts and in vivo and to measure the distribution of axial velocity in the associated flow it is found contrary to a widely held view that the geometry is commonly non planar moreover relatively small values of the parameters which render the geometry non planar appear significantly to affect the velocity distribution the findings suggest that non planarity is an important factor influencing arterial flows including wall shear the implications are not restricted to vascular biology pathology and surgery but may extend to the design of general piping systems

algebraic study of multigrid methods for symmetric definite problems abstract we present a convergence theory based on a simple algebraic assumption which permits us to analyze multilevel iterative methods for symmetric positive definite linear systems coarse grid problems are derived variationally we prove fast convergence of v and w cycles with any positive number of smoothing steps under a discrete analogue of the h 2 and h 1 ga regularity assumption respectively and for a wide class of smoothings including arbitrarily ordered gauss seidel and steepest descent

a velocity pressure streamline diffusion finite element method for navier stokes equations a streamline diffusion finite element method is introduced for the time dependent incompressible navier stokes equations in a bounded domain in r squared and r cubed in the case of a flow with a high reynolds number an error estimate is proved and numerical results are given the method is based on a mixed velocity pressure formulation using the same finite element discretization of space time for the velocity and the pressure spaces which consist of piecewise linear functions together with certain least squares modifications of the galerkin variational formulation giving added stability without sacrificing accuracy

derivative estimation for longitudinal data analysis examining features of blood pressure measured repeatedly during pregnancy estimating velocity and acceleration trajectories allows novel inferences in the field of longitudinal data analysis such as estimating change regions rather than change points and testing group effects on nonlinear change in an outcome ie a nonlinear interaction in this article we develop derivative estimation for 2 standard approaches polynomial mixed models and spline mixed models we compare their performance with an established method principal component analysis through conditional expectation through a simulation study we then apply the methods to repeated blood pressure bp measurements in a uk cohort of pregnant women where the goals of analysis are to i identify and estimate regions of bp change for each individual and ii investigate the association between parity and bp change at the population level the penalized spline mixed model had the lowest bias in our simulation study and we identified evidence for bp change regions in over 75 of pregnant women using mean velocity difference revealed differences in bp change between women in their first pregnancy compared with those who had at least 1 previous pregnancy we recommend the use of penalized spline mixed models for derivative estimation in longitudinal data analysis

robust multigrid algorithms for the navier strokes equations abstract anisotropies occur naturally in computational fluid dynamics where the simulation of small scale physical phenomena such as boundary layers at high reynolds numbers causes the grid to be highly stretched leading to a slowdown in convergence of multigrid methods several approaches aimed at making multigrid a robust solver have been proposed and analyzed in the literature using the scalar diffusion equation however they have rarely been applied to solving more complicated models such as the incompressible navier stokes equations this paper contains the first published numerical results of the behavior of two popular robust multigrid approaches alternating plane smoothers combined with standard coarsening and plane implicit smoothers combined with semi coarsening for solving the 3 d incompressible navier stokes equations in the simulation of the driven cavity and a boundary layer over a flat plate on a stretched grid grid size grid stretching and reynolds number are the factors considered in evaluating the robustness of the multigrid methods both approaches yield large increases in convergence rates over cell implicit smoothers on stretched grids the combination of plane implicit smoothers and semi coarsening was found to be fully robust in the flat plate simulation up to reynolds numbers 10 6 and the best alternative in the driven cavity simulation for reynolds numbers above 10 3 the alternating plane approach exhibits a better behavior for lower reynolds numbers below 10 3 in the driven cavity simulation

a front tracking method for computation of interfacial flows with soluble surfactants a finite difference front tracking method is developed for computations of interfacial flows with soluble surfactants the method is designed to solve the evolution equations of the interfacial and bulk surfactant concentrations together with the incompressible navier stokes equations using a non linear equation of state that relates interfacial surface tension to surfactant concentration at the interface the method is validated for simple test cases and the computational results are found to be in a good agreement with the analytical solutions the method is then applied to study the cleavage of drop by surfactant a problem proposed as a model for cytokinesis h p greenspan on the dynamics of cell cleavage j theor biol 65 1 1977 79 h p greenspan on fluid mechanical simulations of cell division and movement j theor biol 70 1 1978 125 finally the method is used to model the effects of soluble surfactants on the motion of buoyancy driven bubbles in a circular tube and the results are found to be in a good agreement with available experimental data

new convergence estimates for multigrid algorithms in this paper new convergence estimates are proved for both symmetric and nonsymmetric multigrid algorithms applied to symmetric positive definite problems our theory relates the convergence of multigrid algorithms to a regularity and approximation parameter cap alpha epsilon 0 1 and the number of relaxations m we show that for the symmetric and nonsymmetric nu cycles the multigrid iteration converges for any positive m at a rate which deteriorates no worse than 1 cj sup 1 sup cap alpha sup sup cap alpha where j is the number of grid levels we then define a generalized nu cycle algorithm which involves exponentially increasing for example doubling the number of smoothings on successively coarser grids we show that the resulting symmetric and nonsymmetric multigrid iterations converge for any cap alpha with rates that are independent of the mesh size the theory is presented in an abstract setting which can be applied to finite element multigrid and finite difference multigrid methods

a general framework for nonlinear multigrid inversion a variety of new imaging modalities such as optical diffusion tomography require the inversion of a forward problem that is modeled by the solution to a three dimensional 3d partial differential equation for these applications image reconstruction is particularly difficult because the forward problem is both nonlinear and computationally expensive to evaluate we propose a general framework for nonlinear multigrid inversion that is applicable to a wide variety of inverse problems the multigrid inversion algorithm results from the application of recursive multigrid techniques to the solution of optimization problems arising from inverse problems the method works by dynamically adjusting the cost functionals at different scales so that they are consistent with and ultimately reduce the finest scale cost functional in this way the multigrid inversion algorithm efficiently computes the solution to the desired fine scale inversion problem importantly the new algorithm can greatly reduce computation because both the forward and inverse problems are more coarsely discretized at lower resolutions an application of our method to bayesian optical diffusion tomography with a generalized gaussian markov random field image prior model shows the potential for very large computational savings numerical data also indicates robust convergence with a range of initialization conditions for this nonconvex optimization problem

