information theoretic determination of minimax rates of convergence we present some general results determining minimax bounds on statistical risk for density estimation based on certain information theoretic considerations these bounds depend only on metric entropy conditions and are used to identify the minimax rates of convergence

advances in multi domain lattice boltzmann grid refinement grid refinement has been addressed by different authors in the lattice boltzmann method community the information communication and reconstruction on grid transitions is of crucial importance from the accuracy and numerical stability point of view while a decimation is performed when going from the fine to the coarse grid a reconstruction must performed to pass form the coarse to the fine grid in this context we introduce a decimation technique for the copy from the fine to the coarse grid based on a filtering operation we show this operation to be extremely important because a simple copy of the information is not sufficient to guarantee the stability of the numerical scheme at high reynolds numbers then we demonstrate that to reconstruct the information a local cubic interpolation scheme is mandatory in order to get a precision compatible with the order of accuracy of the lattice boltzmann method these two fundamental extra steps are validated on two classical 2d benchmarks the 2d circular cylinder and the 2d dipole wall collision the latter is especially challenging from the numerical point of view since we allow strong gradients to cross the refinement interfaces at a relatively high reynolds number of 5000 a very good agreement is found between the single grid and the refined grid cases the proposed grid refinement strategy has been implemented in the parallel open source library palabos

a geometrical method for removing edge effects from kernel type nonparametric regression estimators abstract we introduce a simple geometric method for removing edge effects from kernel type nonparametric regression estimators it involves reflecting the data set in two estimated points thereby generating a new data set with three times the range of the original data the usual kernel type estimator may be applied to the new enlarged data set without any danger of edge effects this technique is applicable generally to both regularly spaced and randomly spaced designs and admits a natural analog of leave one out cross validation the new cross validation algorithm may be extended to the very ends of the design interval unlike its more conventional counterpart which must be downweighted at the ends of the interval to avoid edge effects

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

ideal spatial adaptation by wavelet shrinkage summary with ideal spatial adaptation an oracle furnishes information about how best to adapt a spatially variable estimator whether piecewise constant piecewise polynomial variable knot spline or variable bandwidth kernel to the unknown function estimation with the aid of an oracle offers dramatic advantages over traditional linear estimation by nonadaptive kernels however it is a priori unclear whether such performance can be obtained by a procedure relying on the data alone we describe a new principle for spatially adaptive estimation selective wavelet reconstruction we show that variable knot spline fits and piecewise polynomial fits when equipped with an oracle to select the knots are not dramatically more powerful than selective wavelet reconstruction with an oracle we develop a practical spatially adaptive method riskshrink which works by shrinkage of empirical wavelet coefficients riskshrink mimics the performance of an oracle for selective wavelet reconstruction as well as it is possible to do so a new inequality in multivariate normal decision theory which we call the oracle inequality shows that attained performance differs from ideal performance by at most a factor of approximately 2 log n where n is the sample size moreover no estimator can give a better guarantee than this within the class of spatially adaptive procedures riskshrink is essentially optimal relying only on the data it comes within a factor log 2 n of the performance of piecewise polynomial and variableknot spline methods equipped with an oracle in contrast it is unknown how or if piecewise polynomial methods could be made to function this well when denied access to an oracle and forced to rely on data alone

an immersed boundary method with direct forcing for the simulation of particulate flows we present an improved method for computing incompressible viscous flow around suspended rigid particles using a fixed and uniform computational grid the main idea is to incorporate peskin s regularized delta function approach acta numerica 11 2002 1 into a direct formulation of the fluid solid interaction force in order to allow for a smooth transfer between eulerian and lagrangian representations while at the same time avoiding strong restrictions of the time step this technique was implemented in a finite difference and fractional step context a variety of two and three dimensional simulations are presented ranging from the flow around a single cylinder to the sedimentation of 1000 spherical particles the accuracy and efficiency of the current method are clearly demonstrated

proteus a coupled iterative force correction immersed boundary cascaded lattice boltzmann solver for moving and deformable boundary applications abstract many realistic fluid flow problems are characterised by high reynolds numbers and complex moving or deformable geometries in our previous study we presented a novel coupling between an iterative force correction immersed boundary and a multi domain cascaded lattice boltzmann method falagkaris et al and investigated flows around rigid bodies at reynolds numbers up to 1 0 5 here we extend its application to flows around moving and deformable bodies with prescribed motions emphasis is given on the influence of the internal mass on the computation of the aerodynamic forces including deforming boundary applications where the rigid body approximation is no longer valid both the rigid body and the internal lagrangian points approximations are examined the resulting solver has been applied to viscous flows around an in line oscillating cylinder a pitching foil a plunging sd7003 airfoil and a plunging and flapping naca 0014 airfoil good agreement with experimental results and other numerical schemes has been obtained it is shown that the internal lagrangian points approximation accurately captures the internal mass effects in linear and angular motions as well as in deforming motions at reynolds numbers up to 4 1 0 4 in all cases the aerodynamic loads are significantly affected by the internal fluid forces

analysis of the absorbing layers for the weakly compressible lattice boltzmann methods abstract it has been demonstrated that lattice boltzmann methods lbms are very efficient for computational aeroacoustics caa in order to address the issue of absorbing acoustic boundary conditions for lbm three kinds of damping terms are proposed and added to the right hand side of the lbm governing equations according to the classical theory these terms play an important role to damp and minimize the acoustic wave reflections from computational boundaries the corresponding macroscopic equations with the damping terms are recovered for analyzing the macroscopic behaviors of the these damping terms and determining the critical absorbing strength the dissipative and dispersive properties of the proposed absorbing layer terms are then further analyzed considering the linearized lbm equations they are explored in the wave number spaces via the von neumann analysis the related damping strength critical values and the optimal absorbing term are discussed finally some benchmark problems are implemented to assess the theoretical results

