accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

robust bounded influence tests in general parametric models abstract we introduce robust tests for testing hypotheses in a general parametric model these are robust versions of the wald scores and likelihood ratio tests and are based on general m estimators their asymptotic properties and influence functions are derived it is shown that the stability of the level is obtained by bounding the self standardized sensitivity of the corresponding m estimator furthermore optimally bounded influence tests are derived for the wald and scores type tests applications to real and simulated data sets are given to illustrate the tests performance

reweighted ls estimators converge at the same rate as the initial estimator the problem of combining high efficiency with high breakdown properties for regression estimators has piqued the interest of statisticians for some time one proposal specifically suggested by rousseeuw and leroy is to use the least median of squares estimator omit observations whose residuals are larger than some constant cut off value and apply least squares to the remaining observations although this proposal does retain high breakdown point it actually converges no faster than the initial estimator in fact the reweighted least squares estimator is asymptotically a constant times the initial estimator if the initial estimator converges at a rate strictly slower than n 1 2

geometrical computations explain projection patterns of long range horizontal connections in visual cortex neurons in primary visual cortex respond selectively to oriented stimuli such as edges and lines the long range horizontal connections between them are thought to facilitate contour integration while many physiological and psychophysical findings suggest that collinear or association field models of good continuation dictate particular projection patterns of horizontal connections to guide this integration process significant evidence of interactions inconsistent with these hypotheses is accumulating we first show that natural random variations around the collinear and association field models cannot account for these inconsistencies a fact that motivates the search for more principled explanations we then develop a model of long range projection fields that formalizes good continuation based on differential geometry the analysis implicates curvature s in a fundamental way and the resulting model explains both consistent data and apparent outliers it quantitatively predicts the typically ignored spread in projection distribution its nonmonotonic variance and the differences found among individual neurons surprisingly and for the first time this model also indicates that texture and shading continuation can serve as alternative and complementary functional explanations to contour integration because current anatomical data support both curve and texture integration models equally and because both are important computationally new testable predictions are derived to allow their differentiation and identification

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

optimum robust testing in linear models robust tests for linear models are derived via wald type tests that are based on asymptotically linear estimators for a robustness criterion the maximum asymptotic bias of the level of the test for distributions in a shrinking contamination neighborhood is used by also regarding the asymptotic power of the test admissible robust tests and most efficient robust tests are derived for the greatest efficiency the determinant of the covariance matrix of the underlying estimator is minimized also most robust tests are derived it is shown that at the classical d optimal designs the most robust tests and the most efficient robust tests have a very simple form moreover the d optimal designs provide the highest robustness and the highest efficiency under robustness constraints across all designs so d optimal designs are also the optimal designs for robust testing two examples are considered for which the most robust tests and the most efficient robust tests are given

modeling a no slip flow boundary with an external force field abstract a novel technique related to peskin s immersed boundary approach is used to introduce solid surfaces into a simulated flow field the navier stokes equations permit the presence of an externally imposed body force that may vary in space and time forces are chosen to lie along a desired surface and to have a magnitude and direction opposing the local flow such that the flow is brought to rest on an element of the surface for unsteady viscous flow the direct calculation of the needed force is facilitated by a feedback scheme in which the velocity is used to iteratively determine the desired value in particular we determine the surface body force from the relation f x 5 t t 0 u x 5 t dt u x 5 t for surface points x 5 velocity u time t and negative constants and examples are presented which include 2d flow around cylinders 3d turbulent channel flow where one boundary is simulated with a force field and turbulent channel flow over a riblet covered surface while the new method may be applied to complex geometries on a non cartesian mesh we have chosen to use a simple cartesian grid all simulations are done with a spectral code in a single computational domain without any mapping of the mesh

