estimation of functional regression model via functional dimension reduction abstract the functional linear regression model corresponding to a scalar response and a functional predictor is becoming increasingly common since the predictor is infinite dimensional some form of dimension reduction is essential there are two popular dimension reduction methods such as expanding the functional predictor or regression parameter function on the functional principal component basis or on a fixed bases such as b spline wavelet in the present paper we estimate the functional linear model by using the functional sufficient dimension reduction fsdr basis compared to the existing methods the proposed method is appealing because the fsdr basis is related to both the functional predictor and the response variable whereas the functional principal component basis is only related to the functional predictor and the fixed basis b spline wavelet is independent from both the functional predictor and the response variable our techniques involve methods for giving a new expansion for the predictor for giving a specific expression for the regression parameter function for estimating the fsdr space by a new method and some asymptotical properties about the regression parameter function and the prediction for the test samples numerical studies including both simulation studies and applications on real life data are presented to demonstrate the accuracy of the proposed method

on the estimation of a probability density function by the maximum penalized likelihood method abstract a class of probability density estimates can be obtained by penalizing the likelihood by a functional which depends on the roughness of the logarithm of the density the limiting case of the estimates as the amount of smoothing increasing has a natural form which makes the method attractive for data analysis and which provides a rationale for a particular choice of roughness penalty the estimates are shown to be the solution of an unconstrained convex optimization problem and mild natural conditions are given for them to exist rates of consistency in various norms and conditions for asymptotic normality and approximation by a gaussian process are given thus breaking new ground in the theory of maximum penalized likelihood density estimation author

an exact method for the calculation of certain titchmarsh weyl m functions we consider the second order linear differential equation where q is real valued in the case we calculate the titchmarsh weyl m function associated with

adaptive global testing for functional linear models this article studies global testing of the slope function in functional linear regression models a major challenge in functional global testing is to choose the dimension of projection when approximating the functional regression model by a finite dimensional multivariate linear regression model we develop a new method that simultaneously tests the slope vectors in a sequence of functional principal components regression models the sequence of models being tested is determined by the sample size and is an integral part of the testing procedure our theoretical analysis shows that the proposed method is uniformly powerful over a class of smooth alternatives when the signal to noise ratio exceeds the detection boundary the methods and results reflect the deep connection between the functional linear regression model and the gaussian sequence model we also present an extensive simulation study and a real data example to illustrate the finite sample performance of our method supplementary materials for

streamwise oscillations of a cylinder beneath a free surface free surface effects on vortex formation modes a computational study of a viscous incompressible two fluid model with an oscillating cylinder is investigated at a reynolds number of 200 and at a dimensionless displacement amplitude of a 0 13 and for the dimensionless forcing cylinder oscillation frequency to natural vortex shedding frequency ratios f f 0 1 5 2 5 3 5 specifically two dimensional flow past a circular cylinder subject to forced in line oscillations beneath a free surface is considered the method is based on a finite volume discretization of the two dimensional continuity and unsteady navier stokes equations when a solid body is present on a fixed cartesian grid two fluid model based on improved volume of fluid method is used to discretize the free surface interface the study focuses on the laminar asymmetric flow structure in the near wake region and lock on phenomena at a froude number of 0 2 and for the dimensionless cylinder submergence depths h 0 25 0 5 and 0 75 the equivorticity patterns and pressure distribution contours are used for the numerical flow visualization the code validations in special cases show good comparisons with previous numerical results

a new interface tracking method the polygonal area mapping method we present a new method the polygonal area mapping pam method for tracking a non diffusive immiscible material interface between two materials in two dimensional incompressible flows this method represents material areas explicitly as piecewise polygons traces characteristic points on polygon boundaries along pathlines and calculates new material areas inside interface cells via polygon clippings in a discrete manner the new method has very little spatial numerical diffusion and tracks the interface singularities naturally and accurately in addition to high accuracy the pam method can be directly used on either a structured rectangular mesh or an unstructured mesh without any modifications the mass conservation is enforced by heuristic algorithms adjusting the volume of material polygons the results from a set of widely used benchmark tests show that the pam method is superior to existing volume of fluid vof methods

mean and variance of implicitly defined biased estimators such as penalized maximum likelihood applications to tomography many estimators in signal processing problems are defined implicitly as the maximum of some objective function examples of implicitly defined estimators include maximum likelihood penalized likelihood maximum a posteriori and nonlinear least squares estimation for such estimators exact analytical expressions for the mean and variance are usually unavailable therefore investigators usually resort to numerical simulations to examine the properties of the mean and variance of such estimators this paper describes approximate expressions for the mean and variance of implicitly defined estimators of unconstrained continuous parameters we derive the approximations using the implicit function theorem the taylor expansion and the chain rule the expressions are defined solely in terms of the partial derivatives of whatever objective function one uses for estimation as illustrations we demonstrate that the approximations work well in two tomographic imaging applications with poisson statistics we also describe a plug in approximation that provides a remarkably accurate estimate of variability even from a single noisy poisson sinogram measurement the approximations should be useful in a wide range of estimation problems

a hybrid particle level set method for improved interface capturing in this paper we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field our method uses lagrangian marker particles to rebuild the level set in regions which are underresolved this is often the case for flows undergoing stretching and tearing the overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method our method compares favorably with volume of fluid methods in the conservation of mass and purely lagrangian schemes for interface resolution the method is presented in three spatial dimensions

an efficient estimator for the generalized semilinear model abstract the generalized semilinear model gslim extends the generalized linear model by allowing the addition of an unknown smooth function to the predictor parameter producing a semiparametric model some previously proposed score based estimators for the parametric component in this model are not n consistent we propose efficient estimators for both the parametric and nonparametric components of the gslim the estimator for the parametric component is the solution to an estimated efficient score equation and is based on recent theoretical advances in semiparametric estimation the estimator for the nonparametric component is the maximizer of an estimated penalized likelihood of the form considered by cox and o sullivan a robust standard error estimate maintains valid inference under certain model violations an application and simulation results are given for data with binary outcomes using a modified generalized cross validation algorithm to select the smoothing parameter parameter and standard