the immersed boundary lattice boltzmann method for solving fluid particles interaction problems a new computational method the immersed boundary lattice boltzmann method is presented this method is a combination and utilizes the most desirable features of the lattice boltzmann and the immersed boundary methods the method uses a regular eulerian grid for the flow domain and a lagrangian grid to follow particles that are contained in the flow field the rigid body conditions for the fluid and the particles are enforced by a penalty method which assumes that the particle boundary is deformable with a high stiffness constant the velocity field of the fluid and particles is solved by adding a force density term into the lattice boltzmann equation this novel method preserves the advantages of lbm in tracking a group of particles and at the same time provides an alternative and better approach to treating the solid fluid boundary conditions the method also solves the problems of fluctuation of the forces and velocities on the particles when the bounce back boundary conditions are applied this method enables one to simulate problems with particle deformation and fluid structure deformation its results are validated by comparison with results from other methods

low reynolds number flow around an oscillating circular cylinder using a cell viscousboundary element method flow fields from transversely oscillating circular cylinders in water at rest are studied by numerical solutions of the two dimensional unsteady incompressible navier stokes equations adopting a primitive variable formulation these findings are successfully compared with experimental observations the cell viscous boundary element scheme developed is first validated to examine convergence of solution and the influence of discretization within the numerical scheme of study before the comparisons are undertaken a hybrid approach utilising boundary element and finite element methods is adopted in the cell viscous boundary element method that is cell equations are generated using the principles of a boundary element method with global equations derived following the procedures of finite element methods the influence of key parameters i e reynolds number re keulegan carpenter number kc and stokes number beta on overall flow characteristics and vortex shedding mechanisms are investigated through comparisons with experimental findings and theoretical predictions the latter extends the study into assessment of the values of the drag coefficient added mass or inertia coefficient with key parameters and the variation of lift and in line force results with time derived from the morison s equation the cell viscous boundary element method as described herein is shown to produce solutions which agree very favourably with experimental observations measurements and other theoretical findings

density estimation under random censorship and order restrictions from asymptotic to small samples do a random censorship and or order restrictions e g nonnegativity monotonicity convexity affect estimation of a smooth density under mean integrated squared error mise under mild assumptions the known asymptotic results which are concerned only with rates answer no this answer especially for censored data contradicts practical experience and statistical intuition so what can be said about constants of mise convergence it is shown that asymptotically a censorship does affect the constant and this allows one to find a relationship between sample sizes of directly observed and censored datasets that implies the same precision of estimation and b an order restriction does not affect the constant and thus no isotonic estimation is needed intensive monte carlo simulations show that the lessons of the sharp asymptotics are valuable for small sample sizes also the estimator developed is illustrated both on simulated data and a dataset of lifetimes of conveyer blades used at wastewate

least squares estimators of the mode of a unimodal regression function in this paper we consider nonparametric least squares estimators of the mode of an unknown unimodal regression function we establish almost sure convergence of these estimators with nearly optimal convergence rates under the assumption of the exponential tail for the error distributions

asymptotics in empirical risk minimization in this paper we study a two category classification problem we indicate the categories by labels y 1 and y 1 we observe a covariate or feature x x d consider a collection ha of classifiers indexed by a finite dimensional parameter a and the classifier ha that minimizes the prediction error over this class the parameter a is estimated by the empirical risk minimizer n over the class where the empirical risk is calculated on a training sample of size n we apply the kim pollard theorem to show that under certain differentiability assumptions n converges to a with rate n 1 3 and also present the asymptotic distribution of the renormalized estimator for example let v0 denote the set of x on which given x x the label y 1 is more likely than the label y 1 if x is one dimensional the set v0 is the union of disjoint intervals the problem is then to estimate the thresholds of the intervals we obtain the asymptotic distribution of the empirical risk minimizer when the classifiers have k thresholds where k is fixed we furthermore consider an extension to higher dimensional x assuming basically that v0 has a smooth boundary in some given parametric class we also discuss various rates of convergence when the differentiability conditions are possibly violated here we again restrict ourselves to one dimensional x we show that the rate is n 1 in certain cases and then also obtain the asymptotic distribution for the empirical prediction error

implicit velocity correction based immersed boundary lattice boltzmann method and its applications a version of immersed boundary lattice boltzmann method ib lbm is proposed in this work it is based on the lattice boltzmann equation with external forcing term proposed by guo et al z guo c zheng b shi discrete lattice effects on the forcing term in the lattice boltzmann method phys rev e 65 2002 046308 which can well consider the effect of external force to the momentum and momentum flux as well as the discrete lattice effect in this model the velocity is contributed by two parts one is from the density distribution function and can be termed as intermediate velocity and the other is from the external force and can be considered as velocity correction in the conventional ib lbm the force density external force is explicitly computed in advance as a result we cannot manipulate the velocity correction to enforce the non slip boundary condition at the boundary point in the present work the velocity corrections force density at all boundary points are considered as unknowns which are computed in such a way that the non slip boundary condition at the boundary points is enforced the solution procedure of present ib lbm is exactly the same as the conventional ib lbm except that the non slip boundary condition can be satisfied in the present model while it is only approximately satisfied in the conventional model numerical experiments for the flows around a circular cylinder and an airfoil show that there is no any penetration of streamlines to the solid body in the present results this is not the case for the results obtained by the conventional ib lbm another advantage of the present method is its simple calculation of force on the boundary the force can be directly calculated from the relationship between the velocity correction and the force density

