stability analysis of lattice boltzmann methods the lattice boltzmann equation describes the evolution of the velocity distribution function on a lattice in a manner that macroscopic fluid dynamical behavior is recovered although the equation is a derivative of lattice gas automata it may be interpreted as a lagrangian finite difference method for the numerical simulation of the discrete velocity boltzmann equation that makes use of a bgk collision operator as a result it is not surprising that numerical instability of lattice boltzmann methods have been frequently encountered by researchers we present an analysis of the stability of perturbations of the particle populations linearized about equilibrium values corresponding to a constant density uniform mean flow the linear stability depends on the following parameters the distribution of the mass at a site between the different discrete speeds the bgk relaxation time the mean velocity and the wavenumber of the perturbations this parameter space is too large to compute the complete stability characteristics we report some stability results for a subset of the parameter space for a 7 velocity hexagonal lattice a 9 velocity square lattice and a 15 velocity cubic lattice results common to all three lattices are 1 the bgk relaxation time must be greater than 12 corresponding to positive shear viscosity 2 there exists a maximum stable mean velocity for fixed values of the other parameters and 3 as is increased from 12 the maximum stable velocity increases monotonically until some fixed velocity is reached which does not change for larger

principal component analysis in the local differential privacy model abstract in this paper we study the principal component analysis pca problem under the distributed non interactive local differential privacy model for the low dimensional case i e p n we show the optimal rate of k p n 2 omitting the eigenvalue terms for the private minimax risk of the k dimensional pca using the squared subspace distance as the measurement where n is the sample size and is the privacy parameter for the high dimensional i e p n row sparse case we first give a lower bound of k s log p n 2 on the private minimax risk where s is the underlying sparsity parameter then we provide an efficient algorithm to achieve the upper bound of o s 2 log p n 2 experiments on both synthetic and real world datasets confirm our theoretical guarantees

influential features pca for high dimensional clustering we consider a clustering problem where we observe feature vectors x i in r p i 1 2 ldots n from k possible classes the class labels are unknown and the main interest is to estimate them we are primarily interested in the modern regime of p gg n where classical clustering methods face challenges we propose influential features pca if pca as a new clustering procedure in if pca we select a small fraction of features with the largest kolmogorov smirnov ks scores obtain the first k 1 left singular vectors of the post selection normalized data matrix and then estimate the labels by applying the classical k means procedure to these singular vectors in this procedure the only tuning parameter is the threshold in the feature selection step we set the threshold in a data driven fashion by adapting the recent notion of higher criticism as a result if pca is a tuning free clustering method we apply if pca to 10 gene microarray data sets the method has competitive performance in clustering especially in three of the data sets the error rates of if pca are only 29 or less of the error rates by other methods we have also rediscovered a phenomenon on empirical null by efron j amer statist assoc 99 2004 96 104 on microarray data with delicate analysis especially post selection eigen analysis we derive tight probability bounds on the kolmogorov smirnov statistics and show that if pca yields clustering consistency in a broad context the clustering problem is connected to the problems of sparse pca and low rank matrix recovery but it is different in important ways we reveal an interesting phase transition phenomenon associated with these problems and identify the range of interest for each

controlling chaos experimentally in systems exhibiting large effective lyapunov exponents we investigate experimentally the performance of the ott grebogi and yorke phys rev lett 64 1196 1989 feedback concept to control chaotic motion the experimental systems are a driven pendulum and a driven bronze ribbon both setups have unstable periodic orbits characterized by large effective lyapunov exponents all control vectors for the feedback control are extracted from the experimental data to do this for the pendulum a global model obtained by the flow field analysis of cremers and h ubler z naturforsch 42a 797 1987 is used and for the bronze ribbon linear approximations in embedding space are exploited we analyze the problems that arise due to the amplification of noise by large effective lyapunov exponents in the determination of the control values as well as in the performance of the experimental control successful control can be achieved in our experiments by applying the local control method which allows a quasicontinuous adjustment of the control parameter in contrast to adjusting the control parameter only once per return time of the poincar e map

phase transitions for high dimensional clustering and related problems consider a two class clustering problem where we observe x i ell i mu z i z i stackrel mathit i i d sim n 0 i p 1 leq i leq n the feature vector mu in r p is unknown but is presumably sparse the class labels ell i in 1 1 are also unknown and the main interest is to estimate them we are interested in the statistical limits in the two dimensional phase space calibrating the rarity and strengths of useful features we find the precise demarcation for the region of impossibility and region of possibility in the former useful features are too rare weak for successful clustering in the latter useful features are strong enough to allow successful clustering the results are extended to the case of colored noise using le cam s idea on comparison of experiments we also extend the study on statistical limits for clustering to that for signal recovery and that for global testing we compare the statistical limits for three problems and expose some interesting insight we propose classical pca and important features pca if pca for clustering for a threshold t 0 if pca clusters by applying classical pca to all columns of x with an l 2 norm larger than t we also propose two aggregation methods for any parameter in the region of possibility some of these methods yield successful clustering we discover a phase transition for if pca for any threshold t 0 let xi t be the first left singular vector of the post selection data matrix the phase space partitions into two different regions in one region there is a t such that cos xi t ell rightarrow 1 and if pca yields successful clustering in the other cos xi t ell leq c 0 0 our results require delicate analysis especially on post selection random matrix theory and on lower bound arguments

on consistency and sparsity for principal components analysis in high dimensions principal components analysis pca is a classic method for the reduction of dimensionality of data in the form of n observations or cases of a vector with p variables contemporary datasets often have p comparable with or even much larger than n our main assertions in such settings are a that some initial reduction in dimensionality is desirable before applying any pca type search for principal modes and b the initial reduction in dimensionality is best achieved by working in a basis in which the signals have a sparse representation we describe a simple asymptotic model in which the estimate of the leading principal component vector via standard pca is consistent if and only if p n n 0 we provide a simple algorithm for selecting a subset of coordinates with largest sample variances and show that if pca is done on the selected subset then consistency is recovered even if p n n