bounded influence and high breakdown point testing procedures in linear models abstract three classes of testing procedures based on one step high breakdown point bounded influence estimators for testing subhypotheses in linear models are developed these are drop in dispersion wald type and score type tests the asymptotic distributions of these testing procedures are obtained under the null hypothesis and under contiguous alternatives their stability properties are studied in terms of their influence functions and breakdown points it is shown that the tests have bounded influence functions for the wald type tests the level and power breakdowns are determined by the breakdown point of the parameter estimate and the associated variance covariance matrix the drop in dispersion test exhibits high level breakdown but not high power breakdown point similar behavior is exhibited by the score type tests but slight modifications can be made in the construction of the test statistics to ensure high breakdown points in terms of both level and power an example is given to illustrat

robust wald type tests of one sided hypotheses in the linear model abstract let us consider the linear model y x e in the usual matrix notation where the errors are iid our main objective is to develop robust wald type tests for a large class of hypotheses on by robust we mean robustness in terms of size and power against long tailed error distributions first assume that the error distribution is symmetric about the origin let l and be the least squares and a robust estimator of assume that they are asymptotically normal about with covariance matrices 2 xtx 1 and 2 xtx 1 respectively so could be an m estimator or a high breakdown point estimator robust wald type tests based on denoted by rw are studied here for testing a large class of one sided hypotheses on it is shown that the asymptotic null distribution of rw and that of the usual wald type statistic based on l denoted by w are the same this is a useful result since the critical values and procedures for computing the p values for w are directly applicable to rw as well a more impo

direct simulations of turbulent flow using finite difference schemes abstract this paper presents finite difference solutions to the evolution of small amplitude disturbances and incompressible fully developed turbulent channel flow the main objective of the paper is to provide a comprehensive comparison between the results obtained using finite difference and spectral methods an advantage of finite difference schemes over the highly accurate spectral methods lies in the ease with which they can be applied to complex geometries the finite difference methods used include a kinetic energy conserving type of central difference scheme and a high order accurate upwind difference scheme unlike the central difference scheme the upwind difference scheme was found not to require a kinetic energy conservation property to control aliasing error the dissipative nature of the upwind scheme results in a damping of the higher frequency content as a result very little energy is aliased back the computed data including first and second order statistics for the turbulent channel flow case are found to compare well with experimental data and earlier spectral simulations it appears that the high order accurate upwind scheme is a good candidate for direct simulations of turbulent flows over complex geometries

breakdown robustness of tests abstract power and level breakdown functions of a test statistic are introduced to obtain a unified analysis of local stability as typified by the influence function and global stability as illustrated by the breakdown point the power breakdown function gives the amount of contamination of each alternative distribution that can carry the test statistic to a null value the level breakdown function gives the amount of contamination of a null distribution that can carry the test statistic to each value in the alternative space in many cases the breakdown function bears an inverse relationship to the maximum bias as a function of the contamination as the maximum bias is not invariant to one to one transformations however equivalent test statistics can have different scales for bias a distinct advantage of the inverse scale is the invariance of the breakdown functions to such transformations the resulting stability curves depend on the parameters in a way analogous to the power curve the slopes o

robust alternatives to the f test in mixed linear models based on mm estimates mixed linear models are commonly used to analyze data in many settings these models are generally fitted by means of restricted maximum likelihood techniques relying heavily on normality the sensitivity of the resulting estimators and related tests to this underlying assumption has been identified as a weakness that can even lead to wrong interpretations very recently a highly robust estimator based on a scale estimate that is an s estimator has been proposed for general mixed linear models it has the advantage of being easy to compute and allows the computation of a robust score test however this proposal cannot be used to define a likelihood ratio type test that is certainly the most direct route to robustify an f test as the latter is usually a key tool of hypothesis testing in mixed linear models we propose two new robust estimators that allow the desired extension they also lead to resistant wald type tests useful for testing contrasts and covariate effects we study their properties theoretically and by means of simulations the analysis of a real data set illustrates the advantage of the new approach in the presence of outlying observations

an immersed boundary method with formal second order accuracy and reduced numerical viscosity a formally second order accurate immersed boundary method is presented and tested in this paper we apply this new scheme to simulate the flow past a circular cylinder and study the effect of numerical viscosity on the accuracy of the computation by comparing the numerical results with those of a first order method the numerical evidence shows that the new scheme has less numerical viscosity and is therefore a better choice for the simulation of high reynolds number flows with immersed boundaries