basic smoothing procedures for the multigrid treatment of elliptic 3d operators from 2d multigrid it is a fundamental insight that pointwise relaxation combined with standard coarsening gives no reasonable smoothing effect if the given elliptic operator shows appreciably anisotropic behavior corresponding questions and practical consequences are discussed in this paper for the 3d case in the general anisotropic 3d case even line relaxation is not sufficient if standard coarsening is maintained instead plane relaxation is necessary in certain cases if plane relaxation is applied correctly and performed by use of appropriate 2d multigrid the resulting 3d multigrid method has an asymptotic complexity of o n where n number of 3d grid points and is highly efficient even for moderate values of n in contrast to the common opinion plane relaxation turns out to be a simple and general smoothing concept for standard elliptic 3d problems

on the shear thinning and viscoelastic effects of blood flow under various flow rates the aim of this paper is to describe and discuss the results of numerical comparative study performed in order to demonstrate and quantify some of the most relevant non newtonian characteristics of blood flow in medium sized blood vessels namely its shear thinning and viscoelastic behavior the models studied in this work are the classical newtonian and oldroyd b models as well as their generalized shear thinning modifications numerical tests are performed on three dimensional geometries namely an idealized axisymmetric stenosis and a realistic stenosed carotid bifurcation reconstructed from medical images the numerical solution of the system of governing equations is obtained by a finite volume method on a structured grid model sensitivity tests are achieved with respect to the characteristic flow rate to evaluate its impact on the observed non newtonian effects

methodology and convergence rates for functional linear regression in functional linear regression the slope parameter is a function therefore in a nonparametric context it is determined by an infinite number of unknowns its estimation involves solving an ill posed problem and has points of contact with a range of methodologies including statistical smoothing and deconvolution the standard approach to estimating the slope function is based explicitly on functional principal components analysis and consequently on spectral decomposition in terms of eigenvalues and eigen functions we discuss this approach in detail and show that in certain circumstances optimal convergence rates are achieved by the pca technique an alternative approach based on quadratic regularisation is suggested and shown to have advantages from some points of view

finite element error estimation for quasi newtonian fluid structure interaction problems we consider a monolithic scheme for fluid structure interaction problems involving an incompressible quasi newtonian fluid the monolithic formulation is obtained using the arbitrary lagrangian eulerian ale method with matching conditions at the fluid structure interface the stability and error analysis are performed for the finite element approximation finally some numerical experiments that confirm the theoretical analysis are presented

continuously additive models for nonlinear functional regression summary we introduce continuously additive models which can be viewed as extensions of additive regression models with vector predictors to the case of infinite dimensional predictors this approach produces a class of flexible functional nonlinear regression models where random predictor curves are coupled with scalar responses in continuously additive modelling integrals taken over a smooth surface along graphs of predictor functions relate the predictors to the responses in a nonlinear fashion we use tensor product basis expansions to fit the smooth regression surface that characterizes the model in a theoretical investigation we show that the predictions obtained from fitting continuously additive models are consistent and asymptotically normal we also consider extensions to generalized responses the proposed class of models outperforms existing functional regression models in simulations and real data examples

conditional quantile analysis when covariates are functions with application to growth data summary motivated by the conditional growth charts problem we develop a method for conditional quantile analysis when predictors take values in a functional space the method proposed aims at estimating conditional distribution functions under a generalized functional regression framework this approach facilitates balancing of model flexibility and the curse of dimensionality for the infinite dimensional functional predictors its good performance in comparison with other methods both for sparsely and for densely observed functional covariates is demonstrated through theory as well as in simulations and an application to growth curves where the method proposed can for example be used to assess the entire growth pattern of a child by relating it to the predicted quantiles of adult height

estimating and depicting the structure of a distribution of random functions summary we suggest a nonparametric approach to making inference about the structure of distributions in a potentially infinite dimensional space for example a function space and displaying information about that structure it is suggested that the simplest way of presenting the structure is through modes and density ascent lines the latter being the projections into the sample space of the curves of steepest ascent up the surface of a functionaldata density modes are always points in the sample space and ascent lines are always one parameter structures even when the sample space is determined by an infinite number of parameters they are therefore relatively easily depicted our methodology is based on a functional form of an iterative data sharpening algorithm

nonparametric maximum likelihood estimation of the structural mean of a sample of curves a random sample of curves can be usually thought of as noisy realisations of a compound stochastic process x t z w t where z t produces random amplitude variation and w t produces random dynamic or phase variation in most applications it is more important to estimate the so called structural mean t e z t than the crosssectional mean e x t but this estimation problem is difficult because the process z t is not directly observable in this paper we propose a nonparametric maximum likelihood estimator of t this estimator is shown to be surd n consistent and asymptotically normal under the assumed model and robust to model misspecification simulations and a realdata example show that the proposed estimator is competitive with landmark registration often considered the benchmark and has the advantage of avoiding time consuming and often infeasible individual landmark identification

numerical investigation of the non newtonian blood flow in a bifurcation model with a non planar branch the pulsatile flow of non newtonian fluid in a bifurcation model with a non planar daughter branch is investigated numerically by using the carreau yasuda model to take into account the shear thinning behavior of the analog blood fluid the objective of this study is to deal with the influence of the non newtonian property of fluid and of out of plane curvature in the non planar daughter vessel on wall shear stress wss oscillatory shear index osi and flow phenomena during the pulse cycle the non newtonian property in the daughter vessels induces a flattened axial velocity profile due to its shear thinning behavior the non planarity deflects flow from the inner wall of the vessel to the outer wall and changes the distribution of wss along the vessel in particular in systole phase downstream of the bifurcation the velocity profiles are shifted toward the flow divider and low wss and high shear stress temporal oscillations characterized by osi occur on the outer wall region of the daughter vessels close to the bifurcation secondary motions become stronger with the addition of the out of plane curvature induced by the bending of the vessel and the secondary flow patterns swirl along the non planar daughter vessel a significant difference between the non newtonian and the newtonian pulsatile flow is revealed during the pulse cycle however reasonable agreement between the non newtonian and the rescaled newtonian flow is found calculated results for the pulsatile flow support the view that the non planarity of blood vessels and the non newtonian properties of blood are an important factor in hemodynamics and may play a significant role in vascular biology and pathophysiology