open loop optimal control of a flapping wing using an adjoint lattice boltzmann method abstract we present the usage of an adjoint lattice boltzmann method lbm for open loop control of two dimensional flapping wing motion movement of the wing is parametrised with periodic b splines while the wing interacts with the surrounding flow via an immersed boundary ib method multi objective optimisation is performed using a gradient optimisation algorithm for which sensitivities are calculated with an adjoint method the objectives selected were the mean lift force and mechanical power to achieve performance suitable for optimisation we also present an efficient gpu implementation of the lbm and adjoint lbm the immersed boundary approach employed for the lbm is verified against results from the literature while for the flapping case it is compared with two different finite volume method fvm approaches the obtained pareto front of optimal designs shows a clear discrepancy between the power consumption and the mean lift force a significant improvement of the basic wing design is demonstrated and highlights the applicability of adjoint lbm simulations in complex open loop control problems

grid refinement for lattice bgk models lattice boltzmann models proposed at the end of the 1980s as the noise free version of lattice gas models are based on gas kinetic representation of fluid flow their recent modifications the lattice bgk models provide especially simple effective and stable algorithms for the solution of hydrodynamical problems a local second order grid refinement scheme for the lattice bgk model is proposed in this work the refinement scheme and a boundary fitting scheme for complicated geometries are applied to simulate a benchmark problem of flow past a cylinder in a channel with small and moderate reynolds numbers

estimation of a unimodal distribution function on considere le probleme de l estimation efficace d une fonction de repartition lorsque presque rien n est connu a son sujet sauf qu elle est unimodale

estimation of a k monotone density limit distribution theory and the spline connection we study the asymptotic behavior of the maximum likelihood and least squares estimators of a kmonotone density g0 at a fixed point x0 when k 2 we find that the jth derivative of the estima tors at x0 converges at the rate n k j 2k 1 for j 0 k 1 the limiting distribution depends on an almost surely uniquely de fined stochastic process hk that stays above below the k fold in tegral of brownian motion plus a deterministic drift when k is even odd both the mle and lse are known to be splines of degree k 1 with simple knots establishing the order of the random gap n n where n denote two successive knots is a key ingredient of the proof of the main results we show that this gap problem can be solved if a conjecture about the upper bound on the error in a particular hermite interpolation via odd degree splines holds

approximations of semicontinuous functions with applications to stochastic optimization and statistical estimation upper semicontinuous usc functions arise in the analysis of maximization problems distributionally robust optimization and function identification which includes many problems of nonparametric statistics we establish that every usc function is the limit of a hypo converging sequence of piecewise affine functions of the difference of max type and illustrate resulting algorithmic possibilities in the context of approximate solution of infinite dimensional optimization problems in an effort to quantify the ease with which classes of usc functions can be approximated by finite collections we provide upper and lower bounds on covering numbers for bounded sets of usc functions under the attouch wets distance the result is applied in the context of stochastic optimization problems defined over spaces of usc functions we establish confidence regions for optimal solutions based on sample average approximations and examine the accompanying rates of convergence examples from nonparametric statistics illustrate the results

covering numbers for real valued function classes we find tight upper and lower bounds on the growth rate for the covering numbers of functions of bounded variation in the spl lscr sub 1 metric in terms of all the relevant constants we also find upper and lower bounds on covering numbers for general function classes over the family of spl lscr sub 1 dp metrics in terms of a scale sensitive combinatorial dimension of the function class

recent developments in nonparametric density estimation advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where restrictions are imposed upon the class of densities or on the form of the likelihood function nonparametric density estimators that are data adaptive and lead to locally smoothed estimators are also discussed these include variable partition histograms estimators based on statistically equivalent blocks nearest neighbor estimators variable kernel estimators and adaptive kernel estimators for the multivariate case extensions of methods of univariate density estimation are usually straightforward but can be computationally expensive a method of multivariate density estimation that did not spring from a univariate generalization is described namely projection pursuit density estimation in which both dimensionality reduction and density estimation can be pursued at the same time finally some areas of related research are mentioned such as nonparametric estimation of functionals of a density robust parametric estimation semiparametric models and density estimation for censored and incomplete data directional and spherical data and density estimation for dependent sequences of observations

asymptotic normality of the l_1 error of the grenander estimator groeneboom introduced a jump process that can be used among other things to study the asymptotic properties of the grenander estimator of a monotone density in this paper we derive the asymptotic normality of a suitably rescaled version of the l 1 error of the grenander estimator using properties of this jump process

locally adaptive regression splines in this paper least squares penalized regression estimates withtotal variation penalities are considered it is shown that theseestimators are least squares splines with locally data adaptive placed knotpoints algorithms and asymptotic properties are discussed

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

theory and applications of an alternative lattice boltzmann grid refinement algorithm this contribution proposes an alternative lattice boltzmann grid refinement algorithm that overcomes the drawbacks that plague existing approaches we demonstrate that this algorithm is accurate and applicable for all values of the relaxation time we also show that this algorithm can significantly speed up the flow settlement process by using a hierarchy of grid levels the stationary regime can be approached up to a thousand times faster than with a single grid resolution

discrete lattice effects on the forcing term in the lattice boltzmann method we show that discrete lattice effects must be considered in the introduction of a force into the lattice boltzmann equation a representation of the forcing term is then proposed with the representation the navier stokes equation is derived from the lattice boltzmann equation through the chapman enskog expansion several other existing force treatments are also examined

iterative hessian sketch fast and accurate solution approximation for constrained least squares we study randomized sketching methods for approximately solving least squares problem with a general convex constraint the quality of a least squares approximation can be assessed in different ways either in terms of the value of the quadratic objective function cost approximation or in terms of some distance measure between the approximate minimizer and the true minimizer solution approximation focusing on the latter criterion our first main result provides a general lower bound on any randomized method that sketches both the data matrix and vector in a least squares problem as a surprising consequence the most widely used least squares sketch is sub optimal for solution approximation we then present a new method known as the iterative hessian sketch and show that it can be used to obtain approximations to the original least squares problem using a projection dimension proportional to the statistical complexity of the least squares minimizer and a logarithmic number of iterations we illustrate our general theory with simulations for both unconstrained and constrained versions of least squares including l1 regularization and nuclear norm constraints we also numerically demonstrate the practicality of our approach in a real face expression classification experiment