trade off between curvature tuning and position invariance in visual area v4 humans can rapidly recognize a multitude of objects despite differences in their appearance the neural mechanisms that endow high level sensory neurons with both selectivity to complex stimulus features and tolerance or invariance to identity preserving transformations such as spatial translation remain poorly understood previous studies have demonstrated that both tolerance and selectivity to conjunctions of features are increased at successive stages of the ventral visual stream that mediates visual recognition within a given area such as visual area v4 or the inferotemporal cortex tolerance has been found to be inversely related to the sparseness of neural responses which in turn was positively correlated with conjunction selectivity however the direct relationship between tolerance and conjunction selectivity has been difficult to establish with different studies reporting either an inverse or no significant relationship to resolve this we measured v4 responses to natural scenes and using recently developed statistical techniques we estimated both the relevant stimulus features and the range of translation invariance for each neuron focusing the analysis on tuning to curvature a tractable example of conjunction selectivity we found that neurons that were tuned to more curved contours had smaller ranges of position invariance and produced sparser responses to natural stimuli these trade offs provide empirical support for recent theories of how the visual system estimates 3d shapes from shading and texture flows as well as the tiling hypothesis of the visual space for different curvature values

the use and interpretation of residuals based on robust estimation abstract residual plots and diagnostic techniques are important tools for examining the fit of a regression model in the case of least squares fits plots of residuals provide a visual assessment of the adequacy of various aspects of the fitted model an important question is whether plots of robust residuals can be interpreted in the same manner as their least squares counterparts this article addresses this problem for two popular classes of robust estimates m estimates and gm estimates of the mallows and schweppe types first order properties of the residuals and fitted values are derived under correct and misspecified models these properties are insightful on the general interpretability of robust residual plots and on their ability to detect curvature in misspecified models the results of a simulation study consisting of tests for randomness and curvature in residual plots supports these theoretical properties standardization of robust residuals is also presented

characterizing responses of translation invariant neurons to natural stimuli maximally informative invariant dimensions the human visual system is capable of recognizing complex objects even when their appearances change drastically under various viewing conditions especially in the higher cortical areas the sensory neurons reflect such functional capacity in their selectivity for complex visual features and invariance to certain object transformations such as image translation due to the strong nonlinearities necessary to achieve both the selectivity and invariance characterizing and predicting the response patterns of these neurons represents a formidable computational challenge a related problem is that such neurons are poorly driven by randomized inputs such as white noise and respond strongly only to stimuli with complex high order correlations such as natural stimuli here we describe a novel two step optimization technique that can characterize both the shape selectivity and the range and coarseness of position invariance from neural responses to natural stimuli one step in the optimization is finding the template as the maximally informative dimension given the estimated spatial location where the response could have been triggered within each image the estimates of the locations that triggered the response are updated in the next step under the assumption of a monotonic relationship between the firing rate and stimulus projections on the template at a given position the most likely location is the one that has the largest projection on the estimate of the template the algorithm shows quick convergence during optimization and the estimation results are reliable even in the regime of small signal to noise ratios when we apply the algorithm to responses of complex cells in the primary visual cortex v1 to natural movies we find that responses of the majority of cells were significantly better described by translation invariant models based on one template compared with position specific models with several relevant features

robust bounded influence tests in linear models abstract a robust test that we call an aligned generalized m test for testing subhypotheses in general linear models is developed and its asymptotic properties are studied the test is a robustification of the well known f test and it is an elegant alternative to ronchetti s 1982 class of tests p values associated with it can be approximated readily using existing chi square tables the test is based on an appropriately constructed quadratic form and uses the generalized m estimators of the parameters in the reduced model under the null hypothesis the asymptotic distribution is a central chi square and under contiguous alternatives it is a noncentral chi square with the same degrees of freedom the test can be viewed as a generalization of sen s 1982 m test for linear models the influence function of the test is bounded the bound not only applies to the influence of residuals but to the influence of position in the factor space as well on the other hand sen s test has bounded influence only