modified multigrid for 3d elliptic equations with cross derivatives a portable fortran code that solves the general nonseparable three dimensional linear elliptic partial differential equation pde with cross derivative terms on a rectangular region is described boundary conditions can be any combination of periodic specified or mixed derivative a multigrid scheme modified to handle complexities introduced by cross derivative terms and nonnormal derivative components in boundary conditions is utilized successful application is illustrated with an analytically prescribed abstract example and simulation of an idealized geophysical fluid flow

functional data analysis for sparse longitudinal data we propose a nonparametric method to perform functional principal components analysis for the case of sparse longitudinal data the method aims at irregularly spaced longitudinal data where the number of repeated measurements available per subject is small in contrast classical functional data analysis requires a large number of regularly spaced measurements per subject we assume that the repeated measurements are located randomly with a random number of repetitions for each subject and are determined by an underlying smooth random subject specific trajectory plus measurement errors basic elements of our approach are the parsimonious estimation of the covariance structure and mean function of the trajectories and the estimation of the variance of the measurement errors the eigenfunction basis is estimated from the data and functional principal components score estimates are obtained by a conditioning step this conditional estimation method is conceptually simple and straightforward to implement

optimal alternating direction implicit preconditioners for conjugate gradient methods in this work the extrapolated alternating direction implicit e adi preconditioners for the conjugate gradient cg method are applied for the solution of the second order elliptic partial differential equation pde in a rectangle under dirichlet boundary conditions it is shown that the spectral condition number for the discretized model problem when the adi cg method is used is an order of magnitude smaller than that of the cg method the pde is approximated by uniform 5 and 9 point stencils and it is shown that the determination of the optimal condition number is equivalent to that of the determination of the optimal eadi scheme analytic expressions for the optimal acceleration and extrapolation parameters are obtained in both cases of the 5 and 9 point schemes those for the 5 point complete known ones while those for the 9 point are novel finally numerical examples show that the eadi preconditioners proposed make the cg methods compete well against most of the fastest well known methods

fourier analysis of gmres m preconditioned by multigrid this paper deals with convergence estimates of gmres m saad and schultz siam j sci statist comput 7 1986 pp 856 869 preconditioned by multigrid brandt math comp 31 1977 pp 333 390 hackbusch multi grid methods and applications springer berlin 1985 fourier analysis is a well known and useful tool in the multigrid community for the prediction of two grid convergence rates brandt math comp 31 1977 pp 333 390 stuben and trottenberg in multigrid methods lecture notes in math 960 k stuben and u trottenberg eds springer berlin pp 1 176 this analysis is generalized here to the situation in which multigrid is a preconditioner since it is possible to obtain the whole spectrum of the two grid iteration matrix a preconditioned krylov subspace acceleration method like gmres m implicitly builds up a minimal residual polynomial the determination of the polynomial coefficients is easily possible and can be done explicitly since from fourier analysis a simple block diagonal two grid iteration matrix results based on the gmres m polynomial sharp theoretical convergence estimates can be obtained which are compared with estimates based on the spectrum of the iteration matrix several numerical scalar test problems are computed in order to validate the theoretical predictions

comparison of second and fourth order discretizations for multigrid poisson solvers we combine a compact high order difference approximation with multigrid v cycle algorithm to solve the two dimensional poisson equation with dirichlet boundary conditions this scheme along with several different orderings of grid space and projection operators is compared with the five point formula to show the dramatic improvement in computed accuracy on serial and vector machines

a study of wall shear stress in 12 aneurysms with respect to different viscosity models and flow conditions recent computational fluid dynamics cfd studies relate abnormal blood flow to rupture of cerebral aneurysms however it is still debated how to model blood flow with sufficient accuracy common assumptions made include newtonian behaviour of blood traction free outlet boundary conditions and inlet boundary conditions based on available literature these assumptions are often required since the available patient specific data is usually restricted to the geometry of the aneurysm and the surrounding vasculature however the consequences of these assumptions have so far been inadequately addressed this study investigates the effects of 4 different viscosity models 2 different inflow conditions and 2 different outflow conditions in 12 middle cerebral artery aneurysms the differences are quantified in terms of 3 different wall shear stress wss metrics involving maximal wss average wss and proportion of aneurysm sac area with low wss the results were compared with common geometrical metrics such as volume aspect ratio size ratio and parent vessel diameter and classifications in terms of sex and aneurysm type the results demonstrate strong correlations between the different viscosity models and boundary conditions the correlation between the different wss metrics range from weak to medium no strong correlations were found between the different wss metrics and the geometrical metrics or classifications

fast and robust sixth order multigrid computation for the three dimensional convection diffusion equation we present a sixth order explicit compact finite difference scheme to solve the three dimensional 3d convection diffusion equation we first use a multiscale multigrid method to solve the linear systems arising from a 19 point fourth order discretization scheme to compute the fourth order solutions on both a coarse grid and a fine grid then an operator based interpolation scheme combined with an extrapolation technique is used to approximate the sixth order accurate solution on the fine grid since the multigrid method using a standard point relaxation smoother may fail to achieve the optimal grid independent convergence rate for solving convection diffusion equations with a high reynolds number we implement the plane relaxation smoother in the multigrid solver to achieve better grid independency supporting numerical results are presented to demonstrate the efficiency and accuracy of the sixth order compact soc scheme compared with the previously published fourth order compact foc scheme