an interface capturing method with a continuous function the thinc method with multi dimensional reconstruction an interface capturing method with a continuous function is proposed within the framework of the volume of fluid vof method being different from the traditional vof methods that require a geometrical reconstruction and identify the interface by a discontinuous heaviside function the present method makes use of the hyperbolic tangent function known as one of the sigmoid type functions in the tangent of hyperbola interface capturing thinc method f xiao y honma k kono a simple algebraic interface capturing scheme using hyperbolic tangent function int j numer methods fluids 48 2005 1023 1040 to retrieve the interface in an algebraic way from the volume fraction data of multi component materials instead of the 1d reconstruction in the original thinc method a multi dimensional hyperbolic tangent function is employed in the present new approach the present scheme resolves moving interface with geometric faithfulness and compact thickness and has at least the following advantages 1 the geometric reconstruction is not required in constructing piecewise approximate functions 2 besides a piecewise linear interface curved quadratic surface can be easily constructed as well and 3 the continuous multi dimensional hyperbolic tangent function allows the direct calculations of derivatives and normal vectors numerical benchmark tests including transport of moving interface and incompressible interfacial flows are presented to validate the numerical accuracy for interface capturing and to show the capability for practical problems such as a stationary circular droplet a drop oscillation a shear induced drop deformation and a rising bubble

truncated linear models for functional data summary a conventional linear model for functional data involves expressing a response variable y in terms of the explanatory function x t via the model where a is a scalar b is an unknown function and is a compact interval however in some problems the support of b or x say is a proper and unknown subset of and is a quantity of particular practical interest motivated by a real data example involving particulate emissions we develop methods for estimating we give particular emphasis to the case where 0 and suggest two methods for estimating a b and jointly we introduce techniques for selecting tuning parameters and we explore properties of our methodology by using both simulation and the real data example mentioned above additionally we derive theoretical properties of the methodology and discuss implications of the theory our theoretical arguments give particular emphasis to the problem of identifiability

smoothing splines estimators for functional linear regression the paper considers functional linear regression where scalar responses y1 yn are modeled in dependence of random functions x1 xn we propose a smoothing splines estimator for the functional slope parameter based on a slight modification of the usual penalty theoretical analysis concentrates on the error in an out ofsample prediction of the response for a new random function xn 1 it is shown that rates of convergence of the prediction error depend on the smoothness of the slope function and on the structure of the predictors we then prove that these rates are optimal in the sense that they are minimax over large classes of possible slope functions and distributions of the predictive curves for the case of models with errors in variables the smoothing spline estimator is modified by using a denoising correction of the covariance matrix of discretized curves the methodology is then applied to a real case study where the aim is to predict the maximum of the concentration of ozone by using the curve of this concentration measured the preceding day

low rank covariance function estimation for multidimensional functional data multidimensional function data arise from many fields nowadays the covariance function plays an important role in the analysis of such increasingly common data in this article we propose a novel

smoothing spline semiparametric density models density estimation plays a fundamental role in many areas of statistics and machine learning parametric nonparametric and semiparametric density estimation methods have been proposed in the lite

multicategory support vector machines two category support vector machines svm have been very popular in the machine learning community for classification problems solving multicategory problems by a series of binary classifiers is quite common in the svm paradigm however this approach may fail under various circumstances we propose the multicategory support vector machine msvm which extends the binary svm to the multicategory case and has good theoretical properties the proposed method provides a unifying framework when there are either equal or unequal misclassification costs as a tuning criterion for the msvm an approximate leave one out cross validation function called generalized approximate cross validation is derived analogous to the binary case the effectiveness of the msvm is demonstrated through the applications to cancer classification using microarray data and cloud classification with satellite radiance profiles

soft and hard classification by reproducing kernel hilbert space methods reproducing kernel hilbert space rkhs methods provide a unified context for solving a wide variety of statistical modelling and function estimation problems we consider two such problems we are given a training set yi ti i 1 n where yi is the response for the ith subject and ti is a vector of attributes for this subject the value of yi is a label that indicates which category it came from for the first problem we wish to build a model from the training set that assigns to each t in an attribute domain of interest an estimate of the probability pj t that a future subject with attribute vector t is in category j the second problem is in some sense less ambitious it is to build a model that assigns to each t a label which classifies a future subject with that t into one of the categories or possibly none of the above the approach to the first of these two problems discussed here is a special case of what is known as penalized likelihood estimation the approach to the second problem is known as the support vector machine we also note some alternate but closely related approaches to the second problem these approaches are all obtained as solutions to optimization problems in rkhs many other problems in particular the solution of ill posed inverse problems can be obtained as solutions to optimization problems in rkhs and are mentioned in passing we caution the reader that although a large literature exists in all of these topics in this inaugural article we are selectively highlighting work of the author former students and other collaborators

modelling merging and fragmentation in multiphase flows with surfer we introduce a new numerical method called surfer for the simulation of two and three dimensional flows with several fluid phases and free interfaces between them we consider incompressible fluids obeying the navier stokes equation with newtonian viscosity in the bulk of each phase capillary forces are taken into account even when interfaces merge or break up fluid interfaces are advanced in time using an exactly volume conserving variant of the volume of fluid algorithm thus allowing for full symmetry between fluid phases the navier stokes equation is solved using staggered finite differences on a mac grid and a split explicit time differencing scheme while incompressibility is enforced using an iterative multigrid poisson solver capillary effects are represented as a stress tensor computed from gradients of the volume fraction function this formulation is completely independent of the topology of interfaces and relatively easy to implement in 3d it also allows exact momentum conservation in the discretized algorithm numerical spurious effects or parasite currents are noticed and compared to similar effects in boltzmann lattice gas methods for immiscible fluids simulations of droplets pairs colliding in 2d and in 3d are shown interface reconnection is performed easily despite the large value of capillary forces during reconnection