a compact difference scheme for the navier stokes equations in vorticity velocity formulation this paper presents a new numerical method for solving the incompressible unsteady navier stokes equations in vorticity velocity formulation the method is applicable to spatial simulations of transitional and turbulent boundary layer flows it is based on a compact difference discretization of the streamwise and wall normal derivatives in cartesian coordinates a fourier collocation approach is used for the spanwise derivatives important new features of the numerical method are the use of nonequidistant differences in the wall normal direction the use of split compact differences in the streamwise direction a new fast iteration for a semi implicit time integration of the wall normal diffusion terms and an improvement of the buffer domain technique to prevent reflections of waves at the outflow boundary results of test calculations are presented to verify the improvements obtained by the use of these new techniques

robust inference for generalized linear models by starting from a natural class of robust estimators for generalized linear models based on the notion of quasi likelihood we define robust deviances that can be used for stepwise model selection as in the classical framework we derive the asymptotic distribution of tests based on robust deviances and we investigate the stability of their asymptotic level under contamination the binomial and poisson models are treated in detail two applications to real data and a sensitivity analysis show that the inference obtained by means of the new techniques is more reliable than that obtained by classical estimation and testing procedures

comparison of information and variance maximization strategies for characterizing neural feature selectivity this paper compares several statistical methods for analyzing neural feature selectivity with natural stimuli despite the non gaussian character of correlations in natural stimuli several relevant stimulus dimensions can be found by maximizing either information or as is demonstrated here variance in the case of information the relevance of each dimension is quantified by a kullback leibler divergence between the full input probability distribution and that across inputs associated with positive neural responses both projected onto that dimension we demonstrate that least square matching of the nonlinear prediction based on several dimensions relevant to the recorded spike trains yields an optimization scheme similar to information maximization the relevant dimensions are found as those that capture the most variance in neural response the variance along a stimulus dimension is given by a renyi divergence of order 2 instead of the kullback leibler divergence used for maximizing information statistical errors expected for the two schemes are shown to be similar through both analytical and numerical calculations however in the asymptotic limit of large spike numbers maximizing information results in smaller errors than variance optimization numerical simulations for model cells with different noise levels show that this trend persists and possibly increases when the number of spikes decreases this makes the problem of finding relevant dimensions one of the examples where information theoretic approaches are no more data limited than the variance based measures variance and information optimization also outperform methods based on the spike triggered average for all numbers of spikes and neural noise levels

a high order immersed interface method for simulating unsteady incompressible flows on irregular domains immersed boundary methods and immersed interface methods are becoming increasingly popular for the computation of unsteady flows around complex geometries using a cartesian grid while good results both qualitative and quantitative have been obtained most of the methods rely on low order corrections to account for the immersed boundary the objective of the present work is to present as an alternative a high order modified immersed interface method for the 2d unsteady incompressible navier stokes equations in stream function vorticity formulation the method employs an explicit fourth order runge kutta time integration scheme fourth order compact finite differences for computation of spatial derivatives and a nine point fourth order compact discretization of the poisson equation for computation of the stream function corrections to the finite difference schemes are used to maintain high formal accuracy at the immersed boundary as confirmed by analytical tests to validate the method in its application to incompressible flows several physically relevant test cases are computed including uniform flow past a circular cylinder and tollmien schlichting waves in a boundary layer

robust tests in regression models with omnibus alternatives and bounded influence a robust approach for testing the parametric form of a regression function versus an omnibus alternative is introduced this generalizes existing robust methods for testing subhypotheses in a regression model the new test is motivated by developments in modern smoothing based testing procedures and can be viewed as a robustification of a smoothing based conditional moment test it is asymptotically normal under both the null hypothesis and local alternatives the robustified test retains the omnibus property of the corresponding smoothing test that is it is consistent for any fixed smooth alternative in an infinite dimensional space it is shown that the bias of the asymptotic level under shrinking local contamination is bounded only if the second order hampel s influence function is bounded the test s performance is demonstrated through both monte carlo simulations and application to an agricultural dataset