on the importance of blood rheology for bulk flow in hemodynamic models of the carotid bifurcation abstract here we present a study on the impact of assumptions on image based hemodynamic simulations of healthy carotid bifurcations in particular we evaluate to which extent assumptions on blood rheology influence bulk flow features driven by the fact that few studies have provided adequate insights into the influence of assumptions to confidently model the 4d hemodynamics within the bifurcation the final goal is to complement integrate and extend with a quantitative characterization of the bulk flow the description currently adopted to classify altered hemodynamics which is based on wall shear stress wss hemodynamic simulations of two image based carotid bifurcation geometries were carried out assuming a reference newtonian viscosity two non newtonian rheology models and newtonian viscosities based on characteristic shear rates wss based and lagrangian based metrics for helical flow quantification and for vorticity dynamics quantification were calculated our findings suggest that the assumption of newtonian rheology 1 could be reasonable for bulk flow metrics differences from non newtonian behavior are lower than 10 2 influences at different levels the wss based indicators depending on the bifurcation model even if in our study it is lower than the major source of uncertainty as recognized by the literature i e uncertainty on geometry reconstruction

on the multigrid f cycle in a recent paper a bound was derived on the convergence of the multigrid v cycle for the case when the solution is in the sobolev space h 1 a but not in h 1 a a 62 a showing that the convergence factor approaches one only as 1 o k a 1 a for a large number of levels k we now extend the technique to obtain the asymptotically better bound 1 o k 1 a 2 a on the multigrid f cycle we also show that in many cases for practical values of k one gets the same bound for the f cycle as for the v cycle with a 1

a gmres based plane smoother in multigrid to solve 3d anisotropic fluid flow problems for a discretization of the 3d steady incompressible navier stokes equations a solution method is presented for solving flow problems on stretched grids the discretization is a vertex centered finite volume discretization with a flux splitting approach for the convective terms second order accuracy is obtained with the well known defect correction technique b koren j comput phys 87 25 1990 the solution method used is multigrid for which a plane smoother is presented for obtaining good convergence in flow domains with severely stretched grids a matrix is set up in a plane which is solved iteratively with a preconditioned gmres method here a stop criterion for gmres is tested which reduces the number of inner iterations compared to an exact plane solver without affecting the multigrid convergence rates the performance of the solution method is shown for a poisson model problem and for 3d incompressible channel flow examples

mudpack 2 multigrid software for approximating elliptic partial differential equations on uniform grids with any resolution we describe a multigrid software package mudpack 2 that approximates elliptic partial differential equations in two and three dimensions on uniform meshes without restrictions on the number of grid points in each direction mudpack 2 retains all the functionality of the original mudpack

aspects of multigrid methods for problems in three dimensions smoothing methods are analyzed for use in multigrid methods applied to flow problems in three dimensions a three dimensional version of incomplete block factorization is described the anisotropic diffusion and convection diffusion equations are used as model problems unlike the two dimensional case none of the smoothing methods considered is suitable for all relevant values of the physical parameters

dynamic curvature strongly affects wall shear rates in a coronary artery bifurcation model this study was motivated by the need for a better understanding of coronary artery blood flow patterns and their possible role in atherosclerosis formation of particular interest in this study was the effects of the dynamic deformation due to myocardial contraction on wall shear rate patterns in the coronary arteries a better understanding of these effects on wall shear rate in a bifurcation geometry and an evaluation of the importance of these effects was desired a three dimensional computer model of a bifurcation lying on the surface of a sphere with time varying radius of curvature was employed to simulate the motion and deformation of the arteries the results indicated low mean shear rates along the myocardial wall and very high shear rate variations over 100 of the static mean shear rate along the outer wall the results obtained using a quasi static analysis were found to underestimate the dynamic wall shear rate variation along the myocardial and outer walls it was concluded that dynamic geometry effects are important in determining sites of low mean and oscillating wall shear that have been associated with atherogenesis in curved bifurcating arteries

functional linear regression analysis for longitudinal data we propose nonparametric methods for functional linear regression which are designed for sparse longitudinal data where both the predictor and response are functions of a covariate such as time predictor and response processes have smooth random trajectories and the data consist of a small number of noisy repeated measurements made at irregular times for a sample of subjects in longitudinal studies the number of repeated measurements per subject is often small and may be modeled as a discrete random number and accordingly only a finite and asymptotically nonincreasing number of measurements are available for each subject or experimental unit we propose a functional regression approach for this situation using functional principal component analysis where we estimate the functional principal component scores through conditional expectations this allows the prediction of an unobserved response trajectory from sparse measurements of a predictor trajectory the resulting technique is flexible and allows for different patterns regarding the timing of the measurements obtained for predictor and response trajectories asymptotic properties for a sample of n subjects are investigated under mild conditions as n oo and we obtain consistent estimation for the regression function besides convergence results for the components of functional linear regression such as the regression parameter function we construct asymptotic pointwise confidence bands for the predicted trajectories a functional coefficient of determination as a measure of the variance explained by the functional regression model is introduced extending the standard r 2 to the functional case the proposed methods are illustrated with a simulation study longitudinal primary biliary liver cirrhosis data and an analysis of the longitudinal relationship between blood pressure and body mass index

the influence of the non newtonian properties of blood on the flow in large arteries unsteady flow in a 90 curved tube a numerical and experimental investigation of unsteady entry ow in a 903 curved tube is presented to study the impact of the non newtonian properties of blood on the velocity distribution the time dependent ow rate for the newtonian and the nonnewtonian blood analog uid were identical for the numerical computation a carreau yasuda model was employed to accommodate the shear thinning behavior of the xanthan gum solution the viscoelastic properties were not taken into account the experimental results indicate that signi cant di erences between the newtonian and non newtonian uid are present the numerical results for both the newtonian and the non newtonian uid agree well with the experimental results since viscoelasticity was not included in the numerical code shear thinning behavior of the blood analog uid seems to be the dominant non newtonian property even under unsteady ow conditions finally a comparison between the non newtonian uid model and a newtonian uid at a rescaled reynolds number is presented the rescaled reynolds number based on a characteristic rather than the high shear rate viscosity of the xanthan gum solution was about three times as low as the original reynolds number comparison reveals that the character of ow of the non newtonian uid is simulated quite well by using the appropriate reynolds number 1999 elsevier science ltd all rights reserved

