a simple two sample test in high dimensions based on l2 norm testing the equality of two means is a fundamental inference problem for high dimensional data the hotelling s t2 test either performs poorly or becomes inapplicable several modifications have b

covariance regularization by thresholding this paper considers regularizing a covariance matrix of p variables estimated from n observations by hard thresholding we show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense the variables are gaussian or sub gaussian and log p n 0 and obtain explicit rates the results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity we discuss an intuitive resampling scheme for threshold selection and prove a general cross validation result that justifies this approach we also compare thresholding to other covariance estimators in simulations and on an example from climate data 1 introduction estimation of covariance matrices is important in a number of areas of statistical analysis including dimension reduction by principal component analysis pca classification by linear or quadratic discriminant analysis lda and qda establishing independence and conditional independence relations in the context of graphical models and setting confidence intervals on linear functions of the means of the components in recent years many application areas where these tools are used have been dealing with very high dimensional datasets and sample sizes can be very small relative to dimension examples include genetic data brain imaging spectroscopic imaging climate data and many others it is well known by now that the empirical covariance matrix for samples of size n from a p variate gaussian distribution np p is not a good estimator of the population covariance if p is large many results in random matrix theory illustrate this from the classical mary law 29 to the more recent work of johnstone and his students on the theory of the largest eigenvalues 12 23 30 and associated eigenvectors 24 however with the exception of a method for estimating the covariance spectrum 11 these probabilistic results do not offer alternatives to the sample covariance matrix alternative estimators for large covariance matrices have therefore attracted a lot of attention recently two broad classes of covariance estimators have emerged those that rely on a natural ordering among variables and assume that variables

convergence analysis and numerical implementation of a second order numerical scheme for the three dimensional phase field crystal equation abstract in this paper we analyze and implement a second order in time numerical scheme for the three dimensional phase field crystal pfc equation the numerical scheme was proposed in hu et al 2009 with the unique solvability and unconditional energy stability established however its convergence analysis remains open we present a detailed convergence analysis in this article in which the maximum norm estimate of the numerical solution over grid points plays an essential role moreover we outline the detailed multigrid method to solve the highly nonlinear numerical scheme over a cubic domain and various three dimensional numerical results are presented including the numerical convergence test complexity test of the multigrid solver and the polycrystal growth simulation

multivariate location estimation using extension of r estimates through u statistics type approach we consider a class of u statistics type estimates for multivariate location the estimates extend some r estimates to multivariate data in particular the class of estimates includes the multivariate median considered by gini and galvani 1929 and haldane 1948 and a multivariate extension of the well known hodges lehmann 1963 estimate we explore large sample behavior of these estimates by deriving a bahadur type representation for them in the process of developing these asymptotic results we observe some interesting phenomena that closely resemble the famous shrinkage phenomenon observed by stein 1956 in high dimensions interestingly the phenomena that we observe here occur even in dimension d 2

accurate navier stokes investigation of transitional and turbulent flows in a circular pipe a new fast accurate and roundoff error robust numerical technique for integrating unsteady incompressible navier stokes equations in cylindrical coordinates is presented the algorithm is based on a special change of dependent variables which avoids the singularity problem and provides high accuracy and computational efficiency accuracy and stability of the method are thoroughly tested for the model problem of transitional and turbulent flows in an infinite circular pipe verification of the algorithm includes two issues first spectral characteristics of the hagen poiseuille flow stability problem are compared with those of the discrete linearized navier stokes operator secondly the results of direct navier stokes simulation of all stages of laminar turbulent transition in a circular pipe at reynolds number of 4000 are presented time evolution of finite amplitude disturbances of laminar flow was calculated until the statistically stationary turbulent flow regime was established in addition to common statistical analysis the possibility of turbulence description by means of velocity fields having certain symmetries is examined thus the algorithm presented seems to be a ready to use robust tool for accurate investigation and further parametric studies of both transition mechanisms and fully developed turbulent flow regimes