approximation dans les espaces metriques et theorie de l estimation we investigate the relations between the speed of estimation and the metric structure of the parameter space especially in the case when its metric dimension is infinite given some distance d on generally hellinger distance in the case of n i i d variables we consider the minimax risk for n observations r n q mathop inf sup limits t n theta in theta text ie theta d q theta t n t n being any estimate of we shall look for functions r such that for positive constants c 1 q and c 2 q c 1 r q n r n q c2 r q n r n is the speed of estimation and we shall show under fairly general conditions including i i d variables and regular cases of markov chains and stationnary gaussian processes that r n is determined up to multiplicative constants by the metric structure of we shall also give a construction for some sort of universal estimates the risk of which is bounded by c 2 r q n in all cases where the preceding theory applies

a smoothing technique for discrete delta functions with application to immersed boundary method in moving boundary simulations the effects of complex boundary conditions on flows are represented by a volume force in the immersed boundary methods the problem with this representation is that the volume force exhibits non physical oscillations in moving boundary simulations a smoothing technique for discrete delta functions has been developed in this paper to suppress the non physical oscillations in the volume forces we have found that the non physical oscillations are mainly due to the fact that the derivatives of the regular discrete delta functions do not satisfy certain moment conditions it has been shown that the smoothed discrete delta functions constructed in this paper have one order higher derivative than the regular ones moreover not only the smoothed discrete delta functions satisfy the first two discrete moment conditions but also their derivatives satisfy one order higher moment condition than the regular ones the smoothed discrete delta functions are tested by three test cases a one dimensional heat equation with a moving singular force a two dimensional flow past an oscillating cylinder and the vortex induced vibration of a cylinder the numerical examples in these cases demonstrate that the smoothed discrete delta functions can effectively suppress the non physical oscillations in the volume forces and improve the accuracy of the immersed boundary method with direct forcing in moving boundary simulations

the grenader estimator a nonasymptotic approach on examine certaines proprietes non asymtotiques de l estimateur de grenander d une densite decroissante f cet estimation est defini comme la pente du plus petit majorant concave de la fonction de repartition empirique on montre que son risque mesure avec perte l 1 est borne par une certaine fonctionnelle qui depend de f et du nombre d observations n le comportement asymptotique de la fonctionnelle quand n tend vers l infini est en accord avec les performances asymptotiques connues de l estimation de grenander

generalized local equilibrium in the cascaded lattice boltzmann method by realizing the insufficient degree of galilean invariance of the traditional multiple relaxation time collision operators geier et al phys rev e 73 066705 2006 proposed to relax differently the moments shifted by the macroscopic velocity leading to the so called cascaded lattice boltzmann method lbm this paper points out that a the cascaded lbm essentially consists in adopting a generalized local equilibrium in the frame at rest b this new equilibrium does not affect the consistency of lbm and finally c if the raw moments are relaxed in the frame at rest as usual and the number of relaxation frequencies is reduced the proposed derivation leads to the two relaxation time collisional operator with proper polynomial equilibrium

rational approximate solutions and error bounds for the nonsymmetric riccati matrix differential equation in this paper we determine an interval where the exact solution of the nonsymmetric riccati matrix differential equation is defined rational approximate solutions are whose error is smaller than a prefixed number e 0 are constructed in terms of the data

bounds on variance for unimodal distributions we show a direct relationship between the variance and the differential entropy for subclasses of symmetric and asymmetric unimodal distributions by providing an upper bound on variance in terms of entropy power combining this bound with the well known entropy power lower bound on variance we prove that the variance of the appropriate subclasses of unimodal distributions can be bounded below and above by the scaled entropy power as the differential entropy decreases the variance is sandwiched between two exponentially decreasing functions in the differential entropy this establishes that for the subclasses of unimodal distributions the differential entropy can be used as a surrogate for concentration of the distribution

the riccati transformation in the solution of boundary value problems the multiple shooting method and stabilized march have long been advocated for solving boundary value problems even though certain difficulties arise for problems with rapidly increasing and decreasing fundamental solution components we interpret these difficulties in terms of stability of the underlying problem and describe the stabilized march in terms of associated projections onto subspaces relating the riccati transformation to a specific implementation of the stabilized march leads naturally to an alternative method which involves only stable initial value problems and overcomes major difficulties of these other methods the potential of the method is demonstrated in some numerical examples

navier stokes simulation with constraint forces finite difference method for particle laden flows and complex geometries building on an idea of fogelson and peskin j comput phys 79 50 1988 we describe the implementation and verification of a simulation technique for systems of non brownian particles in fluids at reynolds numbers up to about 20 on the particle scale this direct simulation technique fills a gap between simulations in the viscous regime and high reynolds number modeling it also combines sufficient computational accuracy with numerical efficiency and allows studies of several thousand in principle arbitrarily shaped extended and hydrodynamically interacting particles on regular work stations we verify the algorithm in two and three dimensions for i single falling particles and ii a fluid flowing through a bed of fixed spheres in the context of sedimentation we compute the volume fraction dependence of the mean sedimentation velocity the results are compared with experimental and other numerical results both in the viscous and inertial regime and we find very satisfactory agreement

clustering based on pairwise distances when the data is of mixed dimensions in the context of clustering we consider a generative model in a euclidean ambient space with clusters of different shapes dimensions sizes and densities in an asymptotic setting where the number of points becomes large we obtain theoretical guaranties for some emblematic methods based on pairwise distances a simple algorithm based on the extraction of connected components in a neighborhood graph hierarchical clustering with single linkage and the spectral clustering method of ng jordan and weiss the methods are shown to enjoy some near optimal properties in terms of separation between clusters and robustness to outliers the local scaling method of zelnik manor and perona is shown to lead to a near optimal choice for the scale in the first and third methods we also provide a lower bound on the spectral gap to consistently choose the correct number of clusters in the spectral method

a fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies application to particulate flow abstract in this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies the simulation methods rest essentially on the combination of a lagrange multiplier based fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region b finite element approximations of the navier stokes equations occurring in the global model c time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model the above methodology is particularly well suited to the direct numerical simulation of particulate flow such as the flow of mixtures of rigid solid particles and incompressible viscous fluids possibly non newtonian we conclude this article with the presentation of the results of various numerical experiments including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions

an immersed boundary lattice boltzmann method for the simulation of the flow past an impulsively started cylinder we present a lattice boltzmann method coupled with an immersed boundary technique for the simulation of bluff body flows the lattice boltzmann method for the modeling of the navier stokes equations is enhanced by a forcing term to account for the no slip boundary condition on a non grid conforming boundary we investigate two alternatives of coupling the boundary forcing term with the grid nodes namely the direct and the interpolated forcing techniques the present lb ib methods are validated in simulations of the incompressible flow past an impulsively started cylinder at low and moderate reynolds numbers we present diagnostics such as the near wall vorticity field and the drag coefficient and comparisons with previous computational and experimental works and assess the advantages and drawbacks of the two techniques

asymptotically optimal cells for a historgram on examine les proprietes de l histogramme lorsque les cellules sont arbitraires on obtient une expression asymptotique pour l infimum de l erreur quadratique moyenne integree mise sur toutes les partitions

rademacher averages and phase transitions in glivenko cantelli classes we introduce a new parameter which may replace the fat shattering dimension using this parameter we are able to provide improved complexity estimates for the agnostic learning problem with respect to any l sub p norm moreover we show that if fat sub spl epsi f o spl epsi sup p then f displays a clear phase transition which occurs at p 2 the phase transition appears in the sample complexity estimates covering numbers estimates and in the growth rate of the rademacher averages associated with the class as a part of our discussion we prove the best known estimates on the covering numbers of a class when considered as a subset of l sub p spaces we also estimate the fat shattering dimension of the convex hull of a given class both these estimates are given in terms of the fat shattering dimension of the original class

lower compactness estimates for scalar balance laws we study the compactness in l 1 of the semigroup st t 0 of entropy weak solutions to strictly convex scalar conservation laws in one space dimension the compactness of st for each t 0 was established by p d lax 1 upper estimates for the kolmogorov s entropy of the image through st of bounded sets c in l 1 l 1 which is denoted by

estimation of unimodal densities without smoothness assumptions the grenander estimator of a decreasing density which is defined as the derivative of the concave envelope of the empirical c d f is known to be a very good estimator of an unknown decreasing density on the half line mathbb r when this density is not assumed to be smooth it is indeed the maximum likelihood estimator and one can get precise upper bounds for its risk when the loss is measured by the mathbb l 1 distance between densities moreover if one restricts oneself to the compact subsets of decreasing densities bounded by h with support on 0 l the risk of this estimator is within a fixed factor of the minimax risk the same is true if one deals with the maximum likelihood estimator for unimodal densities with known mode when the mode is unknown the maximum likelihood estimator does not exist any more we shall provide a general purpose estimator together with a computational algorithm for estimating nonsmooth unimodal densities its risk is the same as the risk of the grenander estimator based on the knowledge of the true mode plus some lower order term it can also cope with small departures from unimodality

on the risk of histograms for estimating decreasing densities on cherche a concevoir un estimateur simple et a obtenir des bornes superieures pour son risque qui soient dans un facteur 10 de la borne inferieure de birge 1987

estimation of a function under shape restrictions applications to reliability this paper deals with a nonparametric shape respecting estimation method for u shaped or unimodal functions a general upper bound for the nonasymptotic l 1 risk of the estimator is given the method is applied to the shape respecting estimation of several classical functions among them typical intensity functions encountered in the reliability field in each case we derive from our upper bound the spatially adaptive property of our estimator with respect to the l 1 metric it approximately behaves as the best variable binwidth histogram of the function under estimation

best asymptotic normality of the kernel density entropy estimator for smooth densities in the random sampling setting we estimate the entropy of a probability density distribution by the entropy of a kernel density estimator using the double exponential kernel under mild smoothness and moment conditions we show that the entropy of the kernel density estimator equals a sum of independent and identically distributed i i d random variables plus a perturbation which is asymptotically negligible compared to the parametric rate n sup 1 2 an essential part in the proof is obtained by exhibiting almost sure bounds for the kullback leibler divergence between the kernel density estimator and its expected value the basic technical tools are doob s submartingale inequality and convexity jensen s inequality

review papers recent developments in nonparametric density estimation abstract advances in computation and the fast and cheap computational facilities now available to statisticians have had a significant impact upon statistical research and especially the development of nonparametric data analysis procedures in particular theoretical and applied research on nonparametric density estimation has had a noticeable influence on related topics such as nonparametric regression nonparametric discrimination and nonparametric pattern recognition this article reviews recent developments in nonparametric density estimation and includes topics that have been omitted from review articles and books on the subject the early density estimation methods such as the histogram kernel estimators and orthogonal series estimators are still very popular and recent research on them is described different types of restricted maximum likelihood density estimators including order restricted estimators maximum penalized likelihood estimators and sieve estimators are discussed where re

the cumulant lattice boltzmann equation in three dimensions theory and validation we propose analyze and validate a lattice boltzmann model with a cumulant collision operator the new model is analytically and numerically shown to poses smaller errors than a moment based multiple relaxation time lattice boltzmann model we demonstrate the usability of the cumulant lattice boltzmann model by simulations of flow around a sphere for reynolds numbers from 200 to 105