statistical tools to analyze data representing a sample of curves the paper is concerned with data representing a sample of smooth curves which can be considered as independent realizations of an underlying biological chemical ldots process such samples of curves often possess the following features there is a typical structural pattern common to all curves of the sample on the other hand individual realizations of the typical shape show different dynamics and intensity in particular typical peaks are shifted from individual to individual differences in dynamics complicate the analysis of samples of curves for example the cross sectional average usually does not reflect an average pattern due to shifts structure is smeared or might even disappear our approach consists in synchronizing the individual curves before determining the average or any further statistics pointwise averaging of the synchronized curves then leads to an average curve which represents the common structure with average dynamics and average intensity the method requires the introduction of new statistical objects they are defined mathematically their properties are discussed and possible estimators are proposed the asymptotic bias and variance of the estimators are derived an application to visually evoked brain potentials illustrates the approach

some tools for functional data analysis multivariate data analysis permits the study of observations which are finite sets of numbers but modern data collection situations can involve data or the processes giving rise to them which are functions functional data analysis involves infinite dimensional processes and or data the paper shows how the theory of l splines can support generalizations of linear modelling and principal components analysis to samples drawn from random functions spline smoothing rests on a partition of a function space into two orthogonal subspaces one of which contains the obvious or structural components of variation among a set of observed functions and the other of which contains residual components this partitioning is achieved through the use of a linear differential operator and we show how the theory of polynomial splines can be applied more generally with an arbitrary operator and associated boundary constraints these data analysis tools are illustrated by a study of variation in temperature precipitation patterns among some canadian weather stations

flexible multiple semicoarsening for three dimensional singularly perturbed problems we present robust parallel multigrid based solvers for 3d scalar partial differential equations the robustness is obtained by combining multiple semicoarsening strategies matrix dependent transfer operators and a krylov subspace acceleration the basis for the 3d preconditioner is a 2d method with multiple semicoarsened grids based on the mg s method from c w oosterlee appl numer math 19 1995 pp 115 128 and t washio and c w oosterlee gmd arbeitspapier 949 gmd st augustin germany 1995 the 2d method is generalized to three dimensions with a line smoother in the third dimension the method based on semicoarsening has been parallelized with the grid partitioning technique j linden b steckel and k stuben parallel comput 7 1988 pp 461 475 o a mcbryan et al impact comput sci engrg 3 1991 pp 1 75 and is evaluated as a solver and as a preconditioner on a mimd machine the robustness of the 3d method is shown for finite volume and finite difference discretizations of 3d anisotropic diffusion equations and convection dominated convection diffusion problems

a power law model of blood flow through a tapered overlapping stenosed artery a mathematical model of non newtonian blood flow through a tapered stenotic artery is considered the non newtonian model chosen is characterized by the generalized power law model incorporating the effect of tapering due to the pulsatile nature of blood flow the flow is assumed to be unsteady laminar two dimensional and axisymmetric the governing equations of motion in terms of the viscous shear stress in the cylindrical coordinate system are solved numerically using a finite difference scheme numerical results obtained for the positive taper angle show that the blood flow characteristics such as the axial velocity profiles flow rate and wall shear stress have lower values while the resistive impedances have higher values than the newtonian model

an efficient approach to study the pulsatile blood flow in femoral and coronary arteries by differential quadrature method abstract in this paper flow analysis for a non newtonian third grade blood in coronary and femoral arteries is simulated numerically blood is considered as the third grade non newtonian fluid under periodic body acceleration motion and pulsatile pressure gradient differential quadrature method dqm and crank nicholson method cnm are used to solve the partial differential equation pde governing equation by which a good agreement between them was observed in the results the influences of some physical parameters such as amplitude lead angle and body acceleration frequency on non dimensional velocity and profiles are considered for instance the results show that increasing the amplitude a g and reducing the lead angle of body acceleration make higher velocity profiles in the center line of both arteries

sharp estimates for multigrid rates of convergence with general smoothing and acceleration in this paper we prove the convergence of the multilevel iterative method for solving linear equations that arise from elliptic partial differential equations our theory is presented entirely in terms of the generalized condition number kappa of the matrix a and the smoothing matrix b this leads to a completely algebraic analysis of the method as an iterative technique for solving linear equations the properties of the elliptic equation and discretization procedure enter only when we seek to estimate kappa just as in the case of most standard iterative methods here we consider the fundamental two level iteration and the v and w cycles of the j level iteration j 2 we prove that the v and w cycles converge even when only one smoothing iteration is used we present several examples of the computation of kappa using both fourier analysis and standard finite element techniques we compare the predictions of our theorems with the actual rate of convergence our analysis also shows that

convergence of nonconforming v cycle and f cycle multigrid algorithms for second order elliptic boundary value problems the convergence of v cycle and f cycle multigrid algorithms with a sufficiently large number of smoothing steps is established for nonconforming finite element methods for second order elliptic boundary value problems

pulsatile flow of non newtonian fluids through arterial stenoses the problem of blood flow through stenoses is solved using the incompressible generalized newtonian model the herschel bulkley bingham and power law fluids are incorporated the geometry corresponds to a rigid circular tube with a partial occlusion calculations are performed by a galerkin finite element method for the pulsatile case a predictor corrector time marching scheme is used with an adaptive time step results are obtained for steady and pulsatile physiological flows computations show that the memory effects taken into account in the model affect deeply the flow compared with newtonian reference case the disturbances are stronger by their vorticity intensity and persist after the geometrical obstacle this is especially true for severe stenoses