numerical approximations to a new phase field model for two phase flows of complex fluids abstract we derive a new phase field theory for immiscible mixtures of nematic liquid crystals and viscous fluids using the variational principle coupled with the generalized onsager principle a novel phase transition mechanism is implemented to couple the nematic liquid crystal phase with the viscous fluid phase to arrive at the dissipative hydrodynamic model for incompressible fluid mixtures through a delicate explicit implicit numerical discretization we develop a decoupled linear scheme for a simplified version of the phase field model as well as a coupled nonlinear scheme for the full model both schemes are shown as unconditionally energy stable with consistent discrete dissipative energy laws several numerical examples are presented to show the effectiveness of the new model and the new numerical schemes developed for it

retrieving topological information for phase field models the phase field approach has become a popular tool in modeling interface motion microstructure evolution and more recently the shape transformation of vesicle membranes under elastic bending energy while it is advantageous to employ phase field models in numerical simu lations to automatically handle topological changes to the microstructures or the configurations of vesicle membranes detecting topological events may also become important for many applications such as those in the simulation of blood cells motivated by such considerations a new quantity is formulated to retrieve some topological information based on the phase field formulation and to capture the occurrence of topological events it can also be used as a control method to avoid unphys ical changes of topology due to the numerical methods should it become necessary for particular practical applications through numerical experiments we demonstrate the effectiveness and the robustness of the new quantity in detecting the topology of fluid bubbles and vesicle membranes

linear and unconditionally energy stable schemes for the binary fluid surfactant phase field model abstract in this paper we consider the numerical solution of a binary fluid surfactant phase field model in which the free energy contains a nonlinear coupling entropy a ginzburg landau double well potential and a logarithmic flory huggins potential the resulting system consists of two nonlinearly coupled cahn hilliard type equations we develop a first and a second order time stepping schemes for this system using the invariant energy quadratization approach in particular the system is transformed into an equivalent one by introducing appropriate auxiliary variables and all nonlinear terms are then treated semi explicitly both schemes are linear and lead to symmetric positive definite systems in space at each time step thus they can be efficiently solved we further prove that these schemes are unconditionally energy stable in the discrete sense various 2d and 3d numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes

efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model abstract in this paper we study efficient numerical schemes of the classical phase field elastic bending energy model that has been widely used to describe the shape deformation of biological lipid vesicles in which the free energy of the system consists of an elastic bending energy a surface area constraint and a volume constraint one major challenge in solving such model numerically is how to design appropriate temporal discretizations in order to preserve energy stability with large time step sizes at the semi discrete level we develop a first order and a second order time stepping scheme for this highly nonlinear and stiff parabolic pde system based on the invariant energy quadratization approach in particular the resulted semi discretizations lead to linear systems in space with symmetric positive definite operators at each time step thus can be efficiently solved in addition the proposed schemes are rigorously proved to be unconditionally energy stable various numerical experiments in 2d and 3d are presented to demonstrate the stability and accuracy of the proposed schemes

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

estimation of error variance via ridge regression summarywe propose a novel estimator of error variance and establish its asymptotic properties based on ridge regression and random matrix theory the proposed estimator is valid under both low and high dimensional models and performs well not only in nonsparse cases but also in sparse ones the finite sample performance of the proposed method is assessed through an intensive numerical study which indicates that the method is promising compared with its competitors in many interesting scenarios

decoupled energy stable schemes for phase field models of two phase incompressible flows in this paper we construct two classes based on stabilization and convex splitting of decoupled unconditionally energy stable schemes for cahn hilliard phase field models of two phase incompressible flows at each time step these schemes require solving only a sequence of elliptic equations including a pressure poisson equation furthermore all of these elliptic equations are linear for the schemes based on stabilization making them the first to the best of the authors knowledge totally decoupled linear unconditionally energy stable schemes for phase field models of two phase incompressible flows thus the schemes constructed in this paper are very efficient and easy to implement

on the error estimates for the rotational pressure correction projection methods in this paper we study the rotational form of the pressure correction method that was proposed by timmermans minev and van de vosse we show that the rotational form of the algorithm provides better accuracy in terms of the h1 norm of the velocity and of the l2 norm of the pressure than the standard form