interface reconstruction with least squares fit and split advection in three dimensional cartesian geometry in this paper we present and analyze different volume of fluid vof reconstruction and advection algorithms that approximate the interface separating two immiscible fluids in the three dimensional space the paper describes the improvement of the reconstruction when a least square fit algorithm which minimizes a distance functional is applied its performance is tested for several smooth surfaces against other simple reconstruction methods then eulerian lagrangian and mixed split advection schemes are presented and analyzed in particular one advection method is discussed that conserves mass exactly for a divergence free velocity field thus allowing computations to machine precision

automatic smoothing of regression functions in generalized linear models abstract we consider the penalized likelihood method for estimating nonparametric regression functions in generalized linear models nelder and wedderburn 1972 and present a generalized cross validation procedure for empirically assessing an appropriate amount of smoothing in these estimates asymptotic arguments and numerical simulations are used to show that the generalized cross validatory procedure preforms well from the point of view of a weighted mean squared error criterion the methodology adds to the battery of graphical tools for model building and checking within the generalized linear model framework included are two examples motivated by medical and horticultural applications

some properties of regularized kernel methods in regularized kernel methods the solution of a learning problem is found by minimizing functionals consisting of the sum of a data and a complexity term in this paper we investigate some properties of a more general form of the above functionals in which the data term corresponds to the expected risk first we prove a quantitative version of the representer theorem holding for both regression and classification for both differentiable and non differentiable loss functions and for arbitrary offset terms second we show that the case in which the offset space is non trivial corresponds to solving a standard problem of regularization in a reproducing kernel hilbert space in which the penalty term is given by a seminorm finally we discuss the issues of existence and uniqueness of the solution from the specialization of our analysis to the discrete setting it is immediate to establish a connection between the solution properties of sparsity and coefficient boundedness and some properties of the loss function for the case of support vector machines for classification we also obtain a complete characterization of the whole method in terms of the khun tucker conditions with no need to introduce the dual formulation

approximation of method of regularization estimators la methode de tikhonov est une methode generale d estimation du parametre de regression non parametrique dans un modele lineaire abstrait avec un bruit discret on donne une analyse asymptotique dans laquelle le probleme d estimation discrete est approche par une estimation continue

efficient implementation of thinc scheme in this paper we improve the multi dimensional thinc tangent of hyperbola for interface capturing scheme f xiao y honma t kono a simple algebraic interface capturing scheme using hyperbolic tangent function int j numer meth fluid 48 2005 1023 the thinc scheme is a vof volume of fluid type method in the original thinc scheme one dimensional thinc scheme was straightforwardly used for multi dimensional cases in this paper we propose the wlic weighed line interface calculation method to extend the thinc scheme to multi dimension in the wlic method the interface is reconstructed by taking an average of line interfaces along x y and z coordinates with the weights calculated from surface normal the wlic method can extend the thinc scheme to multi dimension while maintaining simplicity of implementation and achieve a higher accuracy than the original thinc scheme the wlic method can readily extend the thinc scheme to three dimension

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

penalized quasi likelihood estimation in partial linear models consider a partial linear model where the expectation of a random variable y depends on covariates x z through f theta 0 x m 0 z with theta 0 an unknown parameter and m 0 an unknown function we apply the theory of empirical processes to derive the asymptotic properties of the penalized quasi likelihood estimator

an accurate adaptive solver for surface tension driven interfacial flows a method combining an adaptive quad octree spatial discretisation geometrical volume of fluid interface representation balanced force continuum surface force surface tension formulation and height function curvature estimation is presented the extension of these methods to the quad octree discretisation allows adaptive variable resolution along the interface and is described in detail the method is shown to recover exact equilibrium to machine accuracy between surface tension and pressure gradient in the case of a stationary droplet irrespective of viscosity and spatial resolution accurate solutions are obtained for the classical test case of capillary wave oscillations an application to the capillary breakup of a jet of water in air further illustrates the accuracy and efficiency of the method the source code of the implementation is freely available as part of the gerris flow solver

estimation of the survival function for gray s piecewise constant time varying coefficients model gray s extension of cox s proportional hazards ph model for right censored survival data allows for a departure from the ph assumption via introduction of time varying regression coefficients tvc using penalized splines gray s work focused on estimation inference and residual analyses but no estimator for the survival function has been proposed we derive a survival function estimator for one important member of the class of tvc models a piecewise constant time varying coefficients pc tvc model we also derive an estimate for the confidence limits of the survival function accuracy in estimating underlying survival times and survival quantiles is assessed for both cox s and gray s pc tvc model using a simulation study featuring scenarios violating the ph assumption finally an example of the estimated survival functions and the corresponding confidence limits derived from cox s ph and gray s pc tvc model respectively is presented for a liver transplant data set in the second part of the thesis we examine the effect of model misspecification for two classes of regression models for right censored survival data additive and multiplicative models for the conditional hazard rate a particular attention is given to data exhibiting time varying regression coefficients the class of multiplicative models is represented by cox ph model and gray s tvc model respectively and for additive models we use aalen s linear model both gray s tvc model and aalen s linear model incorporate time varying coefficients a simulation study is performed to cross analyze survival data which follows either a multiplicative or an additive model for the conditional hazard rate the effect of misspecifying the true model for the conditional hazard rate is assessed by looking at the power of the individual models to detect an existing effect bias and mean square error observed for each conditional model based estimator of survival we also show that aalen s model formulae is a first order taylor series approximation of that of gray s model which explains the comparably higher flexibility on part of the aalen s model as compared to the cox ph when the gray s tvc model for the data is misspecified

numerical simulation of moving contact line problems using a volume of fluid method abstract moving contact lines are implemented in a volume of fluid scheme with piecewise linear interface construction interfacial tension is treated as a continuous body force computed from numerical derivatives of a smoothed volume of fluid function two methods for implementing the contact angle condition are investigated the first extrapolates the volume of fluid function beyond the flow domain on the basis of the condition that its gradient is perpendicular to the interface and that the normal to the interface at the wall is determined by the contact angle the second method treats the problem as a three phase situation and mimics the classical argument of young it is found that the latter approach introduces an artificial localized flow and the extrapolation method is preferable slip is a crucial factor in the spreading of contact lines the numerical method introduces slip at the discrete level effectively introducing a slip length on the order of the mesh size