lattice boltzmann model for the incompressible navier stokes equation in this paper a lattice boltzmann lb model to simulate incompressible flow is developed the main idea is to explicitly eliminate the terms of o m2 where m is the mach number due to the density fluctuation in the existing lb models in the proposed incompressible lb model the pressure p instead of the mass density is the independent dynamic variable the incompressible navier stokes equations are derived from the incompressible lb model via chapman enskog procedure numerical results of simulations of the plane poiseuille flow driven either by pressure gradient or a fixed velocity profile at entrance as well as of the 2d womersley flow are presented the numerical results are found to be in excellent agreement with theory

invariant grids for reaction kinetics abstract in this paper we construct low dimensional manifolds of reduced description for equations of chemical kinetics from the standpoint of the method of invariant manifold mim mim is based on a formulation of the condition of invariance as an equation and its solution by newton iterations a grid based version of mim is developed the method of invariant grids we describe the newton method and the relaxation method for the invariant grids construction the problem of the grid correction is fully decomposed into the problems of the grid s nodes correction the edges between the nodes appear only in the calculation of the tangent spaces this fact determines high computational efficiency of the method of invariant grids the method is illustrated by two examples the simplest catalytic reaction michaelis menten mechanism and the hydrogen oxidation the algorithm of analytical continuation of the approximate invariant manifold from the discrete grid is proposed generalizations to open systems are suggested the set of methods covered makes it possible to effectively reduce description in chemical kinetics

controlling chaotic dynamical systems using time delay coordinates abstract the ott grebogi yorke control method is analyzed in the case that the attractor is reconstructed from a time series using time delay coordinates it turns out that the control formula of ogy should be modified in order to apply to experimental systems if time dekay coordinates are used we reveal that the experimental surface of section map does not only depend on the actual parameter but also on the preceding one in order to meet this dependence two modifications are introduced which lead to a better performance of the control to compare their control abilities they are applied to simulations of a duffing oscillator the issue of measurement noise is considered

discretizations for the incompressible navier stokes equations based on the lattice boltzmann method a discrete velocity model with spatial and velocity discretization based on a lattice boltzmann method is considered in the low mach number limit a uniform numerical scheme for this model is investigated in the limit the scheme reduces to a finite difference scheme for the incompressible navier stokes equation which is a projection method with a second order spatial discretization on a regular grid the discretization is analyzed and the method is compared to chorin s original spatial discretization numerical results supporting the analytical statements are presented

the effect of lumping and expanding on kinetic differential equations let us consider the differential equation dot bf y t bf f bf y t with an bf f from bf r n to bf r n and suppose that there exists a transformation bf h from bf r n to bf r hat n hat n le n such that hat bf y bf h circ bf y obeys a differential equation dot hat bf y t hat bf f hat bf y t with some function hat bf f then the first equation is said to be lumpable to the second by bf h here mainly the case is investigated when the original differential equation has been induced by a complex chemical reaction we provided a series of necessary and sufficient conditions for the existence of such functions bf h and hat bf f some of them are formulated in terms of bf h and bf f only beyond these conditions our main concern here is how lumping changes those properties of the solutions which are either interesting from the point of view of the qualitative theory of differential equations or from the point

a direct formulation for sparse pca using semidefinite programming given a covariance matrix we consider the problem of maximizing the variance explained by a particular linear combination of the input variables while constraining the number of nonzero coefficients in this combination this problem arises in the decomposition of a covariance matrix into sparse factors or sparse principal component analysis pca and has wide applications ranging from biology to finance we use a modification of the classical variational representation of the largest eigenvalue of a symmetric matrix where cardinality is constrained and derive a semidefinite programming based relaxation for our problem we also discuss nesterov s smooth minimization technique applied to the semidefinite program arising in the semidefinite relaxation of the sparse pca problem the method has complexity o n 4 sqrt log n epsilon where n is the size of the underlying covariance matrix and epsilon is the desired absolute accuracy on the optimal value of the problem

maximal aggregation of polynomial dynamical systems ordinary differential equations odes with polynomial derivatives are a fundamental tool for understanding the dynamics of systems across many branches of science but our ability to gain mechanistic insight and effectively conduct numerical evaluations is critically hindered when dealing with large models here we propose an aggregation technique that rests on two notions of equivalence relating ode variables whenever they have the same solution backward criterion or if a self consistent system can be written for describing the evolution of sums of variables in the same equivalence class forward criterion a key feature of our proposal is to encode a polynomial ode system into a finitary structure akin to a formal chemical reaction network this enables the development of a discrete algorithm to efficiently compute the largest equivalence building on approaches rooted in computer science to minimize basic models of computation through iterative partition refinements the physical interpretability of the aggregation is shown on polynomial ode systems for biochemical reaction networks gene regulatory networks and evolutionary game theory

error analysis for chorin s original fully discrete projection method and regularizations in space and time over twenty five years ago chorin proposed a computationally efficient method for computing viscous incompressible flow which has influenced the development of efficient modern methods and inspired much analytical work using asymptotic error analysis techniques it is now possible to describe precisely the kind of errors that are generated in the discrete solutions from this method and the order at which they occur while the expected convergence rate is seen for velocity the pressure accuracy is degraded by two effects a numerical boundary layer due to the projection step and a global error due to the alternating or parasitic modes present in the discretization of the incompressibility condition the error analysis of the projection step follows the work of e and liu and the analysis of the alternating modes is due to the author the two are combined to show the asymptotic character of the errors in the scheme regularization methods in space and time for recovering full accuracy for the computed pressure are discussed

crises sudden changes in chaotic attractors and transient chaos abstract the occurrence of sudden qualitative changes of chaotic dynamics as a parameter is varied is discussed and illustrated it is shown that such changes may result from the collision of an unstable periodic orbit and a coexisting chaotic attractor we call such collisions crises phenomena associated with crises include sudden changes in the size of chaotic attractors sudden appearances of chaotic attractors a possible route to chaos and sudden destructions of chaotic attractors and their basins this paper presents examples illustrating that crisis events are prevalent in many circumstances and systems and that just past a crisis certain characteristic statistical behavior whose type depends on the type of crisis occurs in particular the phenomenon of chaotic transients is investigated the examples discussed illustrate crises in progressively higher dimension and include the one dimensional quadratic map the two dimensional henon map systems of ordinary differential equations in three dimensions and a three dimensional map in the case of our study of the three dimensional map a new route to chaos is proposed which is possible only in invertible maps or flows of dimension at least three or four respectively based on the examples presented the following conjecture is proposed almost all sudden changes in the size of chaotic attractors and almost all sudden destruction or creations of chaotic attractors and their basins are due to crises