an efficient two grid method for the cahn hilliard equation with the concentration dependent mobility and the logarithmic flory huggins bulk potential abstract in this paper we consider the two grid method for solving cahn hilliard equation with the concentration dependent mobility and the logarithmic flory huggins bulk potential this two grid method consists of two steps first step solve the cahn hilliard equation with full implicit mixed finite element method and newton iteration method on a coarse grid second step solve two linear equations on a fine grid the stability and convergence of the proposed scheme are established finally some numerical experiments are presented

tests for high dimensional data based on means spatial signs and spatial ranks tests based on sample mean vectors and sample spatial signs have been studied in the recent literature for high dimensional data with the dimension larger than the sample size for suitable sequences of alternatives we show that the powers of the mean based tests and the tests based on spatial signs and ranks tend to be same as the data dimension grows to infinity for any sample size when the coordinate variables satisfy appropriate mixing conditions further their limiting powers do not depend on the heaviness of the tails of the distributions this is in striking contrast to the asymptotic results obtained in the classical multivariate setup on the other hand we show that in the presence of stronger dependence among the coordinate variables the spatial sign and rank based tests for high dimensional data can be asymptotically more powerful than the mean based tests if in addition to the data dimension the sample size also grows to infinity the sizes of some mean based tests for high dimensional data studied in the recent literature are observed to be significantly different from their nominal levels this is due to the inadequacy of the asymptotic approximations used for the distributions of those test statistics however our asymptotic approximations for the tests based on spatial signs and ranks are observed to work well when the tests are applied on a variety of simulated and real datasets

an efficient spectral projection method for the navier stokes equations in cylindrical geometries i axisymmetric cases abstract an efficient and accurate numerical scheme is presented for the axisymmetric navier stokes equations in primitive variables in a cylinder the scheme is based on a new spectral galerkin approximation for the space variables and a second order projection scheme for the time variable the new spectral projection scheme is implemented to simulate the unsteady incompressible axisymmetric flow with a singular boundary condition which is approximated to within a desired accuracy by using a smooth boundary condition a sensible comparison is made with a standard second order in time and space finite difference scheme based on a stream function vorticity formulation and with available experimental data the numerical results indicate that both schemes produce very reliable results and that despite the singular boundary condition the spectral projection scheme is still more accurate in terms of a fixed number of unknowns and more efficient in terms of cpu time required for resolving the flow at a fixed reynolds number to within a prescribed accuracy than the finite difference scheme more importantly the spectral projection scheme can be readily extended to three dimensional nonaxisymmetric cases

decoupled energy stable schemes for phase field vesicle membrane model we consider the numerical approximations of the classical phase field vesicle membrane models proposed a decade ago in du et al 2004 6 we first reformulate the model derived from an energetic variational formulation into a form which is suitable for numerical approximation and establish the energy dissipation law then we develop a stabilized decoupled time discretization scheme for the coupled nonlinear system the scheme is unconditionally energy stable and leads to linear and decoupled elliptic equations to be solved at each time step stability analysis and ample numerical simulations are presented thereafter

energy stable predictor corrector schemes for the cahn hilliard equation abstract in this paper we construct a new class of predictor corrector time stepping schemes for the cahn hilliard equation which are linear second order accurate in time unconditionally energy stable and uniquely solvable then we present the stability and error estimates of the semi discrete numerical schemes for solving the cahn hilliard equation with general nonlinear bulk potentials the semi discrete scheme is further discretized using the compact central finite difference method several numerical examples are shown to verify the theoretical results in particular the numerical simulations show that the predictor corrector schemes reach the second order convergence rate at relatively larger time step sizes than the classical linear schemes the numerical strategies and theoretical tools developed in this article could be readily applied to study other phase field models or models that can be cast as gradient flow problems

gauge uzawa methods for incompressible flows with variable density two new gauge uzawa schemes are constructed for incompressible flows with variable density one is in the conserved form while the other is in the convective form it is shown that the first order versions of both schemes in their semi discretized form are unconditionally stable numerical experiments indicate that the first order resp second order versions of the two schemes lead to first order resp second order convergence rate for all variables and that these schemes are suitable for handling problems with large density ratios such as in the situation of air bubble rising in water