second order accurate volume of fluid algorithms for tracking material interfaces we introduce two new volume of fluid interface reconstruction algorithms and compare the accuracy of these algorithms to four other widely used volume of fluid interface reconstruction algorithms we find that when the interface is smooth e g continuous with two continuous derivatives the new methods are second order accurate and the other algorithms are first order accurate we propose a design criteria for a volume of fluid interface reconstruction algorithm to be second order accurate namely that it reproduce lines in two space dimensions or planes in three space dimensions exactly we also introduce a second order unsplit volume of fluid advection algorithm that is based on a second order finite difference method for scalar conservation laws due to bell dawson and shubin we test this advection algorithm by modeling several different interface shapes propagating in two simple incompressible flows and compare the results with the standard second order operator split advection algorithm although both methods are second order accurate when the interface is smooth we find that the unsplit algorithm exhibits noticeably better resolution in regions where the interface has discontinuous derivatives such as at corners

toward efficient and accurate interface capturing on arbitrary hybrid unstructured grids the thinc method with quadratic surface representation and gaussian quadrature abstract a novel interface capturing scheme is proposed to compute moving interface on arbitrary hybrid unstructured grids different from conventional volume of fluid vof schemes that require complicated geometric manipulations for interface reconstructions the present method can be viewed as a hybrid geometric algebraic type vof approach i e the interface is implictly retrieved from an algebraic function that effectively makes use of the geometrical information such as normal direction and curvature of the interface unlike previous versions of the thinc tangent of hyperbola interface capturing method the interface is represented by a quadratic surface for grid cells of arbitrary shapes in this scheme the gaussian quadrature is used to estimate the integration of the cell wise multi dimensional hyperbolic tangent reconstruction function which is then used to retrieve the interface from the volume fraction value of the target cell the gaussian quadrature is also used to compute the numerical fluxes from the reconstruction function numerical accuracy for reconstruction and flux computation can be effectively improved by increasing quadrature points the whole solution procedure follows the finite volume method for advection transport and is thus simple and easy to use for cells of arbitrary shapes as verified in the benchmark tests in this paper the presented scheme so called thinc qq thinc method with quadratic surface representation and gaussian quadrature scheme shows significantly improved geometrical fidelity of interface representation particularly for curved surface despite algorithmic simplicity the solution quality of thinc qq is comparable to other existing vof methods with plic geometrical interface reconstructions and thus an accurate and efficient vof scheme of great practical significance for unstructured grids

short note revisit to the thinc scheme a simple algebraic vof algorithm this short note presents an improved multi dimensional algebraic vof method to capture moving interfaces the interface jump in the thinc tangent of hyperbola for interface capturing scheme is adaptively scaled to a proper thickness according to the interface orientation the numerical accuracy in computing multi dimensional moving interfaces is significantly improved without any geometrical reconstruction the proposed method is extremely simple and easy to use and its numerical accuracy is superior to other existing methods of its kind and comparable to the conventional plic piecewise linear interface calculation type vof schemes

generalized linear models with functional predictors summary we present a technique for extending generalized linear models to the situation where some of the predictor variables are observations from a curve or function the technique is particularly useful when only fragments of each curve have been observed we demonstrate on both simulated and real data sets how this approach can be used to perform linear logistic and censored regression with functional predictors in addition we show how functional principal components can be used to gain insight into the relationship between the response and functional predictors finally we extend the methodology to apply generalized linear models and principal components to standard missing data problems

numerical simulation of spray coalescence in an eulerian framework direct quadrature method of moments and multi fluid method the scope of the present study is eulerian modeling and simulation of polydisperse liquid sprays undergoing droplet coalescence and evaporation the fundamental mathematical description is the williams spray equation governing the joint number density function f v u x t of droplet volume and velocity eulerian multi fluid models have already been rigorously derived from this equation in laurent et al f laurent m massot p villedieu eulerian multi fluid modeling for the numerical simulation of coalescence in polydisperse dense liquid sprays j comput phys 194 2004 505 543 the first key feature of the paper is the application of direct quadrature method of moments dqmom introduced by marchisio and fox d l marchisio r o fox solution of population balance equations using the direct quadrature method of moments j aerosol sci 36 2005 43 73 to the williams spray equation both the multi fluid method and dqmom yield systems of eulerian conservation equations with complicated interaction terms representing coalescence in order to focus on the difficulties associated with treating size dependent coalescence and to avoid numerical uncertainty issues associated with two way coupling only one way coupling between the droplets and a given gas velocity field is considered in order to validate and compare these approaches the chosen configuration is a self similar 2d axisymmetrical decelerating nozzle with sprays having various size distributions ranging from smooth ones up to dirac delta functions the second key feature of the paper is a thorough comparison of the two approaches for various test cases to a reference solution obtained through a classical stochastic lagrangian solver both eulerian models prove to describe adequately spray coalescence and yield a very interesting alternative to the lagrangian solver the third key point of the study is a detailed description of the limitations associated with each method thus giving criteria for their use as well as for their respective efficiency

reconstructing volume tracking a new algorithm for the volume tracking of interfaces in two dimensions is presented the algorithm is based upon a well defined second order geometric solution of a volume evolution equation the method utilizes local discrete material volume and velocity data to track interfaces of arbitrarily complex topology a linearity preserving piecewise linear interface geometry approximation ensures that solutions generated retain second order spatial accuracy second order temporal accuracy is achieved by virtue of a multidimensional unsplit time integration scheme we detail our geometrically based solution method in which material volume fluxes are computed systematically with a set of simple geometric tasks we then interrogate the method by testing its ability to track interfaces through large controlled topology changes whereby an initially simple interface configuration is subjected to vortical flows numerical results for these strenuous test problems provide evidence for the algorithm s improved solution quality and accuracy