a unified convergence theory for abstract multigrid or multilevel algorithms serial and parallel multigrid methods are analyzed in the style of standard iterative methods a basic error bound is derived in terms of residuals on neighboring levels the terms in this bound derive from the iterative methods used as smoothers on each level and the operators used to go from a level to the next coarser level this bound is correct whether the underlying operator is symmetric or nonsymmetric definite or indefinite and singular or nonsingular this paper allows any iterative method as a smoother or rougher in the multigrid cycle while standard multigrid error analysis typically assumes a specific multigrid cycle e g a v w or f cycle analysis for arbitrary multigrid cycles including adaptively chosen ones is provided this theory applies directly to aggregation disaggregation methods used to solve systems of linear equations

high accuracy multigrid solution of the 3d convection diffusion equation we present an explicit fourth order compact finite difference scheme for approximating the three dimensional 3d convection diffusion equation with variable coefficients this 19 point formula is defined on a uniform cubic grid fourier smoothing analysis is performed to show that the smoothing factor of certain relaxation techniques used with the scheme is smaller than 1 we design a parallelization oriented multigrid method for fast solution of the resulting linear system using a four color gauss seidel relaxation technique for robustness and efficiency and a scaled residual injection operator to reduce the cost of multigrid inter grid transfer operator numerical experiments on a 16 processor vector computer are used to test the high accuracy of the discretization scheme as well as the fast convergence and the parallelization or vectorization efficiency of the solution method several test problems are solved and highly accurate solutions of the 3d convection diffusion equations are obtained for small to medium values of the grid reynolds number effects of using different residual projection operators are compared on both vector and serial computers

assessment of blood volume flow in slightly curved arteries from a single velocity profile abstract non invasive estimation of arterial blood volume flow bvf has become a central issue in assessment of cardiovascular risk poiseuille and womersley approaches are commonly used to assess the bvf from centerline velocity but both methods neglect the influence of curvature based on the assumption that the velocity in curved tubes as function of the circumferential position for a given radial position can be approximated by a cosine the bvf can also be estimated by averaging velocities at opposite radial positions referred to as the cosine model ctm this study investigates the accuracy of bvf estimation in slightly curved arteries for bvf waveforms obtained in the brachial artery of 6 volunteers computational fluid dynamics simulations were used to compute the influence of curvature on velocity profiles the bvf was then estimated from the simulation results with the ctm and methods based on poiseuille womersley and using the center stream velocity and the velocity waveform at the position where the maximum velocity is observed and compared to the prescribed bvf the simulations show that the influence of curvature is strongest when the flow decelerates for poiseuille and womersley the time average bvf was underestimated by maximally 10 4 and 7 8 for a radius of curvature of 50 and 100 mm respectively the estimation error is lower for the ctm and equals 4 2 and 1 2 for a radius of curvature of 50 and 100 mm respectively from this study we can conclude that the velocity waveform at the position of the maximum rather than the center stream velocity waveform combined with the womersley method should be chosen the ctm improves current estimation techniques if in vivo velocity distributions are available

three dimensional bayesian optical image reconstruction with domain decomposition most current efforts in near infrared optical tomography are effectively limited to two dimensional reconstructions due to the computationally intensive nature of full three dimensional 3 d data inversion previously we described a new computationally efficient and statistically powerful inversion method apprize automatic progressive parameter reducing inverse zonation and estimation the apprize method computes minimum variance estimates of parameter values here spatially variant absorption due to a fluorescent contrast agent and covariance while simultaneously estimating the number of parameters needed as well as the size shape and location of the spatial regions that correspond to those parameters estimates of measurement and model error are explicitly incorporated into the procedure and implicitly regularize the inversion in a physically based manner the optimal estimation of parameters is bounds constrained precluding infeasible values in this paper the apprize method for optical imaging is extended for application to arbitrarily large 3 d domains through the use of domain decomposition the effect of subdomain size on the performance of the method is examined by assessing the sensitivity for identifying 112 randomly located single voxel heterogeneities in 58 3 d domains also investigated are the effects of unmodeled heterogeneity in background optical properties the method is tested on simulated frequency domain photon migration measurements at 100 mhz in order to recover absorption maps owing to fluorescent contrast agent this study provides a new approach for computationally tractable 3 d optical tomography

mudpack multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations mudpack is a package of portable fortran subprograms which use multigrid iteration for solving real or complex elliptic partial differential equations the solution regions are rectangles in two dimensions and boxes in three dimensions any combination of periodic dirichlet and mixed derivative boundary conditions is allowed the equations are automatically discretized using second order finite differencing the package will vectorize on cray computers examples are given which demonstrate ease of use efficiency and applicability to a wide range of problems

multi grid solution of three dimensional problems with discontinuous coefficients a three dimensional multi grid algorithm is developed and compared to an efficient three dimensional incomplete cholesky conjugate gradient iccg algorithm on a suite of test problems with discontinouous and or anisotropic coefficients the multi grid algorithm used plane relaxation and a definition of coarse grid operators which is an extension of that used for two dimensional problems with discontinuous coefficients numerical results show that the three dimensional results are not nearly as good as the two dimensional results on similar problems

on the implementation of the gmres m method to elliptic equations in meteorology in this paper we consider the relatively new preconditioned generalized minimal residual method restarted every m iterations gmres m for the solution of three dimensional elliptic equations large sparse non symmetric matrices are involved the particular equation of interest is the quasi geostrophic omega equation often used in meteorology to compute vertical motion the gmres m method is tested with different preconditioners for the solution of two and three dimensional elliptic equations the method requires no relaxation parameters and has no restrictions on the size of the 3d grid gmres can be used for more types of matrices than other methods such as sor numerical results show that jacobi preconditioned gmres m performs best for 3d and high resolution problems among five different preconditioners tested while ilij factorization of the partial or whole matrix a as a preconditioner is good for 2d and low resolution problems the sor preconditioners for the gmres m method with optimal relaxation parameters are not as efficient and the best choice for the relaxation parameter in sor preconditioning is not the same as the best choice for the simple sor method an algorithm for using the preconditioned gmres m method is presented