asymptotic analysis of the lattice boltzmann equation in this article we analyze the lattice boltzmann equation lbe by using the asymptotic expansion technique we first relate the lbe to the finite discrete velocity model fdvm of the boltzmann equation with the diffusive scaling the analysis of this model directly leads to the incompressible navier stokes equations as opposed to the compressible navier stokes equations obtained by the chapman enskog analysis with convective scaling we also apply the asymptotic analysis directly to the fully discrete lbe as opposed to the usual practice of analyzing a continuous equation obtained through the taylor expansion of the lbe this leads to a consistency analysis which provides order by order information about the numerical solution of the lbe the asymptotic technique enables us to analyze the structure of the leading order errors and the accuracy of numerically derived quantities such as vorticity it also justifies the use of richardson s extrapolation method as an example a two dimensional taylor vortex flow is used to validate our analysis the numerical results agree very well with our analytic predictions

efficient algorithms for learning functions with bounded variation we show that the class fbv of 0 1 valued functions with total variation at most 1 can be agnostically learned with respect to the absolute loss in polynomial time from o 1 e2 log 1 examples matching a known lower bound to within a constant factor we establish a bound of o 1 m on the expected error of a polynomial time algorithm for learning fbv in the prediction model also matching a known lower bound to within a constant factor applying a known algorithm transformation to our prediction algorithm we obtain a polynomial time pac learning algorithm for fbv with a sample complexity bound of o 1 e log 1 this also matches a known lower bound to within a constant factor

an immersed boundary method based on the gas kinetic bgk scheme for incompressible viscous flow in this paper an immersed boundary ib method based on the gas kinetic bhatnagar gross krook bgk scheme is proposed to simulate the incompressible viscous flow around stationary and moving rigid bodies in the presented ib method a set of lagrangian points represent the solid boundary and will exert force on the surrounding eulerian points which are the cell centers within the finite volume framework this force is calculated by a special iterative procedure and has an effect on the flux at the interface of cell which guarantees the fulfillment of the no slip boundary condition and entirely avoids the flow penetration the flow field is obtained from the bgk scheme with local grid refinement without complex mesh generation and transformation the present technique can be conveniently applied to simulations with complex and moving solid boundaries the second order temporal accuracy and better than first order spatial accuracy in l 2 norm are testified by the simulation of stokes first problem other three different test cases including flows around a stationary circular cylinder two circular cylinders in tandem and an oscillating circular cylinder demonstrate the good capability of the present method expanding the immersed boundary ib method to the finite volume gas kinetic bhatnagar gross krook bgk scheme the flux across the cell s interface will be influenced by the immersed boundary force the flow penetration to the immersed boundary is completely eliminated the present method can be conveniently applied to problems with moving objects

combinatorics of random processes and sections of convex bodies we find a sharp combinatorial bound for the metric entropy of sets in r n and general classes of functions this solves two basic combinatorial conjectures on the empirical processes 1 a class of functions satisfies the uniform central limit theorem if the square root of its combinatorial dimension is integrable 2 the uniform entropy is equivalent to the combinatorial dimension under minimal regularity our method also constructs a nicely bounded coordinate section of a symmetric convex body in r n in the operator theory this essentially proves for all normed spaces the restricted invertibility principle of bourgain and tzafriri

maximum likelihood estimation of smooth monotone and unimodal densities we study the nonparametric estimation of univariate monotone and unimodal densities using the maximum smoothed likelihood approach the monotone estimator is the derivative of the least concave majorant of the distribution corresponding to a kernel estimator we prove that the mapping on distributions with density the derivative of the least concave majorant of is a contraction in all l p norms 1 p and some other distances such as the hellinger and kullback leibler distances the contractivity implies error bounds for monotone density estimation almost the same error bounds hold for unimodal estimation

immersed boundary method for flow around an arbitrarily moving body for the simulation of flow around an arbitrarily moving body an immersed boundary method is developed in a non inertial reference frame that is fixed to the body the navier stokes equation is formulated in a conservative form such that the force terms due to the rotation and the translational and rotational accelerations are included in the nonlinear term in order to satisfy the no slip condition on the body surface momentum forcing and mass source sink are applied on the body surface or inside the body the numerical method is based on a finite volume approach on a staggered mesh together with a fractional step method the present numerical method is applied to both the forced motion and fluid structure interaction problems in the latter we solve fully coupled navier stokes and dynamic equations for the moving body without introducing any iteration four different flow problems are tested and the results are in excellent agreements with previous numerical and experimental ones

second order interpolation of the flow field in the lattice boltzmann method we propose a scheme to reconstruct the flow field up to second order in a cell composed of four lattices nodes in 2 dimensions the information contained in the higher order moments of the distribution functions is used to construct an interpolation scheme of second order for the velocity field

covering numbers for bounded variation functions abstract in this paper we provide upper and lower estimates for the minimal number of functions needed to represent a bounded variation function with an accuracy of epsilon with respect to l 1 distance

maximal width learning of binary functions this paper concerns learning binary valued functions defined on r and investigates how a particular type of regularity of hypotheses can be used to obtain better generalization error bounds we derive error bounds that depend on the sample width a notion analogous to that of sample margin for real valued functions this motivates learning algorithms that seek to maximize sample width

on the minimisation of l sp p error in mode estimation we show that for l p convergence of the mode of a nonparametric density estimator to the mode of an unknown probability density finiteness of the pth moment of the underlying distribution is both necessary and sufficient the basic requirement of existence of finite variance has been overlooked by statisticians who have earlier considered mean square convergence of nonparametric mode estimators they have focussed on mean squared error of the asymptotic distribution rather than on asymptotic mean squared error the effect of bandwidth choice on the rate of l p convergence is analysed and smoothed bootstrap methods are used to develop an empirical approximation to the l p measure of error the resulting bootstrap estimator of l p error may be minimised with respect to the bandwidth of the nonparametric density estimator and in this way an empirical rule may be developed for selecting the bandwidth for mode estimation particular attention is devoted to the problem of selecting the appropriate amount of smoothing in the bootstrap algorithm