a surface marker algorithm coupled to an area preserving marker redistribution method for three dimensional interface tracking in this work we present a new interface tracking algorithm based on surface markers to reconstruct and advect interfaces in three dimensional space the algorithm is coupled to a marker redistribution method which is area preserving along the interface the interface is described by a set of closed lines which define a coarse lagrangian quadrangular mesh in the fixed computational grid fixed markers are located where the lines cross and are maintained during the whole simulation at each time step the interface is first reconstructed and then all markers are advected by following streamlines in the reconstruction step new markers are determined by computing the intersections of the interface lines with the grid cell faces and by adding area conservation markers between fixed and grid intersection markers intersection and conservation markers defined at the previous time step are discarded the method maintains a smooth geometrical description of the interface for both two dimensional and three dimensional tests with accurate volume conservation even in very challenging situations where the fluid bodies progressively deform and stretch developing localized regions with very high curvature and thin fluid filaments the method compares favorably with front capturing methods such as volume of fluid and level set and other hybrid techniques such as the particle level set method

optimal penalized function on function regression under a reproducing kernel hilbert space framework many scientific studies collect data where the response and predictor variables are both functions of time location or some other covariate understanding the relationship between these functiona

convergence characteristics of methods of regularization estimators for nonlinear operator equations the paper studies convergence properties of method of regularization estimators of theta in the abstract nonlinear regression model z i eta theta x i varepsilon i quad i 1 2

nonparametric transformations for both sides of a regression model one way to model heteroscedasticity and skewness of the error distribution in regression is to transform both sides of the regression equation if it is possible to transform the regression equation to result in normally distributed errors then we can obtain more efficient parameter estimates and valid prediction intervals one problem with this approach is that the choice of transformation is usually restricted to the power or shifted power family often there is no scientific basis for this model and the limited flexibility of this parametric family may miss important features of the distribution a more comprehensive approach is to estimate the transformation by using nonparametric methods based on maximizing a penalized likelihood function this maximization problem leads naturally to a spline estimate for the log derivative of the transformation an algorithm for computing the estimate is given along with results on the existence and uniqueness of the estimate this nonparametric method is illustrated by using a non normally distributed fisheries data set

a study of least squares and maximum likelihood for image reconstruction in positron emission tomography positron emission tomography pet is a radiologic tool offering a unique capability for measuring tissue metabolism in vivo a number of biological and physical factors limit the resolution of pet so often the statistical aspects of image reconstruction have an appreciable effect on the quality of information obtained from a study to a first approximation reconstruction involves the solution of a linear inverse problem with a line integral radon type transform standard filtered back projection reconstruction is based on the method of least squares although computationally efficient the method does not enforce positivity constraints leading to undesirable negative artifacts in the results maximum likelihood based approaches to reconstruction do not suffer from this problem but their computational complexity has limited the ability to determine quantitatively the improvements in image quality in this paper asymptotic approximations and numerical simulations are used to examine the least squares and maximum likelihood methods in some detail the studies are carried out for idealized representations of conventional and time of flight tomographs the asymptotic analysis indicates that for a range of sobolev norms the rates of estimation of least squares and maximum likelihood reconstructions are of the same order this is borne out by numerical studies however in these studies maximum likelihood is found to be more efficient than least squares on a conventional distance angle tomograph the root mean square error is on the order of 10 20 smaller for maximum likelihood reconstructions the corresponding reduction in the root mean square error on a time of flight tomograph is in the 5 range similar results are found using more complex region of interest based measures of reconstruction accuracy in addition it is found that much of the effect of maximum likelihood can apparently be realized by postprocessing least squares solutions in an ad hoc manner to reduce negativity artifacts since the postprocessing scheme is computationally fast this suggests that in pet it may be possible to realize the benefits of maximum likelihood without the substantial computational overhead of the em algorithm

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

generalized functional linear models we propose a generalized functional linear regression model for a regression situation where the response variable is a scalar and the predictor is a random function a linear predictor is obtained by forming the scalar product of the predictor function with a smooth parameter function and the expected value of the response is related to this linear predictor via a link function if in addition a variance function is specified this leads to a functional estimating equation which corresponds to maximizing a functional quasi likelihood this general approach includes the special cases of the functional linear model as well as functional poisson regression and functional binomial regression the latter leads to procedures for classification and discrimination of stochastic processes and functional data we also consider the situation where the link and variance functions are unknown and are estimated nonparametrically from the data using a semiparametric quasi likelihood procedure an essential step in our proposal is dimension reduction by approximating the predictor processes with a truncated karhunen loeve expansion we develop asymptotic inference for the proposed class of generalized regression models in the proposed asymptotic approach the truncation parameter increases with sample size and a martingale central limit theorem is applied to establish the resulting increasing dimension asymptotics we establish asymptotic normality for a properly scaled distance between estimated and true functions that corresponds to a suitable l 2 metric and is defined through a generalized covariance operator as a consequence we obtain asymptotic tests and simultaneous confidence bands for the parameter function that determines the model the proposed estimation inference and classification procedures and variants with unknown link and variance functions are investigated in a simulation study we find that the practical selection of the number of components works well with the aic criterion and this finding is supported by theoretical considerations we include an application to the classification of medflies regarding their remaining longevity status based on the observed initial egg laying curve for each of 534 female medflies