fast and high accuracy multiscale multigrid method with multiple coarse grid updating strategy for the 3d convection diffusion equation an improved multiscale multigrid method with multiple coarse grid updating strategy for a three dimensional convection diffusion equation is presented the novelty of the proposed method lies in a fine grid updating strategy arising from the idea of multiple coarse grid computation the new fine grid updating strategy is able to replace the iterative refinement procedure in the existing multiscale multigrid method wang and zhang 2010 16 to obtain high order solutions since the proposed method needs a fourth order compact scheme with unequal meshsize grids a 19 point fourth order compact difference scheme with unequal meshsize in different coordinate directions is also developed for the three dimensional convection diffusion equation numerical results are given to compare the computed accuracy and the computational efficiency of the multiscale multigrid method with the multiple coarse grid updating strategy against the multiscale multigrid method with the iterative refinement procedure

multigrid methods for variational problems general theory for the v cycle in two earlier papers siam j numer anal 19 1982 pp 924 929 21 1984 pp 255 262 we developed an algebraic convergence theory for a class of multigrid methods applied to positive definite self adjoint linear operator equations the purpose of the present paper is to extend these results by eliminating an earlier approximation order restriction developing additional rate estimates and allowing for very general relaxation schemes including those that are nonstationary and nonsymmetric these results apply to most well known iterative methods and preconditioners

multigrid for high dimensional elliptic partial differential equations on non equidistant grids this work presents techniques theory and numbers for multigrid in a general d dimensional setting the main focus of this paper is the multigrid convergence for high dimensional partial differential equations on non equidistant grids such as may be encountered in a sparse grid solution as a model problem we have chosen the anisotropic stationary diffusion equation on a rectangular hypercube we present some techniques for building the general d dimensional adaptations of the multigrid components and propose grid coarsening strategies to handle anisotropies that are induced due to discretization on a non equidistant grid apart from the practical formulas and techniques we present in detail the smoothing analysis of the point omega red black jacobi method for a general multidimensional case we show how relaxation parameters may be evaluated efficiently and used for better convergence this analysis incorporates full and partial doubling and quadrupling coarsening strategies as well as the second and the fourth order finite difference operators finally we present some results derived from numerical experiments based on the test problem

stationary biparametric adi preconditioners for conjugate gradient methods in the present article we determine optimal stationary biparametric adi preconditioners for the conjugate gradient methods when applied for the solution of a model problem second order elliptic pde the pde is approximated by 5 and 9 point stencils as was proved in hadjidimos and m lapidakis optimal alternating direction implicit preconditioners for conjugate gradient methods the problem of determining the best adi preconditioner is equivalent to that of determining the optimal extrapolated e adi method so analytic expressions are found for the optimal acceleration and extrapolation parameters for both discretizations where those for the 9 point stencil ones are new finally numerical examples run using other well known preconditioners show that the adi ones we propose are very competitive

additive modelling of functional gradients summary we consider the problem of estimating functional derivatives and gradients in the framework of a regression setting where one observes functional predictors and scalar responses derivatives are then defined as functional directional derivatives that indicate how changes in the predictor function in a specified functional direction are associated with corresponding changes in the scalar response for a model free approach navigating the curse of dimensionality requires the imposition of suitable structural constraints accordingly we develop functional derivative estimation within an additive regression framework here the additive components of functional derivatives correspond to derivatives of nonparametric one dimensional regression functions with the functional principal components of predictor processes as arguments this approach requires nothing more than estimating derivatives of one dimensional nonparametric regressions and thus is computationally very straightforward to implement while it also provides substantial flexibility fast computation and consistent estimation we illustrate the consistent estimation and interpretation of the resulting functional derivatives and functional gradient fields in a study of the dependence of lifetime fertility of flies on early life reproductive trajectories

the interaction between quasilinear elastodynamics and the navier stokes equations the interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations coupled to one another along the moving material interface through the continuity of the velocity and traction vectors we prove the existence and uniqueness locally in time of strong solutions in sobolev spaces for quasilinear elastodynamics coupled to the incompressible navier stokes equations unlike our approach in 5 for the case of linear elastodynamics we cannot employ a fixed point argument on the nonlinear system itself and are instead forced to regularize it by a particular parabolic artificial viscosity term we proceed to show that with this specific regularization we obtain a time interval of existence which is independent of the artificial viscosity together with a priori estimates we identify the global solution in both phases as well as the interface motion as a weak limit in strong norms of our sequence of regularized problems

a fourth order kernel free boundary integral method for implicitly defined surfaces in three space dimensions abstract the kernel free boundary integral kfbi method is a finite difference version of the traditional boundary integral method for elliptic and parabolic partial differential equations on complex domains it evaluates boundary or volume integrals involved in the solution of boundary integral equations bies by solving equivalent but simple interface problems on regular grids so that the integral kernel or green function is never needed or computed this is the essential difference of the kfbi method from the traditional ones it takes advantage of the well conditioning property of discrete bies so that the number of krylov subspace iterations is essentially independent of discretization parameter or system dimension this paper presents a fourth order kernel free boundary integral method for second order elliptic partial differential equations on complex domains in three space dimensions whose boundaries are given by implicitly defined surfaces it represents the domain boundary and discretizes data on it by intersection points of the surface with cartesian grid lines the approach has a variety of advantages the current work solves simple interface problems with corrected 27 point compact finite difference schemes and calculates the discrete equations with a fast fourier transform based elliptic solver numerical examples show that the proposed method is efficient as well as accurate

fast and robust solvers for pressure correction in bubbly flow problems we consider the numerical simulation of two phase fluid flow where bubbles or droplets of one phase move against a background of the other phase such flows are governed by the navier stokes equations the solution of which may be approximated using a pressure correction approach within such an approach the computational cost is often dominated by the solution of a linear system corresponding to a discrete poisson equation with discontinuous coefficients in this paper we explore the efficient solution of these linear systems using robust multilevel solvers such as deflated variants of the preconditioned conjugate gradient method or robust multigrid techniques we consider these families of methods in more detail and compare their performance in the simulation of bubbly flows some of these methods turn out to be very effective and reduce the amount of work to solve the pressure correction system substantially resulting in efficient calculations for two phase flows on highly resolved grids