effects of density and force discretizations on spurious velocities in lattice boltzmann equation for two phase flows spurious current emerging in the vicinity of phase interfaces is a well known disadvantage of the lattice boltzmann equation lbe for two phase flows previous analysis shows that this unphysical phenomenon comes from the force imbalance at discrete level inherited in lbe guo et al 2011 phys rev e 83 036707 based on the analysis of the lbe free of checkerboard effects in this work we further show that the force imbalance is caused by the different discretization stencils the implicit one from the streaming process and the explicit one from the discretization of the force term particularly the total contribution includes two parts one from the difference between the intrinsically discretized density or ideal gas pressure gradient and the explicit ones in the force term and the other from the explicit discretized chemical potential gradients in the intrinsically discretized force term the former contribution is a special feature of lbe which was not realized previously

an asymptotic induced scheme for nonstationary transport equations in the diffusive limit an asymptotic induced scheme for nonstationary transport equations with the diffusion scaling is developed the scheme works uniformly for all ranges of mean free paths it is based on the asymptotic analysis of the diffusion limit of the transport equation a theoretical investigation of the behavior of the scheme in the diffusion limit is given and an approximation property is proven moreover numerical results for different physical situations are shown and the uniform convergence of the scheme is established numerically

innovated higher criticism for detecting sparse signals in correlated noise higher criticism is a method for detecting signals that are both sparse and weak although first proposed in cases where the noise variables are independent higher criticism also has reasonable performance in settings where those variables are correlated in this paper we show that by exploiting the nature of the correlation performance can be improved by using a modified approach which exploits the potential advantages that correlation has to offer indeed it turns out that the case of independent noise is the most difficult of all from a statistical viewpoint and that more accurate signal detection for a given level of signal sparsity and strength can be obtained when correlation is present we characterize the advantages of correlation by showing how to incorporate them into the definition of an optimal detection boundary the boundary has particularly attractive properties when correlation decays at a polynomial rate or the correlation matrix is toeplitz

finding a sparse vector in a subspace linear sparsity using alternating directions is it possible to find the sparsest vector direction in a generic subspace mathcal s subseteq mathbb r p with text dim left mathcal s right n this problem can be considered a homogeneous variant of the sparse recovery problem and finds connections to sparse dictionary learning sparse pca and many other problems in signal processing and machine learning in this paper we focus on a planted sparse model for the subspace the target sparse vector is embedded in an otherwise random subspace simple convex heuristics for this planted recovery problem provably break down when the fraction of nonzero entries in the target sparse vector substantially exceeds o 1 sqrt n in contrast we exhibit a relatively simple nonconvex approach based on alternating directions which provably succeeds even when the fraction of nonzero entries is omega 1 to the best of our knowledge this is the first practical algorithm to achieve linear scaling under the planted sparse model empirically our proposed algorithm also succeeds in more challenging data models e g sparse dictionary learning

variable selection for model based clustering we consider the problem of variable or feature selection for model based clustering the problem of comparing two nested subsets of variables is recast as a model comparison problem and addressed using approximate bayes factors a greedy search algorithm is proposed for finding a local optimum in model space the resulting method selects variables or features the number of clusters and the clustering model simultaneously we applied the method to several simulated and real examples and found that removing irrelevant variables often improved performance compared with methods based on all of the variables our variable selection method consistently yielded more accurate estimates of the number of groups and lower classification error rates as well as more parsimonious clustering models and easier visualization of results

on the distribution of the largest eigenvalue in principal components analysis let x 1 denote the square of the largest singular value of an n p matrix x all of whose entries are independent standard gaussian variates equivalently x 1 is the largest principal component variance of the covariance matrix x x or the largest eigenvalue of a p variate wishart distribution on n degrees of freedom with identity covariance consider the limit of large p and n with n p gamma ge 1 when centered by mu p sqrt n 1 sqrt p 2 and scaled by sigma p sqrt n 1 sqrt p 1 sqrt n 1 1 sqrt p 1 3 the distribution of x 1 approaches the tracey widom law of order 1 which is defined in terms of the painleve ii differential equation and can be numerically evaluated and tabulated in software simulations show the approximation to be informative for n and p as small as 5 the limit is derived via a corresponding result for complex wishart matrices using methods from random matrix theory the result suggests that some aspects of large p multivariate distribution theory may be easier to apply in practice than their fixed p counterparts

a simple prior free method for non rigid structure from motion factorization this paper proposes a simple prior free method for solving the non rigid structure from motion nrsfm factorization problem other than using the fundamental low order linear combination model assumption our method does not assume any extra prior knowledge either about the non rigid structure or about the camera motions yet it works effectively and reliably producing optimal results and not suffering from the inherent basis ambiguity issue which plagued most conventional nrsfm factorization methods our method is very simple to implement which involves solving a very small sdp semi definite programming of fixed size and a nuclear norm minimization problem we also present theoretical analysis on the uniqueness and the relaxation gap of our solutions extensive experiments on both synthetic and real motion capture data assuming following the low order linear combination model are conducted which demonstrate that our method indeed outperforms most of the existing non rigid factorization methods this work offers not only new theoretical insight but also a practical everyday solution to nrsfm