analyzing neural responses to natural signals maximally informative dimensions we propose a method that allows for a rigorous statistical analysis of neural responses to natural stimuli that are nongaussian and exhibit strong correlations we have in mind a model in which neurons are selective for a small number of stimulus dimensions out of a high dimensional stimulus space but within this subspace the responses can be arbitrarily nonlinear existing analysis methods are based on correlation functions between stimuli and responses but these methods are guaranteed to work only in the case of gaussian stimulus ensembles as an alternative to correlation functions we maximize the mutual information between the neural responses and projections of the stimulus onto low dimensional subspaces the procedure can be done iteratively by increasing the dimensionality of this subspace those dimensions that allow the recovery of all of the information between spikes and the full unprojected stimuli describe the relevant subspace if the dimensionality of the relevant subspace indeed is small it becomes feasible to map the neuron s input output function even under fully natural stimulus conditions these ideas are illustrated in simulations on model visual and auditory neurons responding to natural scenes and sounds respectively

maximum principle preserving schemes for interface problems with discontinuous coefficients new finite difference methods using cartesian grids are developed for elliptic interface problems with variable discontinuous coefficients singular sources and nonsmooth or even discontinuous solutions the new finite difference schemes are constructed to satisfy the sign property of the discrete maximum principle using quadratic optimization techniques the methods are shown to converge under certain conditions using comparison functions the coefficient matrix of the resulting linear system of equations is an m matrix and is coupled with a multigrid solver numerical examples are also provided to show the efficiency of the proposed methods

a cartesian grid embedded boundary method for poisson s equation on irregular domains we present a numerical method for solving poisson s equation with variable coefficients and dirichlet boundary conditions on two dimensional regions the approach uses a finite volume discretization which embeds the domain in a regular cartesian grid we treat the solution as a cell centered quantity even when those centers are outside the domain cells that contain a portion of the domain boundary use conservative differencing of second order accurate fluxes on each cell volume the calculation of the boundary flux ensures that the conditioning of the matrix is relatively unaffected by small cell volumes this allows us to use multigrid iterations with a simple point relaxation strategy we have combined this with an adaptive mesh refinement amr procedure we provide evidence that the algorithm is second order accurate on various exact solutions and compare the adaptive and nonadaptive calculations

on one step gm estimates and stability of inferences in linear regression abstract the folklore on one step estimation is that it inherits the breakdown point of the preliminary estimator and yet has the same large sample distribution as the fully iterated version as long as the preliminary estimate converges faster than n 1 4 where n is the sample size we investigate the extent to which this folklore is valid for one step gm estimators and their associated standard errors in linear regression we find that one step gm estimates based on newton raphson or scoring inherit the breakdown point of high breakdown point initial estimates such as least median of squares provided the usual weights that limit the influence of extreme points in the design space are based on location and scatter estimates with high breakdown points moreover these estimators have bounded influence functions and their standard errors can have high breakdown points the folklore concerning the large sample theory is correct assuming the regression errors are symmetrically distributed and homoscedastic

least median of squares regression abstract classical least squares regression consists of minimizing the sum of the squared residuals many authors have produced more robust versions of this estimator by replacing the square by something else such as the absolute value in this article a different approach is introduced in which the sum is replaced by the median of the squared residuals the resulting estimator can resist the effect of nearly 50 of contamination in the data in the special case of simple regression it corresponds to finding the narrowest strip covering half of the observations generalizations are possible to multivariate location orthogonal regression and hypothesis testing in linear models

a bounded influence high breakdown efficient regression estimator abstract we consider the multiple linear regression model y i x i e i i 1 2 n with random carriers and focus on the estimation of this article s main contribution is to present an estimator that is affine regression and scale equivariant has both a high breakdown point and a bounded influence function and has an asymptotic efficiency greater than 95 versus least squares under gaussian errors we give conditions under which the estimator a one step general m estimator that uses schweppe weights and is based on a high breakdown initial estimator satisfies these properties the major conditions necessary for the estimator are a it must be based on a n consistent initial estimator with a 50 breakdown point and b it must be based on a psi function that is odd bounded and strictly increasing the advantage of this estimator over previous approaches is that it does not downweight high leverage points without first considering how they fit the bulk of the data methods of computi