a lower bound for the risk in estimating the value of a probability density abstract this article considers estimation of f 0 of a density function satisfying a lipshitz condition in a neighborhood of 0 a nonstandard use of the cramer rao inequality yields numerical lower bounds on the minimax squared error risk of any estimator these bounds are then compared with the minimax risk of the asymptotically optimal kernel type estimator the asymptotic bounds obtained as the sample size n are not quite as good as those in donoho and liu in press a b but bounds are presented here also for finite values of n and that paper contains no such bounds for this problem the numerical results reported in table 1 show that the asymptotically optimal kernel estimator performs within a factor of 3 of the minimax bound even for sample size n 30 as n increases the relative performance improves to its limiting value although the convergence is fairly slow

numerical integration of the differential matrix riccati equation two new bernoulli substitution methods for solving the riccati differential equation are tested numerically against direct integration of the riccati equation the chandrasekhar algorithm and the davison maki method on a large set of problems taken from the literature the first of these new methods was developed for the time invariant case and uses the matrix analog of completing the square to transform the problem to a bisymmetric riecati equation whose solution can be given explicitly in terms of a matrix exponential of order n this method is fast and accurate when the extremal solutions of the associated algebraic riccati equation are well separated the second new method was developed as a means of eliminating the instabilities associated with the davison maki algorithm by using reinitialization at each time step the davison maki algorithm can be recast as a recursion which is over three times faster than the original method and is easily shown to be stable for both time invariant and time dependent problems from the results of our study we conclude that the modified davison maki method gives superior performance except for those problems where the number of observers and controllers is small relative to the number of states in which ease the chandrasekhar algorithm is better

proteus a coupled iterative force correction immersed boundary multi domain cascaded lattice boltzmann solver abstract most realistic fluid flow problems are characterized by high reynolds numbers and complex boundaries over the last ten years immersed boundary methods that are able to cope with realistic geometries have been applied to lattice boltzmann lb methods these methods however have normally been applied to low reynolds number problems here we present a novel coupling between an iterative force correction immersed boundary zhang et al 2016 and a multi domain cascaded lb method the iterative force correction immersed boundary method has been selected due to the improved accuracy of the computation while the cascaded lb formulation is used due to its superior stability at high reynolds numbers the coupling is shown to improve both the stability and numerical accuracy of the solution the resulting solver has been applied to viscous flow up to a reynolds number of 100000 passed a naca 0012 airfoil at a 10 degree angle of attack good agreement with results obtained using a body fitted navier stokes solver has been obtained the formulation provides a straight forward and efficient method for modeling realistic geometries and could easily be extended to problems with moving boundaries

cascaded digital lattice boltzmann automata for high reynolds number flow lattice boltzmann methods are of limited applicability for direct numerical simulation of turbulent flow due to instabilities in the zero viscosity limit we observe that this is caused by an insufficient degree of galilean invariance of the relaxation type lattice boltzmann collision operator the cascaded digital lattice boltzmann automata described here provides a method with which to achieve stable collision operators down to the limit of zero viscosity

quantile and probability curves without crossing this paper proposes a method to address the longstanding problem of lack of monotonicity in estimation of conditional and structural quantile functions also known as the quantile crossing problem bassett and koenker 1982 the method consists in sorting or monotone rearranging the original estimated non monotone curve into a monotone rearranged curve we show that the rearranged curve is closer to the true quantile curve than the original curve in finite samples establish a functional delta method for rearrangement related operators and derive functional limit theory for the entire rearranged curve and its functionals we also establish validity of the bootstrap for estimating the limit law of the entire rearranged curve and its functionals our limit results are generic in that they apply to every estimator of a monotone function provided that the estimator satisfies a functional central limit theorem and the function satisfies some smoothness conditions consequently our results apply to estimation of other econometric functions with monotonicity restrictions such as demand production distribution and structural distribution functions we illustrate the results with an application to estimation of structural distribution and quantile functions using data on vietnam veteran status and earnings

global rates of convergence of the mles of log concave and s concave densities we establish global rates of convergence for the maximum likelihood estimators mles of log concave and s concave densities on the main finding is that the rate of convergence of the mle in the hellinger metric is no worse than n 2 5 when 1 s where s 0 corresponds to the log concave case we also show that the mle does not exist for the classes of s concave densities with s 1

bayesian mode regression using mixtures of triangular densities abstract bayesian semiparametric models for mean and median regressions abound but a void for mode regressions exists we fill this gap by nonparametrically modeling the error distribution in such regressions that entails constructing prior distributions on densities which exhibit flexibility while fixing the mode at 0 such priors exist when constraining the mean and median but to our knowledge there is none for the mode our solution with mixtures of triangular distributions results in a conditionally conjugate prior on the space of unimodal untruncated convex densities consistency properties of the resulting modal estimators are studied followed by simulated and real data illustrations

decision theoretic generalizations of the pac model for neural net and other learning applications we describe a generalization of the pac learning model that is based on statistical decision theory in this model the learner receives randomly drawn examples each example consisting of an instance x in x and an outcome y in y and tries to find a hypothesis h x a where h in h that specifies the appropriate action a in a to take for each instance x in order to minimize the expectation of a loss l y a here x y and a are arbitrary sets l is a real valued function and examples are generated according to an arbitrary joint distribution on x times y special cases include the problem of learning a function from x into y the problem of learning the conditional probability distribution on y given x regression and the problem of learning a distribution on x density estimation we give theorems on the uniform convergence of empirical loss estimates to true expected loss rates for certain hypothesis spaces h and show how this implies learnability with bounded sample size disregarding computational complexity as an application we give distribution independent upper bounds on the sample size needed for learning with feedforward neural networks our theorems use a generalized notion of vc dimension that applies to classes of real valued functions adapted from pollard s work and a notion of capacity and metric dimension for classes of functions that map into a bounded metric space supersedes 89 30 and 90 52 also in information and computation vol 100 no 1 september 1992