alternative kinetic theory based lattice boltzmann model for incompressible axisymmetric flows abstract based on the axisymmetric boltzmann equation an incompressible lattice boltzmann model for axisymmetric flows is proposed within the framework of the kinetic theory based model developed by guo et al 2009 while retaining the advantages of the guo et al 2009 model in terms of the solid physics basis and simple source terms involving no gradient calculations the present model further improves the numerical stability and reduces compressibility errors and computational requirements armed with the assumption that the fluid density is a constant and thus the fluid pressure has no direct relation with the density the incompressibility conditions are realized by applying the hermite expansion then the present model employs a novel way of calculating the fluid pressure which is derived from the modified second order moment equation additionally based on the regularized lattice bgk rlbgk model an extra relaxation parameter pertaining to the ghost mode is introduced to enhance the numerical stability of the present model the accuracy and applicability of the present model are verified by both the chapman enskog theoretical analysis and numerical validations it is demonstrated via well acknowledged test cases that the present model is accurate and reliable for incompressible axisymmetric flows and is able to effectively reduce the compressibility errors vis a vis the guo et al 2009

incompressible navier stokes and euler limits of the boltzmann equation we consider solutions of the boltzmann equation in a d dimensional torus d 2 3 for macroscopic times t n 1 t 0 when the space variations are on a macroscopic scale x n 1r n 2 x in the unit torus let u x t be for t t0 a smooth solution of the incompressible navier stokes equations ins for n 2 and of the incompressible euler equation ie for n 2 we prove that has solutions for t t0 which are close to o 2 in a suitable norm to the local maxwellian p 2 t d 2 exp v u x t 2 2t with constant density p and temperature t this is a particular case defined by the choice of initial values of the macroscopic variables of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations for n 3 these equations correspond to variable density ie while for n 2 they involve higher order derivatives of the density

lambertian reflectance and linear subspaces we prove that the set of all lambertian reflectance functions the mapping from surface normals to intensities obtained with arbitrary distant light sources lies close to a 9d linear subspace this implies that in general the set of images of a convex lambertian object obtained under a wide variety of lighting conditions can be approximated accurately by a low dimensional linear subspace explaining prior empirical results we also provide a simple analytic characterization of this linear space we obtain these results by representing lighting using spherical harmonics and describing the effects of lambertian materials as the analog of a convolution these results allow us to construct algorithms for object recognition based on linear methods as well as algorithms that use convex optimization to enforce nonnegative lighting functions we also show a simple way to enforce nonnegative lighting when the images of an object lie near a 4d linear space we apply these algorithms to perform face recognition by finding the 3d model that best matches a 2d query image

lattice boltzmann method and gas kinetic bgk scheme in the low mach number viscous flow simulations both lattice boltzmann method lbm and the gas kinetic bgk scheme are based on the numerical discretization of the boltzmann equation with collisional models such as the bhatnagar gross krook bgk model lbm tracks limited number of particles and the viscous flow behavior emerges automatically from the intrinsic particle stream and collisions process on the other hand the gas kinetic bgk scheme is a finite volume scheme where the time dependent gas distribution function with continuous particle velocity space is constructed and used in the evaluation of the numerical fluxes across cell interfaces currently lbm is mainly used for low mach number nearly incompressible flow simulation for the gas kinetic scheme the application is focusing on the high speed compressible flows in this paper we are going to compare both schemes in the isothermal low mach number flow simulations the methodology for developing both schemes will be clarified through the introduction of operator splitting boltzmann model and operator averaging boltzmann model from the operator splitting boltzmann model the error rooted in many kinetic schemes which are based on the decoupling of particle transport and collision can be easily understood as to the test case we choose to use the 2d cavity flow since it is one of the most extensively studied cases detailed simulation results with different reynolds numbers as well as the benchmark solutions are presented

pattern dynamics in spatiotemporal chaos pattern selection diffusion of defect and pattern competition intermettency abstract a class of coupled map lattices is investigated as a model for the spatiotemporal chaos as the parameters are changed phase transitions among the patterns are observed pattern selection through the chaotic motion of domain is observed chaos is suppressed by the selection selection of the zigzag pattern is studied where localized defects which separate the two anti phase domains are found they change chaotically in time and move randomly in space the diffusion of defect is related with its chaotic motion collapse of zigzag pattern and the defect turbulence are studied in connection with the crisis in high dimensional dynamical systems pattern selection with longer wavelengths is found in the stronger coupling regime transition from pattern selection phase to fully developed turbulence occurs via the intermittency selective flicker like noise and the pareto distribution of lifetime of pattern are observed at the intermittency regime for the quantitative characterization of each phase pattern distributions pattern transition matrix static and dynamical entropies lifetime distribution of pattern spatiotemporal power spectra and lyapunov spectra are calculated phase transition from the pattern selection to fully developed turbulence is investigated with the use of these quantifiers where some order parameters and critical indeces are introduced

a comparative study of lattice boltzmann models for incompressible flow abstract for incompressible flow a comparative study on the four lattice boltzmann lb models the standard model the he luo model guo s model and the present model is performed theoretically the macroscopic equations derived from the involved lb models are compared by the chapman enskog analysis then the analytical framework proposed in m junk s work is applied to investigate the finite difference stencils and the equivalent moment systems pertaining to the concerned lb models conclusions are drawn from the theoretical derivations that the truncated error terms which differ among the concerned lb models have effects on the accuracy of the modeled deviatoric stress moreover the cavity flow in two dimensions is adopted as a benchmark test to confirm the theoretical demonstrations the resulting velocity fields from the present model are more in line with the reference solutions in the region of high deviatoric stress than other three lb models which is consistent with the theoretical expectations and is further confirmed by the comparisons of the truncation error terms in addition we also conclude from the numerical tests that the present model has the advantage of better convergence efficiency but suffers from the worse stability

asymptotic analysis of lattice boltzmann boundary conditions in this article we use a general method for the analysis of finite difference schemes to investigate lattice boltzmann algorithms for navier stokes problems with dirichlet boundary conditions several link based boundary conditions for commonly used lattice boltzmann bgk models are considered with our method the accuracy of the algorithms can be exactly predicted moreover the analytical results can be used to construct new algorithms which is demonstrated with a corrected bounce back rule that requires only local evaluations but still yields second order accuracy for the velocity the analysis is applicable to general geometries and instationary flows