a novel concept for the design of immersed interface methods the objective of this paper is to present a novel robust high order accurate immersed interface method iim for advection diffusion type equations in contrast to other immersed methods that were designed for consistency and accuracy with a posteriori check of the numerical stability we combine local taylor series expansion at irregular grid points with a local stability constraint as part of the design process stability investigations of the iim are employed to demonstrate that the local stability constraint is sufficient for obtaining a globally stable method as long as the neumann number is less than its limiting value one of the key aspects of this iim is that the irregular finite difference stencils can be isolated from the rest of the computational domain to validate our novel immersed interface approach two dimensional and three dimensional test cases for model equations are presented in addition this method is applied to the incompressible navier stokes equations to conduct stability investigations of a boundary layer flow over a rough surface and for investigations of pulsatile stenotic flows stability investigations of wall bounded flows are challenging for immersed methods because the near wall accuracy is important for correctly capturing the characteristics of the hydrodynamic instability mechanisms in particular regarding the wave relation between the wave velocity components close to the wall

a new high order immersed interface method for solving elliptic equations with imbedded interface of discontinuity this paper presents a new high order immersed interface method for elliptic equations with imbedded interface of discontinuity compared with the original second order immersed interface method of r j leveque z li the immersed interface method for elliptic equations with discontinuous coefficients and singular sources siam j numer anal 31 1994 1001 25 the new method achieves arbitrarily high order accuracy for derivatives at an irregular grid point by imposing only two physical jump conditions together with a wider set of grid stencils the new interface difference formulas are expressed in a general explicit form so that they can be applied to different multi dimensional problems without any modification the new interface algorithms of up to o h 4 accuracy have been derived and tested on several one and two dimensional elliptic equations with imbedded interface compared to the standard second order immersed interface method the test results show that the new fourth order immersed interface method leads to a significant improvement in accuracy of the numerical solutions the proposed method has potential advantages in the application to two phase flow because of its high order accuracy and simplicity in applications

the explicit jump immersed interface method finite difference methods for pdes with piecewise smooth solutions many boundary value problems bvps or initial bvps have nonsmooth solutions with jumps along lower dimensional interfaces the explicit jump immersed interface method ejiim was developed following li s fast iterative immersed interface method fiiim recognizing that the foundation for the efficient solution of many such problems is a good solver for elliptic bvps ejiim generalizes the class of problems for which fiiim is applicable it handles interfaces between constant and variable coefficients and extends the immersed interface method iim to bvps on irregular domains with neumann and dirichlet boundary conditions proofs of second order convergence for a one dimensional 1d problem with piecewise constant coefficients and for two dimensional 2d problems with singular sources are given other problems are reduced to the singular sources case with additional equations determining the source strengths the advantages of ejiim are high quality of solutions even on coarse grids and easy adaptation to many problems with complicated geometries while still maintaining the efficiency of the fiiim

asymptotic behavior of general m estimates for regression and scale with random carriers let xini y i be a sequence of independent identically distributed random variables where x i r p and y i r and let r p be an unknown vector such that y i x i u i where u i is independent of x i and has distribution function f u where 0 is an unknown parameter this paper deals with a general class of m estimates of regression and scale defined as solutions of the system sum limits i phi text x i r i x i 0 sum limits i chi r i 0 where r y i xi1 with r p r r and r r this class contains estimators of proposed by huber mallows and krasker and welsch the consistency and asymptotic normality of the general m estimators are proved assuming general regularity conditions on and and assuming the joint distribution of x i y i to fulfill the model only approximately

on m tests in linear models summary for subhypothesis testing in some general linear models some tests based on residuals using m estimators are considered and their asymptotic properties are studied