cellwise conservative unsplit advection for the volume of fluid method we present a cellwise conservative unsplit ccu advection scheme for the volume of fluid method vof in 2d contrary to other schemes based on explicit calculations of the flux balances the ccu advection adopts a cellwise approach where the pre images of the control volumes are traced backwards through the flow map the donating regions of the fluxes are calculated via the streaklines of the grid intersections represented as polygonal chains whose vertices are determined by backward tracing of particles injected in the flow at different times high order accuracy is obtained from the fourth order runge kutta method where intermediate velocities along pathlines are determined with quadratic temporal and bicubic spatial interpolations the volumes of the donating regions are corrected in order to fulfill the discrete continuity of incompressible flows consequently the calculation produces non overlapping donating regions and pre images with conforming edges to their neighbors resulting in the conservativeness and the boundedness liquid volume fraction inside the interval 0 1 of the ccu advection scheme finally the update of the liquid volume fractions is computed from the intersections of the pre image polygons with the reconstructed interfaces the ccu scheme is tested on several benchmark tests for the vof advection together with the standard piecewise linear interface calculation plic the geometrical errors of the ccu compare favorably with other unsplit vof plic schemes finally potential improvements of the vof method with the use of more precise interface representation techniques and the future extension of the ccu scheme to 3d are discussed cellwise vof update through backward tracing of the cell s pre image accuracy enhanced via bicubic spatial interpolation and polygonal chain streakline the ccu advection scheme is fully conservative and bounded no limitation on the courant number geometrical errors dominated by plic interface representation errors

volume of fluid interface tracking with smoothed surface stress methods for three dimensional flows motivated by the need for three dimensional methods for interface calculations that can deal with topology changes we describe a numerical scheme built from a volume of fluid interface tracking technique that uses a piecewise linear interface calculation in each cell momentum balance is computed using explicit finite volume finite differences on a regular cubic grid surface tension is implemented by the continuous surface stress or continuous surface force method examples and verifications of the method are given by comparing simulations to analytical results and experiments for sedimenting droplet arrays and capillary waves at finite reynolds number in the case of a pinching pendant drop both three dimensional and axisymmetric simulations are compared to experiments agreement is found both before and after the reconnections

a monotonicity preserving conservative sharp interface flow solver for high density ratio two phase flows this paper presents a novel approach for solving the conservative form of the incompressible two phase navier stokes equations in order to overcome the numerical instability induced by the potentially large density ratio encountered across the interface the proposed method includes a volume of fluid type integration of the convective momentum transport a monotonicity preserving momentum rescaling and a consistent and conservative ghost fluid projection that includes surface tension effects the numerical dissipation inherent in the volume of fluid treatment of the convective transport is localized in the interface vicinity enabling the use of a kinetic energy conserving discretization away from the singularity two and three dimensional tests are presented and the solutions shown to remain accurate at arbitrary density ratios the proposed method is then successfully used to perform the detailed simulation of a round water jet emerging in quiescent air therefore suggesting the applicability of the proposed algorithm to the computation of realistic turbulent atomization

a new volume of fluid advection algorithm a new vof advection algorithm is presented termed the stream scheme the algorithm uses a linear piecewise method for free surface reconstruction coupled to a unique fully multidimensional method of cell boundary flux integration the performance of the new algorithm is compared against other vof advection algorithms using a variety of standard advection tests in general the performance of the stream scheme appears superior to existing multidimensional and dimensionally split advection algorithms

a mixed markers and volume of fluid method for the reconstruction and advection of interfaces in two phase and free boundary flows in this work we present a new mixed markers and volume of fluid vof algorithm for the reconstruction and advection of interfaces in the two dimensional space the interface is described by using both the volume fraction function c as in vof methods and surface markers which locate the interface within the computational cells the c field and the markers are advected by following the streamlines new markers are determined by computing the intersections of the advected interface with the grid lines then other markers are added inside each cut cell to conserve the volume fraction c a smooth motion of the interface is obtained typical of the marker approach with a good volume conservation as in standard vof methods in this article we consider a few typical two dimensional tests and compare the results of the mixed algorithm with those obtained with vof methods translations rotations and vortex tests are performed showing that many problems of the vof technique can be solved and a good accuracy in the geometrical motion and mass conservation can be achieved

local quasi likelihood with a parametric guide generalized linear models and the quasi likelihood method extend the ordinary regression models to accommodate more general conditional distributions of the response nonparametric methods need no explicit parametric specification and the resulting model is completely determined by the data themselves however nonparametric estimation schemes generally have a slower convergence rate such as the local polynomial smoothing estimation of nonparametric generalized linear models studied in fan heckman and wand j amer statist assoc 90 1995 141 150 in this work we propose a unified family of parametrically guided nonparametric estimation schemes this combines the merits of both parametric and nonparametric approaches and enables us to incorporate prior knowledge asymptotic results and numerical simulations demonstrate the improvement of our new estimation schemes over the original nonparametric counterpart

short note an interface capturing method with a continuous function the thinc method on unstructured triangular and tetrahedral meshes a novel interface capturing method is proposed to compute moving interfaces on unstructured grids with triangular 2d and tetrahedral 3d elements different from the conventional vof volume of fluid method which involves geometric reconstructions of the interface the present method is based on the algebraic reconstruction approach originally developed in the thinc tangent of hyperbola interface capturing scheme by xiao et al 2005 17 a continuous multidimensional hyperbolic tangent function is employed for retrieving the jump like distribution of the indicator function which avoids the explicit geometric representation of the interface and thus substantially reduces the algorithmic complexity in unstructured grids numerical diffusion and smearing are effectively eliminated and the compact thickness of the jump transition layer in the volume fraction is retained throughout the computation even for largely deformed interface the solution quality of the present scheme is comparable to the vof method with plic piecewise linear interface calculation algorithm

variational solution of penalized likelihood problems and smooth curve estimation usual nonparametric regression estimators often show many little wiggles which do not appear to be necessary for a good description of the data the new wp smoother is a maximum penalized likelihood mpl estimate with a novel roughness penalty it penalizes a relative change of curvature this leads to disjoint classes of functions each with a given number n w of inflection points for a wp estimate f x x w 1 x w nw exp h f x which is semiparametric with parameters w j and nonparametric part h f the main mathematical result is a convenient form of the characterizing differential equation for a very general class of mpl estimators

analytical relations connecting linear interfaces and volume fractions in rectangular grids abstract the computational geometric problems arising when a linear interface cuts a cube are considered they are of interest in particular for the calculation of volume fractions or interface positions in three dimensional interface calculations in the volume of fluid vof methods typically the normal vector is known one then wants to compute the volume fraction knowing the interface position or conversely the interface position knowing the volume fraction explicit expressions of general use are given and the algorithms used to search for solutions are described in detail explicit formulas for cubic roots are found to be less than two thirds as time consuming as newton raphson iterations