finding a low rank basis in a matrix subspace for a given matrix subspace how can we find a basis that consists of low rank matrices this is a generalization of the sparse vector problem it turns out that when the subspace is spanned by rank 1 matrices the matrices can be obtained by the tensor cp decomposition for the higher rank case the situation is not as straightforward in this work we present an algorithm based on a greedy process applicable to higher rank problems our algorithm first estimates the minimum rank by applying soft singular value thresholding to a nuclear norm relaxation and then computes a matrix with that rank using the method of alternating projections we provide local convergence results and compare our algorithm with several alternative approaches applications include data compression beyond the classical truncated svd computing accurate eigenvectors of a near multiple eigenvalue image separation and graph laplacian eigenproblems

discrete velocity models of the boltzmann equation a survey on the mathematical aspects of the theory discrete velocity models of the boltzmann equation are of considerable conceptual interest in the kinetic theory of gases and at the same time a fascinating mathematical subject the last decade has seen a lot of work on this subject which is reviewed here we put an emphasis on mathematical aspects in particular the global existence question qualitative properties of solutions explicit solutions and the fluid dynamical limit the mechanical background is discussed more concisely in part i numerous contributions concerning other questions are outlined we try to give a complete reference list up to and including 1985

an alternative lattice boltzmann model for three dimensional incompressible flow abstract in this work an alternative lattice boltzmann lb model for three dimensional 3d incompressible flow is proposed the equilibrium distribution function edf of the present model is directly derived in accordance with the incompressibility conditions by applying the hermite expansion moreover an alternative formula for pressure computation is designed from the second order moment of the distribution function the present 3d lb model inherits the advantageous features of guo s lb model the density is a constant the fluid pressure is independent of density and the navier stokes n s equations for incompressible flow can be derived two benchmark tests flow over a backward facing step and the lid driven cavity flow are applied to validate the present model accurate results for these tests are obtained with the present model and further comparisons with the previous lb models the standard lb model the he luo model and guo s lb model demonstrate that the present model provides better accuracy in the region of high deviatoric stress and such advantage is further enhanced by using the d3q27 lattice

stable core and chaos control in random boolean networks a variant of a simple method for chaos control is applied to achieve control in random boolean networks rbns it is shown that a rbn in the chaotic phase can be forced to behave periodically if a certain quenched fraction of the automata is given a fixed state the system variables every t time steps an analytic relationship between and t is derived using the size of the stable core as an order parameter

invariant measurement of strange sets in terms of cycles we argue that extraction of unstable cycles and their eigenvalues is not only experimentally feasible but is also a theoretically optimal measurement of the invariant properties of a dynamical system

rival approaches to mathematical modelling in immunology in order to formulate quantitatively correct mathematical models of the immune system one requires an understanding of immune processes and familiarity with a range of mathematical techniques selection of an appropriate model requires a number of decisions to be made including a choice of the modelling objectives strategies and techniques and the types of model considered as candidate models the authors adopt a multidisciplinary perspective

penalized model based clustering with application to variable selection variable selection in clustering analysis is both challenging and important in the context of model based clustering analysis with a common diagonal covariance matrix which is especially suitable for high dimension low sample size settings we propose a penalized likelihood approach with an l1 penalty function automatically realizing variable selection via thresholding and delivering a sparse solution we derive an em algorithm to fit our proposed model and propose a modified bic as a model selection criterion to choose the number of components and the penalization parameter a simulation study and an application to gene function prediction with gene expression profiles demonstrate the utility of our method

truncation error analysis of lattice boltzmann methods a truncation error analysis is performed for models based on the lattice boltzmann lb equation this analysis involves two steps the recursive application of the lb equation and a taylor series expansion unlike previous analytical studies of lb methods the present work does not assume an asymptotic relationship between the temporal and spatial discretization parameters or between the probability distribution function f and its equilibrium distribution feq effective finite difference stencils are derived for both the distribution function and the primitive variables i e density and velocity the governing partial differential equations are also recovered the associated truncation errors are derived and the results are validated by numerical simulation of analytic flows analysis of the truncation errors elucidates the roles of the kinetic theory relaxation parameter and the discretization parameters x and t the effects of initial and boundary conditions are also addressed and are shown to significantly affect the overall accuracy of the method

simulation of cavity flow by the lattice boltzmann method a detailed analysis is presented to demonstrate the capabilities of the lattice boltzmann method thorough comparisons with other numerical solutions for the two dimensional driven cavity flow show that the lattice boltzmann method gives accurate results over a wide range of reynolds numbers studies of errors and convergence rates are carried out compressibility effects are quantified for different maximum velocities and parameter ranges are found for stable simulations the paper s objective is to stimulate further work using this relatively new approach for applied engineering problems in transport phenomena utilizing parallel computers

chapman enskog analysis for finite volume formulation of lattice boltzmann equation the classical chapman enskog expansion is performed for the recently proposed finite volume formulation of lattice boltzmann equation lbe method d v patil k n lakshmisha finite volume tvd formulation of lattice boltzmann simulation on unstructured mesh j comput phys 228 2009 5262 5279 first a modified partial differential equation is derived from a numerical approximation of the discrete boltzmann equation then the multi scale small parameter expansion is followed to recover the continuity and the navier stokes ns equations with additional error terms the expression for apparent value of the kinematic viscosity is derived for finite volume formulation under certain assumptions the attenuation of a shear wave taylor green vortex flow and driven channel flow are studied to analyze the apparent viscosity relation

numerical schemes for hyperbolic conservation laws with stiff relaxation terms hyperbolic systems often have relaxation terms that give them a partially conservative form and that lead to a long time behavior governed by reduced systems that are parabolic in nature in this article it is shown by asymptotic analysis and numerical examples that semidiscrete high resolution methods for hyperbolic conservation laws fail to capture this asymptotic behavior unless the small relaxation rate is resolved by a fine spatial grid we introduce a modification of higher order godunov methods that possesses the correct asymptotic behavior allowing the use of coarse grids large cell peclet numbers the idea is to build into the numerical scheme the asymptotic balances that lead to this behavior numerical experiments on 2 2 systems verify our analysis

derivation of the lattice boltzmann method by means of the discrete ordinate method for the boltzmann equation in this note we show that the lattice boltzmann bgk method in common use can be derived from the boltzmann equation itself according to the present derivation it is apparent that the equation thus derived can be reduced to the navier stokes equation at a limit of small knudsen number since the equation thus derived is the boltzmann equation itself hence the present derivation method gives us a simple and flexible recipe to construct the lattice boltzmann method and makes it easy to construct an extended lattice boltzmann method in various ways 15 refs 1 fig