a universally acceptable smoothing factor for kernel density estimates we define a minimum distance estimate of the smoothing factor for kernel density estimates based on a methodology first developed by yatracos it is shown that if f nh denotes the kernel density estimate on r d for an i i d sample of size n drawn from an unknown density f where h is the smoothing factor and if f n is the kernel estimate with the same kernel and with the proposed new data based smoothing factor then under a regularity condition on the kernel k formula math this is the first published smoothing factor that can be proven to have this property

accurate numerical integration of stiff differential riccati equations abstract in this paper accurate numerical integration methods for four times continuously differentiable coefficient stiff riccati matrix differential equations are proposed first a discrete numerical solution is constructed at a net of points of the predetermined existence interval using implicit one step matrix methods finally using linear b spline matrix functions a continuous numerical solution whose error is uniformly upper bounded by a prefixed error in the existence domain is constructed

algorithm dependent generalization bounds for multi task learning often tasks are collected for multi task learning mtl because they share similar feature structures based on this observation in this paper we present novel algorithm dependent generalization bounds for mtl by exploiting the notion of algorithmic stability we focus on the performance of one particular task and the average performance over multiple tasks by analyzing the generalization ability of a common parameter that is shared in mtl when focusing on one particular task with the help of a mild assumption on the feature structures we interpret the function of the other tasks as a regularizer that produces a specific inductive bias the algorithm for learning the common parameter as well as the predictor is thereby uniformly stable with respect to the domain of the particular task and has a generalization bound with a fast convergence rate of order mathcal o 1 n where n is the sample size of the particular task when focusing on the average performance over multiple tasks we prove that a similar inductive bias exists under certain conditions on the feature structures thus the corresponding algorithm for learning the common parameter is also uniformly stable with respect to the domains of the multiple tasks and its generalization bound is of the order mathcal o 1 t where t is the number of tasks these theoretical analyses naturally show that the similarity of feature structures in mtl will lead to specific regularizations for predicting which enables the learning algorithms to generalize fast and correctly from a few examples

the limit distribution of the l error of grenander type estimators let f be a nonincreasing function defined on 0 1 under standard regularity conditions we derive the asymptotic distribution of the supremum norm of the difference between f and its grenander type estimator on sub intervals of 0 1 the rate of convergence is found to be of order n log n 1 3 and the limiting distribution to be gumbel

risk bounds in isotonic regression nonasymptotic risk bounds are provided for maximum likelihood type isotonic estimators of an unknown nondecreasing regression function with general average loss at design points these bounds are optimal up to scale constants and they imply uniform n 1 3 consistency of the ell p risk for unknown regression functions of uniformly bounded variation under mild assumptions on the joint probability distribution of the data with possibly dependent observations

stabilization of the lattice boltzmann method by the h theorem a numerical test for a one dimensional benchmark shock tube problem we implement the lattice boltzmann method based on the h theorem i karlin a ferrante and h c ottinger europhys lett 47 182 1999 results of simulation demonstrate significant improvement of stability as compared to realizations without explicit entropic estimations

quantitative compactness estimates for hamilton jacobi equations we study quantitative compactness estimates in mathbf w 1 1 rm loc for the map s t t 0 that is associated with the given initial data u 0 in rm lip mathbb r n for the corresponding solution s t u 0 of a hamilton jacobi equation u t h big nabla x u big 0 qquad t geq 0 quad x in mathbb r n with a uniformly convex hamiltonian h h p we provide upper and lower estimates of order 1 varepsilon n on the kolmogorov varepsilon entropy in mathbf w 1 1 of the image through the map st of sets of bounded compactly supported initial data estimates of this type are inspired by a question posed by lax course on hyperbolic systems of conservation laws xxvii scuola estiva di fisica matematica ravello 2002 within the context of conservation laws and could provide a measure of the order of resolution of a numerical method implemented for this equation

estimating a density under order restrictions nonasymptotic minimax risk on considere la classe de toutes les densites unimodales definies sur un intervalle de longueur l et borne par h on etudie le risque minimax sur cette classe lorsqu on estime en utilisant n observations i i d la perte etant mesuree par la distance l 1 entre l estimateur et la densite reelle

the importance of convexity in learning with squared loss we show that if the closure of a function class f under the metric induced by some probability distribution is not convex then the sample complexity for agnostically learning f with squared loss using only hypotheses in f is spl omega ln 1 spl delta spl epsiv sup 2 where 1 spl delta is the probability of success and spl epsiv is the required accuracy in comparison if the class f is convex and has finite pseudodimension then the sample complexity is o 1 spl epsiv ln 1 spl epsiv ln 1 b if a nonconvex class f has finite pseudodimension then the sample complexity for agnostically learning the closure of the convex hull of f is o 1 spl epsiv 1 spl epsiv ln 1 spl epsiv ln 1 spl delta hence for agnostic learning learning the convex hull provides better approximation capabilities with little sample complexity penalty

a riccati transformation method for solving linear bvps i theoretical aspects a method of solving linear two point boundary value problems bvps for first order systems of ordinary differential equations odes is considered of particular interest is the class of singularly perturbed bvps the method attempts to split the integration of the rapidly increasing decreasing solution components typical for such problems into different initial value problems ivps the case of separated boundary conditions bcs is considered and their ability to detect solution components that eventually increase decrease is exploited by using the riccati transformation the original bvp is then reformulated as three ivps one of them being a matrix riccati equation the riccati equation is nonlinear and a multiple embedding setting must be used the entire method is then fit theoretically into a multiple shooting like framework and the original bvp as a chain of subproblems is studied in this way a generalized decoupling result is presented which enables the study of stability to be brough