a geometrical area preserving volume of fluid advection method a new class of algorithms that preserve mass exactly for incompressible flows on a cartesian mesh are presented they amount to piecewise linear area preserving mappings of tessellations of the plane they are equivalent to volume of fluid vof advection methods which are decomposed into an eulerian implicit scheme in one direction followed by a lagrangian explicit step in the other one it is demonstrated that mass conservation is exact for incompressible flows and that there are no undershoots or overshoots of the volume fraction which thus always remains constrained between 0 and 1

consistency of support vector machines and other regularized kernel classifiers it is shown that various classifiers that are based on minimization of a regularized risk are universally consistent i e they can asymptotically learn in every classification task the role of the loss functions used in these algorithms is considered in detail as an application of our general framework several types of support vector machines svms as well as regularization networks are treated our methods combine techniques from stochastics approximation theory and functional analysis

asymptotic analysis of penalized likelihood and related estimators a general approach to the first order asymptotic analysis ofpenalized likelihood and related estimators is described the method gives expansions for the systematic and random error asymptotic convergence rates in a family of spectral norms are obtained the theory applies to a broad range of function estimation prob erns including non paxametric dellsity hazard and generalized regression curve estimation some examples are provided ams 1980 subject classifications primary 62 g05 secondary 62j05 41 a35 41 a25 47 a53 45 llo 45 m05

minimax adaptive tests for the functional linear model we introduce two novel procedures to test the nullity of the slope function in the functional linear model with real output the test statistics combine multiple testing ideas and random projections of the input data through functional principal component analysis interestingly the procedures are completely data driven and do not require any prior knowledge on the smoothness of the slope nor on the smoothness of the covariate functions the levels and powers against local alternatives are assessed in a nonasymptotic setting this allows us to prove that these procedures are minimax adaptive up to an unavoidable log log n multiplicative term to the unknown regularity of the slope as a side result the minimax separation distances of the slope are derived for a large range of regularity classes a numerical study illustrates these theoretical results

nonlinearly smoothed em density estimation with automated smoothing parameter selection for nonparametric deconvolution problems abstract we study a nonparametric deconvolution density estimation problem the estimator is obtained by an em algorithm for a smoothed maximum likelihood estimation problem which has a unique continuous solution we present an implementation of the procedure incorporating a data driven discrepancy principle for selecting the smoothing parameter simulations illustrate the good properties of the resulting estimator when the unknown distribution is smooth and has regularly varying thin tails comparisons with a fourier kernel deconvolution method are made for the case of normal noise we show that under mild smoothness conditions the estimator based on the data driven smoothing parameter is strongly consistent

minimax and adaptive prediction for functional linear regression this article considers minimax and adaptive prediction with functional predictors in the framework of functional linear model and reproducing kernel hilbert space minimax rate of convergence for the excess prediction risk is established it is shown that the optimal rate is determined jointly by the reproducing kernel and the covariance kernel in particular the alignment of these two kernels can significantly affect the difficulty of the prediction problem in contrast the existing literature has so far focused only on the setting where the two kernels are nearly perfectly aligned this motivates us to propose an easily implementable data driven roughness regularization predictor that is shown to attain the optimal rate of convergence adaptively without the need of knowing the covariance kernel simulation studies are carried out to illustrate the merits of the adaptive predictor and to demonstrate the theoretical results

a high order projection method for tracking fluid interfaces in variable density incompressible flows we present a numerical method for computing solutions of the incompressible euler or navier stokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties the method is based on a second order projection method for variable density flows using an approximate projection formulation the boundary between the fluids is tracked with a second order volume of fluid interface tracking algorithm we present results for viscious rayleigh taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm we also present computational results for the rayleigh taylor instability in air helium and for bubbles and drops in an air water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts

two component mixture cure rate model with spline estimated nonparametric components in some survival analysis of medical studies there are often long term survivors who can be considered as permanently cured the goals in these studies are to estimate the noncured probability of the whole population and the hazard rate of the susceptible subpopulation when covariates are present as often happens in practice to understand covariate effects on the noncured probability and hazard rate is of equal importance the existing methods are limited to parametric and semiparametric models we propose a two component mixture cure rate model with nonparametric forms for both the cure probability and the hazard rate function identifiability of the model is guaranteed by an additive assumption that allows no time covariate interactions in the logarithm of hazard rate estimation is carried out by an expectation maximization algorithm on maximizing a penalized likelihood for inferential purpose we apply the louis formula to obtain point wise confidence intervals for noncured probability and hazard rate asymptotic convergence rates of our function estimates are established we then evaluate the proposed method by extensive simulations we analyze the survival data from a melanoma study and find interesting patterns for this study

estimation in functional linear quantile regression this paper studies estimation in functional linear quantile regression in which the dependent variable is scalar while the covariate is a function and the conditional quantile for each fixed quantile index is modeled as a linear functional of the covariate here we suppose that covariates are discretely observed and sampling points may differ across subjects where the number of measurements per subject increases as the sample size also we allow the quantile index to vary over a given subset of the open unit interval so the slope function is a function of two variables typically time and quantile index likewise the conditional quantile function is a function of the quantile index and the covariate we consider an estimator for the slope function based on the principal component basis an estimator for the conditional quantile function is obtained by a plug in method since the so constructed plug in estimator not necessarily satisfies the monotonicity constraint with respect to the quantile index we also consider a class of monotonized estimators for the conditional quantile function we establish rates of convergence for these estimators under suitable norms showing that these rates are optimal in a minimax sense under some smoothness assumptions on the covariance kernel of the covariate and the slope function empirical choice of the cutoff level is studied by using simulations