relaxation scheme for a lattice boltzmann type discrete velocity model and numerical navier stokes limit a discrete velocity model based on a lattice boltzmann approximation is considered in the low mach number limit a numerical scheme for this model working uniformly in the incompressible navier stokes limit is constructed the scheme is induced by the asymptotic analysis of the navier stokes limit and works uniformly for all ranges of mean free paths in the limit the scheme reduces to an explicit finite difference scheme for the incompressible navier stokes equation the chorin projection method with mac grid numerical results are presented and the uniform convergence of the scheme is established numerically

an algorithm to compute a sparse basis of the null space leta be a realm n matrix with full row rankm in many algorithms in engineering and science such as the force method in structural analysis the dual variable method for the navier stokes equations or more generally null space methods in quadratic programming it is necessary to compute a basis matrixb for the null space ofa hereb isn r r n m of rankr withab 0 in many instancesa is large and sparse and often banded the purpose of this paper is to describe and test a variation of a method originally suggested by topcu and called the turnback algorithm for computing a banded basis matrixb two implementations of the algorithm are given one using gaussian elimination and the other using orthogonal factorization by givens rotations the fortran software was executed on an ibm 3081 computer with an fps 164 attached array processor at the triangle universities computing center and on a cyber 205 vector computer test results on a variety of structural analysis problems including two and three dimensional frames plane stress plate bending and mixed finite element problems are discussed these results indicate that both implementations of the algorithm yielded a well conditioned banded basis matrixb whena is well conditioned however the orthogonal implementation yielded a better conditionedb for large illconditioned problems

kinetic schemes in the case of low mach numbers originally kinetic schemes have been used as numerical methods to solve the system of compressible euler equations in gas dynamics the main idea in the approach is to construct the numerical flux function based on a microscopical description of the gas in this article the schemes are investigated in the case of isentropic euler equations and low mach numbers expanding the microscopical velocity distribution naturally leads to new kinetic schemes with strong resemblance to lattice boltzmann methods by adjusting the parameters of the kinetic scheme the numerical viscosity can be used to reproduce a given physical viscosity in this way a finite difference solver for the incompressible navier stokes equation is obtained its close relation to the lattice boltzmann approach is highlighted

the spectral norm of random inner product kernel matrices we study an inner product kernel random matrix model whose empirical spectral distribution was shown by xiuyuan cheng and amit singer to converge to a deterministic measure in the large n and p limit we provide an interpretation of this limit measure as the additive free convolution of a semicircle law and a marchenko pastur law by comparing the tracial moments of this matrix to an additive deformation of a wigner matrix we establish that for odd kernel functions the spectral norm of this matrix converges almost surely to the edge of the limiting spectrum our study is motivated by the analysis of a covariance thresholding procedure for the statistical detection and estimation of sparse principal components and our results characterize the limit of the largest eigenvalue of the thresholded sample covariance matrix in the null setting

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

controlling transient chaos the method of ott grebogi and yorke 1990 is extended to control transient chaos the controlled signal then exhibits a periodic behaviour which is qualitatively different from that of the actual attractor the time needed to achieve control is shown to be constant and to lie in the order of magnitude of the transient lifetime the number of controlled trajectories however decreases the maximum perturbation according to a power law applicability to experimental situations and comparison with permanent chaotic cases are discussed

controlling chaotic dynamical systems abstract we describe a method that converts the motion on a chaotic attractor to a desired attracting time periodic motion by making only small time dependent perturbations of a control parameter the time periodic motion results from the stabilization of one of the infinite number of previously unstable periodic orbits embedded in the attractor the present paper extends that of ott grebogi and yorke phys rev lett 64 1990 1196 allowing for a more general choice of the feedback matrix and implementation to higher dimensional systems the method is illustrated by an application to the control of a periodically impulsively kicked dissipative mechanical system with two degrees of freedom resulting in a four dimensional map the double rotor map a key issue addressed is that of the dependence of the average time to achieve control on the size of the perturbations and on the choice of the feedback matrix

minimax estimation via wavelet shrinkage we attempt to recover an unknown function from noisy sampled data using orthonormal bases of compactly supported wavelets we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients the shrinkage can be tuned to be nearly minimax over any member of a wide range of triebel and besov type smoothness constraints and asymptotically minimax over besov bodies with p q linear estimates cannot achieve even the minimax rates over triebel and besov classes with p 2 so the method can significantly outperform every linear method e g kernel smoothing spline sieve in a minimax sense variants of our method based on simple threshold nonlinear estimators are nearly minimax our method possesses the interpretation of spatial adaptivity it reconstructs using a kernel which may vary in shape and bandwidth from point to point depending on the data least favorable distributions for certain of the triebel and besov scales generate objects with sparse wavelet transforms many real objects have similarly sparse transforms which suggests that these minimax results are relevant for practical problems sequels to this paper which was first drafted in november 1990 discuss practical implementation spatial adaptation properties universal near minimaxity and applications to inverse problems

some simple chaotic flows a systematic examination of general three dimensional autonomous ordinary differential equations with quadratic nonlinearities has uncovered 19 distinct simple examples of chaotic flows with either five terms and two nonlinearities or six terms and one nonlinearity the properties of these systems are described including their critical points lyapunov exponents and fractal dimensions

cooperative catalysis and chemical chaos a chemical model for the lorenz equations abstract a chemical reaction mechanism is presented that gives the lorenz equations after the axes are translated so that the range of all the variables now representing concentrations is in the positive quadrant the resulting set of nonlinear equations can be interpreted in terms of straightforward chemical reactions the mechanism requires some species to be in excess constant concentration and for some steps to be a slow rate determining reaction followed by a very fast instantaneous reaction the mechanism does not involve autocatalysis but rather cooperative catalysis where the reactants mutually catalyze appropriate reactions