a new convergence proof for the multigrid method including the v cycle for a positive definite finite element equation we describe a multigrid iteration and prove convergence under natural assumptions on the discretization and the elliptic problem hitherto existing convergence proofs require a sufficiently large number of smoothing iterations and exclude the v cycle the presented proof applies to procedures with any number of smoothing iterations and to the v cycle

combined compact difference scheme for linear second order partial differential equations with mixed derivative a combined compact difference scheme is proposed for linear second order partial differential equations with mixed derivative the scheme is based on a nine point stencil at the interior with sixth order accurate local truncation error fourier analysis is used to analyze the spectral resolution of the proposed scheme numerical tests demonstrate at least sixth order convergence rate with dirichlet boundary condition and fifth order with robin boundary condition a bonus is that high reynolds numbers do not interfere with the order of accuracy

kinematic splitting algorithm for fluid structure interaction in hemodynamics abstract in this paper we study a kinematic splitting algorithm for fluid structure interaction problems this algorithm belongs to the class of loosely coupled fluid structure interaction schemes we will present stability analysis for a coupled problem of non newtonian shear dependent fluids in moving domains with viscoelastic boundaries fluid flow is described by the conservation laws with nonlinearities in convective and diffusive terms for simplicity of presentation the structure is modelled by the generalized string equation but the results presented in the paper may be generalized to more complex structure models the arbitrary lagrangian eulerian approach is used in order to take into account moving computational domain numerical experiments including numerical error analysis and comparison of hemodynamic parameters for newtonian and non newtonian fluids demonstrate reliability of the proposed scheme

analysis of a streamline diffusion finite element method for the stokes and navier stokes equations for the stokes equations with convection and the incompressible navier stokes equations the authors analyze a streamline diffusion finite element method that is capable of balancing both the convection and the pressure thus allowing the use of arbitrary pairs of velocity pressure spaces for the linear problem the authors obtain for all mesh peclet numbers optimal error estimates in natural norms including in particular the l 2 norm of the pressure the same holds for the nonlinear problem which close to a regular branch of solutions i e the linearized operator is an isomorphism of the norm of the inverse of which still depends on the reynolds number consequently the dependence of the error constants on the reynolds number is not completely resolved in this case

multigrid methods convergence theory in a variational framework to solve the symmetric variational problem u in h a u v l v forall v in h multigrid methods are constructed using a sequence of nested subspaces u k k 0 to m of h the convergence is studied relatively to the energy norm i e in the hilbert space h a cdot cdot and is based on the 2 level case m 1 for the m 1 level case rather sharp bounds for the convergence factor can be constructed recursively and their nondependence on the number of levels is established under certain hypothesis

a short note on standard parallel multigrid algorithms for 3d problems we present some results concerning parallel multigrid mg algorithms for 3d problems for certain practically important anisotropic cases plane relaxation is needed for smoothing if standard mg coarsening etc is used parallel versions of these sophisticated mg algorithms are described and their time complexity and multiprocessor efficiency is studied

a front tracking method for the computations of multiphase flow direct numerical simulations of multiphase flows using a front tracking method are presented the method is based on writing one set of governing equations for the whole computational domain and treating the different phases as one fluid with variable material properties interfacial terms are accounted for by adding the appropriate sources as functions at the boundary separating the phases the unsteady navier stokes equations are solved by a conventional finite volume method on a fixed structured grid and the interface or front is tracked explicitly by connected marker points interfacial source terms such as surface tension are computed on the front and transferred to the fixed grid advection of fluid properties such as density is done by following the motion of the front the method has been implemented for fully three dimensional flows as well as for two dimensional and axisymmetric ones first the method is described for the flow of two or more isothermal phases the representation of the moving interface and its dynamic restructuring as well as the transfer of information between the moving front and the fixed grid are discussed applications and extensions of the method to homogeneous bubbly flows atomization flows with variable surface tension solidification and boiling are then presented

a simple cell centered multigrid method for 3d interface problems abstract a multigrid algorithm is presented for cell centered discretizations of interface problems instead of constructing the coarse grid operators by means of the galerkin approximation the coarse grid operators are obtained by discretization on the coarse grids the advantage of this approach is that we obtain m matrices on all grids and that the sparsity pattern of the fine grid matrix is retained on all grids moreover the coarse grid operators are very easy to construct numerical results of several test problems are presented

peter hall functional data analysis and random objects functional data analysis has become a major branch of nonparametric statistics and is a fast evolving field peter hall has made fundamental contributions to this area and its theoretical underpinnings he wrote more than 25 papers in functional data analysis between 1998 and 2016 and from 2005 on was a tenured faculty member with a 25 appointment in the department of statistics at the university of california davis this article describes aspects of his appointment and academic life in davis and also some of his major results in functional data analysis along with a brief history of this area it concludes with an outlook on new types of functional data and an emerging field of random objects that subsumes functional data analysis as it deals with more complex data structures

geometric multigrid with applications to computational fluid dynamics an overview is given of the development of the geometric multigrid method with emphasis on applications in computational fluid dynamics over the last ten years both compressible and incompressible flow problems and their corresponding multigrid solution methods are discussed the state of the art is described with respect to methods employed in industry as well as the multigrid efficiency obtained in academic applications

multigrid smoothing factors for red black gauss seidel relaxation applied to a class of elliptic operators analytic formulae are obtained for the smoothing factors yielded by gauss seidel relaxation in two color ordering for a class of scalar elliptic operators block and point relaxations in conjunction with full or partial coarsening are encompassed for operators with general constant positive coefficients in general dimensions and for an arbitrary number of relaxation sweeps it is found that there is no direct dependence of the smoothing factors on the dimension and that the effect of the number of relaxation sweeps on the smoothing factor is usually independent of the operator coefficients and of the relaxation scheme the results are compared with computed results of two level analyses smoothing strategies implied by the formulae are discussed