short note a simple and conservative operator splitting semi lagrangian volume of fluid advection scheme a semi lagrangian operator splitting high performance volume of fluid advection scheme slosh vof is developed in the backward time integration a directional splitting strategy is adopted which greatly simplifies the definition and locating of the departure volume and reduces it to a grid cell expanded or compressed in one grid direction the vof value in the departure cell is estimated using a geometrical interpolation algorithm with a piecewise linear interface calculation plic scheme for the interfacial cells the proposed slosh vof method is unconditionally stable and very large time steps can be used to significantly speed up the overall computations it is conceptually simple and can be very easily implemented due to the direction split advection its performance is evaluated through several benchmark advection tests the slosh vof results are comparable to those from an eulerian vof method in terms of interface position errors and improved with regard to mass conservation even with cfl numbers increased up to one order of magnitude

a statistical measure of regularity for the study of wind generated wave field images the study of water waves has generated a wealth of sophisticated modeling developments in applied mathematics empirical observation capabilities have created a need for novel data analysis tools this article is motivated by consideration of wind generated wave field image data from a wave tank facility a quantitative measure of wave field regularity is developed the methodology is based on decomposition of the wave field into simple plane waves an adaptation of projection pursuit the percent variance explained as a function of the number of terms in the plane wave representation is used to define a variogram and regularity of the wave field is assessed in terms of the weighted difference between the observed variogram and the expected variogram for a completely random field the proposed regularity measure is illustrated by application to image data from wind generated waves the results suggest that the regularity measure is a function of parameters describing the generation of the wave field wi

camouflaged deconvolution with application to blood curve modeling in fdg pet studies abstract camouflaged deconvolution arises when the kernel in a simple convolution model is not completely specified we consider a situation in which the same fixed signal is repeatedly measured by separate convolutions with imprecisely known kernels we develop a regularization methodology for application to these problems the method involves simultaneous estimation of the target signal and the unknown parameters of the convolution kernels cross validation is used to determine the degree of smoothness of the solution we use simulation studies matched to the application to evaluate statistical performance these simulations find that the convergence of the regularization estimator is largely unaffected by the lack of information about the convolution kernels we illustrate the methodology by application to a blood curve modeling problem arising in the context of positron emission tomography pet studies with flurodeoxyglucose fdg a commonly used glucose tracer the results show promise towards the

effective dimension reduction for sparse functional data we propose a method of effective dimension reduction for functional data emphasizing the sparse design where one observes only a few noisy and irregular measurements for some or all of the subjects the proposed method borrows strength across the entire sample and provides a way to characterize the effective dimension reduction space via functional cumulative slicing our theoretical study reveals a bias variance trade off associated with the regularizing truncation and decaying structures of the predictor process and the effective dimension reduction space a simulation study and an application illustrate the superior finite sample performance of the method

a coupled level set and volume of fluid method for computing 3d and axisymmetric incompressible two phase flows we present a coupled level set volume of fluid clsvof method for computing 3d and axisymmetric incompressible two phase flows this method combines some of the advantages of the volume of fluid method with the level set method to obtain a method which is generally superior to either method alone we present direct comparisons between computations made with the clsvof method and computations made with the level set method the volume of fluid method and the boundary integral method we also compare our computations to the exact solution for an oscillating ellipse due to lamb and experimental results obtained for a rising gas bubble in liquid obtained by hnat and buckmaster our computational examples focus on flows in which surface tension forces and changes in topology are dominant features of the flow

nonparametric spectral density estimation using penalized whittle likelihood abstract the penalized likelihood approach is not well developed in time series analysis even though it has been applied successfully in a number of nonparametric function estimation problems chow and grenander proposed a penalized likelihood type approach to the nonparametric estimation of the spectral density of gaussian processes in this article this estimator is extended to more general stationary processes its practical implementation is developed in some detail and some asymptotic rates of convergence are established its performance is also compared to more widely used alternatives in the field a computational algorithm involving an iterative least squares initialized by the log periodogram is first developed then motivated by an asymptotic linearization an estimator of the integrated squared error between the estimated and true log spectral densities is proposed from this a data dependent procedure for selection of the amount of smoothing is constructed the methodology is illustrated

a reproducing kernel hilbert space approach to functional linear regression we study in this paper a smoothness regularization method for functional linear regression and provide a unified treatment for both the prediction and estimation problems by developing a tool on simultaneous diagonalization of two positive definite kernels we obtain shaper results on the minimax rates of convergence and show that smoothness regularized estimators achieve the optimal rates of convergence for both prediction and estimation under conditions weaker than those for the functional principal components based methods developed in the literature despite the generality of the method of regularization we show that the procedure is easily implementable numerical results are obtained to illustrate the merits of the method and to demonstrate the theoretical developments

an improved plic vof method for tracking thin fluid structures in incompressible two phase flows a drawback common to plic schemes is the difficulty of reconstructing the interface when filaments of a thickness smaller than the cell size are present in this work we present an improvement of the reconstruction method proposed by lopez et al a volume of fluid method based on multidimensional advection and spline interface reconstruction j comput phys 195 2004 718 for two dimensional flows which allows tracking fluid structures thinner than the cell size by allowing the interface to be represented in each cell by two non contiguous linear segments the method is based on using markers that are placed every time step at the mid points of the reconstructed cell interface segments the markers are used to improve the orientation and location of the interface segments although the method essentially remains of a vof type numerical tests show that a substantial improvement in accuracy is achieved over previous volume of fluid methods and that the proposed method compares well even with a recent hybrid markers and volume of fluid method

conservative volume of fluid method for free surface simulations on cartesian grids this paper contributes to the state of the art in cartesian grid methods through development of new advection and reconstruction volume of fluid vof algorithms which are applicable to two and three dimensional flows a computationally efficient and second order vof reconstruction method is presented which uses no inversions to determine the interface normal direction next the lack of conservation of fluid volume in previous vof advection methods are shown to be due to improper treatment of one dimensional stretching in the velocity field this paper uses simple explicit time stepping and a cell center estimate of the volume fraction in the dilatation term to achieve a completely conservative advection method the new methods are simple robust and shown to out perform existing approaches for canonical test problems relevant to breaking wave flows