localized functional principal component analysis we propose localized functional principal component analysis lfpca looking for orthogonal basis functions with localized support regions that explain most of the variability of a random process the lfpca is formulated as a convex optimization problem through a novel deflated fantope localization method and is implemented through an efficient algorithm to obtain the global optimum we prove that the proposed lfpca converges to the original functional principal component analysis fpca when the tuning parameters are chosen appropriately simulation shows that the proposed lfpca with tuning parameters chosen by cross validation can almost perfectly recover the true eigenfunctions and significantly improve the estimation accuracy when the eigenfunctions are truly supported on some subdomains in the scenario that the original eigenfunctions are not localized the proposed lfpca also serves as a nice tool in finding orthogonal basis functions that balance between interpretability and the capability of exp

on the finite difference based lattice boltzmann method in curvilinear coordinates the lattice boltzmann method is a microscopic based approach for solving the fluid flow problems at the macroscopic scales the presently popular method uses regularly spaced lattices and cannot handle curved boundaries with desirable flexibility to circumvent such difficulties a finite difference based lattice boltzmann method fdlbm in curvilinear coordinates is explored using body fitted coordinates with non uniform grids several test cases including the impulsively started cylindrical couette flow steady state cylindrical couette flow steady flow over flat plates and steady flow over a circular cylinder are used to examine various issues related to the fdlbm the effect of boundary conditions for the distribution functions on the solution the merits between second order central difference and upwind schemes for advection terms and the effect of the reynolds number are investigated favorable results are obtained using fdlbm in curvilinear coordinates indicating that the method is potentially capable of solving finite reynolds number flow problems in complex geometries

model complexity reduction of chemical reaction networks using mixed integer quadratic programming the model complexity reduction problem of large chemical reaction networks under isobaric and isothermal conditions is considered with a given detailed kinetic mechanism and measured data of the key species over a finite time horizon the complexity reduction is formulated in the form of a mixed integer quadratic optimization problem where the objective function is derived from the parametric sensitivity matrix the proposed method sequentially eliminates reactions from the mechanism and simultaneously tunes the remaining parameters until the pre specified tolerance limit in the species concentration space is reached the computational efficiency and numerical stability of the optimization are improved by a pre reduction step followed by suitable scaling and initial conditioning of the hessian involved the proposed complexity reduction method is illustrated using three well known case studies taken from reaction kinetics literature 2012 elsevier ltd all rights reserved

on combinatorial testing problems we study a class of hypothesis testing problems in which upon observing the realization of an n dimensional gaussian vector one has to decide whether the vector was drawn from a standard normal distribution or alternatively whether there is a subset of the components belonging to a certain given class of sets whose elements have been contaminated that is have a mean different from zero we establish some general conditions under which testing is possible and others under which testing is hopeless with a small risk the combinatorial and geometric structure of the class of sets is shown to play a crucial role the bounds are illustrated on various examples

sparse principal component analysis based on least trimmed squares sparse principal component analysis pca is used to obtain stable and interpretable principal components pcs from high dimensional data a robust sparse pca method is proposed to handle potentia

domain synchronization of a cyclic lotka volterra reaction diffusion mechanism abstract the domain synchronization of a cyclic lotka volterra reaction diffusion mechanism is studied in a dilute solution this is done by performing a number of molecular dynamics simulations and by setting up a mathematical model that can account for the dynamics seen in the simulations the simulations show that below a certain critical length the domain is synchronized but for larger domains the system quickly becomes unsynchronized a mathematical model based on the law of mass action and fick s law will not reveal this critical length however if an ad hoc gaussian noise term is added to the model it resemble the synchronization phenomena seen in the molecular dynamics simulations

the effect of clustering on the lotka volterra model abstract using the lotka volterra model as a reference for an oscillating system the effects of nonideality excluded volume and clustering are investigated it is found that these two different types of interaction have opposite effects on the stability of the system and that for a range of parameters stable limit cycles can result even limited clustering such as dimer formation is sufficient to give rise to limit cycle behavior lattice models are also investigated where clustering is treated using the bethe approximation

approximate reduction of heterogenous nonlinear models with differential hulls we present a model reduction technique for a class of nonlinear ordinary differential equation ode models of heterogeneous systems where heterogeneity is expressed in terms of classes of state variables having the same dynamics structurally but which are characterized by distinct parameters to this end we first build a system of differential inequalities that provides lower and upper bounds for each original state variable but such that it is homogeneous in its parameters then we use two methods for exact aggregation of odes to exploit this homogeneity yielding a smaller model of size independent of the number of heterogeneous classes we apply this technique to two case studies a multiclass queuing network and a model of epidemics spread

higher criticism for detecting sparse heterogeneous mixtures higher criticism or second level significance testing is a multiplecomparisons concept mentioned in passing by tukey it concerns a situation where there are many independent tests of significance and one is interested in rejecting the joint null hypothesis tukey suggested comparing the fraction of observed significances at a given level to the expected fraction under the joint null in fact he suggested standardizing the difference of the two quantities and forming a z score the resulting z score tests the significance of the body of significance tests we consider a generalization where we maximize this z score over a range of significance levels 0 0 we are able to show that the resulting higher criticism statistic is effective at resolving a very subtle testing problem testing whether n normal means are all zero versus the alternative that a small fraction is nonzero the subtlety of this sparse normal means testing problem can be seen from work of ingster and jin who studied such problems in great detail in their studies they identified an interesting range of cases where the small fraction of nonzero means is so small that the alternative hypothesis exhibits little noticeable effect on the distribution of the p values either for the bulk of the tests or for the few most highly significant tests in this range when the amplitude of nonzero means is calibrated with the fraction of nonzero means the likelihood ratio test for a precisely specified alternative would still succeed in separating the two hypotheses we show that the higher criticism is successful throughout the same region of amplitude sparsity where the likelihood ratio test

optimal estimation and rank detection for sparse spiked covariance matrices this paper considers a sparse spiked covariance matrix model in the high dimensional setting and studies the minimax estimation of the covariance matrix and the principal subspace as well as the minimax rank detection the optimal rate of convergence for estimating the spiked covariance matrix under the spectral norm is established which requires significantly different techniques from those for estimating other structured covariance matrices such as bandable or sparse covariance matrices we also establish the minimax rate under the spectral norm for estimating the principal subspace the primary object of interest in principal component analysis in addition the optimal rate for the rank detection boundary is obtained this result also resolves the gap in a recent paper by berthet and rigollet ann stat 41 4 1780 1815 2013 where the special case of rank one is considered

blind source separation by sparse decomposition in a signal dictionary the blind source separation problem is to extract the underlying source signals from a set of linear mixtures where the mixing matrix is unknown this situation is common in acoustics radio medical signal and image processing hyperspectral imaging and other areas we suggest a two stage separation process a priori selection of a possibly overcomplete signal dictionary for instance a wavelet frame or a learned dictionary in which the sources are assumed to be sparsely representable followed by unmixing the sources by exploiting the their sparse representability we consider the general case of more sources than mixtures but also derive a more efficient algorithm in the case of a nonovercomplete dictionary and an equal numbers of sources and mixtures experiments with artificial signals and musical sounds demonstrate significantly better separation than other known techniques

higher criticism thresholding optimal feature selection when useful features are rare and weak in important application fields today genomics and proteomics are examples selecting a small subset of useful features is crucial for success of linear classification analysis we study feature selection by thresholding of feature z scores and introduce a principle of threshold selection based on the notion of higher criticism hc for i 1 2 p let i denote the two sided p value associated with the ith feature z score and i denote the ith order statistic of the collection of p values the hc threshold is the absolute z score corresponding to the p value maximizing the hc objective i p i i p 1 i p we consider a rare weak rw feature model where the fraction of useful features is small and the useful features are each too weak to be of much use on their own hc thresholding hct has interesting behavior in this setting with an intimate link between maximizing the hc objective and minimizing the error rate of the designed classifier and very different behavior from popular threshold selection procedures such as false discovery rate thresholding fdrt in the most challenging rw settings hct uses an unconventionally low threshold this keeps the missed feature detection rate under better control than fdrt and yields a classifier with improved misclassification performance replacing cross validated threshold selection in the popular shrunken centroid classifier with the computationally less expensive and simpler hct reduces the variance of the selected threshold and the error rate of the constructed classifier results on standard real datasets and in asymptotic theory confirm the advantages of hct

minimax sparse principal subspace estimation in high dimensions we study sparse principal components analysis in high dimensions where p the number of variables can be much larger than n the number of observations and analyze the problem of estimating the subspace spanned by the principal eigenvectors of the population covariance matrix we prove optimal non asymptotic lower and upper bounds on the minimax subspace estimation error under two different but related notions of lq subspace sparsity for 0 q 1 our upper bounds apply to general classes of covariance matrices and they show that lq constrained estimates can achieve optimal minimax rates without restrictive spiked covariance conditions 1 introduction principal components analysis pca was introduced in the early 20th century pearson 1901 hotelling 1933 and is arguably the most well known and widely used technique for dimension reduction it is part of the mainstream statistical repertoire and is routinely used in numerous and diverse areas of application however contemporary applications often involve much higher dimensional data than envisioned by the early developers of pca in such high dimensional situations where the number of variables p is of the same order or much larger than the number of observations n serious difficulties emerge standard pca can produce inconsistent estimates of the principal directions of variation and lead to unreliable conclusions johnstone and lu 2009 paul 2007 nadler 2008 the principal directions of variation correspond to the eigenvectors of the covariance matrix and in high dimensions consistent estimation of the eigenvectors is generally not possible without additional assumptions about the covariance matrix or its eigenstructure much of the recent development in pca has focused on methodology that applies the concept of sparsity to eigenvector estimation some examples include jolliffe trendafilov and uddin

controlling the false discovery rate a practical and powerful approach to multiple testing summary the common approach to the multiplicity problem calls for controlling the familywise error rate fwer this approach though has faults and we point out a few a different approach to problems of multiple significance testing is presented it calls for controlling the expected proportion of falsely rejected hypotheses the false discovery rate this error rate is equivalent to the fwer when all hypotheses are true but is smaller otherwise therefore in problems where the control of the false discovery rate rather than that of the fwer is desired there is potential for a gain in power a simple sequential bonferronitype procedure is proved to control the false discovery rate for independent test statistics and a simulation study shows that the gain in power is substantial the use of the new procedure and the appropriateness of the criterion are illustrated with examples

uniformly accurate schemes for hyperbolic systems with relaxation we develop high resolution shock capturing numerical schemes for hyperbolic systems with relaxation in such systems the relaxation time may vary from order 1 to much less than unity when the relaxation time is small the relaxation term becomes very strong and highly stiff and underresolved numerical schemes may produce spurious results usually one cannot decouple the problem into separate regimes and handle different regimes with different methods thus it is important to have a scheme that works uniformly with respect to the relaxation time using the broadwell model of the nonlinear boltzmann equation we develop a second order scheme that works effectively with a fixed spatial and temporal discretization for all ranges of the mean free path formal uniform consistency proof for a first order scheme and numerical convergence proof for the second order scheme are also presented we also make numerical comparisons of the new scheme with some other schemes this study is motivated by the reentry problem in hypersonic computations

computational barriers in minimax submatrix detection this paper studies the minimax detection of a small submatrix of elevated mean in a large matrix contaminated by additive gaussian noise to investigate the tradeoff between statistical performance and computational cost from a complexity theoretic perspective we consider a sequence of discretized models which are asymptotically equivalent to the gaussian model under the hypothesis that the planted clique detection problem cannot be solved in randomized polynomial time when the clique size is of smaller order than the square root of the graph size the following phase transition phenomenon is established when the size of the large matrix p to infty if the submatrix size k theta p alpha for any alpha in 0 2 3 computational complexity constraints can incur a severe penalty on the statistical performance in the sense that any randomized polynomial time test is minimax suboptimal by a polynomial factor in p if k theta p alpha for any alpha in 2 3 1 minimax optimal detection can be attained within constant factors in linear time using schatten norm loss as a representative example we show that the hardness of attaining the minimax estimation rate can crucially depend on the loss function implications on the hardness of support recovery are also obtained