diagonal likelihood ratio test for equality of mean vectors in high dimensional data we propose a likelihood ratio test framework for testing normal mean vectors in high dimensional data under two common scenarios the one sample test and the two sample test with equal covariance matrices we derive the test statistics under the assumption that the covariance matrices follow a diagonal matrix structure in comparison with the diagonal hotelling s tests our proposed test statistics display some interesting characteristics in particular they are a summation of the log transformed squared t statistics rather than a direct summation of those components more importantly to derive the asymptotic normality of our test statistics under the null and local alternative hypotheses we do not need the requirement that the covariance matrices follow a diagonal matrix structure as a consequence our proposed test methods are very flexible and readily applicable in practice simulation studies and a real data analysis are also carried out to demonstrate the advantages of our likelihood ratio test methods

a time stepping scheme involving constant coefficient matrices for phase field simulations of two phase incompressible flows with large density ratios we present an efficient time stepping scheme for simulations of the coupled navier stokes cahn hilliard equations for the phase field approach the scheme has several attractive characteristics i it is suitable for large density ratios and numerical experiments with density ratios up to 1000 have been presented ii it involves only constant time independent coefficient matrices for all flow variables which can be pre computed during pre processing so it effectively overcomes the performance bottleneck induced by variable coefficient matrices associated with the variable density and variable viscosity iii it completely de couples the computations of the velocity pressure and the phase field function strategy for spectral element type spatial discretizations to overcome the difficulty associated with the large spatial order of the cahn hilliard equation is also discussed ample numerical simulations demonstrate that the current algorithm together with the navier stokes cahn hilliard phase field approach is an efficient and effective method for studying two phase flows involving large density ratios moving contact lines and interfacial topology changes

tests for high dimensional regression coefficients with factorial designs we propose simultaneous tests for coefficients in high dimensional linear regression models with factorial designs the proposed tests are designed for the large p small n situations where the conventional f test is no longer applicable we derive the asymptotic distribution of the proposed test statistic under the high dimensional null hypothesis and various scenarios of the alternatives which allow power evaluations we also evaluate the power of the f test for models of moderate dimension the proposed tests are employed to analyze a microarray data on yorkshire gilts to find significant gene ontology terms which are significantly associated with the thyroid hormone after accounting for the designs of the experiment

stabilized semi implicit numerical schemes for the cahn hilliard like equation with variable interfacial parameter abstract we present stabilized semi implicit schemes for the mmc tdgl equation a cahn hilliard like equation with variable interfacial parameter by introducing two stabilization terms we decompose the 4th order variable coefficient pde into two decoupled helmholtz type equations with the constant coefficients the proposed scheme is proved to be stable which means the decay of energy is preserved due to the periodic boundary condition the fast fourier transformation method is applied to accelerate the computations to improve the stability of the proposed scheme we present a nonlinear stabilized semi implicit scheme which can be simulated efficiently with fixed point iteration for the nonlinear terms some numerical experiments are carried out to simulate the phase transition to verify the total energy decay and to observe the influence of some parameters the strictly energy stable scheme saves much cpu time due to the constant coefficients with fft method without losing the accuracy it is the key novelty

stability analysis of large time stepping methods for epitaxial growth models numerical methods for solving the continuum model of the dynamics of the molecular beam epitaxy mbe require very large time simulation and therefore large time steps become necessary the main purpose of this work is to construct and analyze highly stable time discretizations which allow much larger time steps than those of a standard implicit explicit approach to this end an extra term which is consistent with the order of the time discretization is added to stabilize the numerical schemes then the stability properties of the resulting schemes are established rigorously numerical experiments are carried out to support the theoretical claims the proposed methods are also applied to simulate the mbe models with large solution times the power laws for the coarsening process are obtained and are compared with previously published results

a novel linear second order unconditionally energy stable scheme for a hydrodynamic q tensor model of liquid crystals abstract the hydrodynamic q tensor model has been used for studying flows of liquid crystals and liquid crystal polymers it can be derived from a variational approach together with the generalized onsager principle in which the total energy decreases in time in this paper we develop a novel linear second order semi discrete scheme in time to solve the governing system of equations in the model the scheme is developed following the novel energy quadratization strategy so that it is linear and unconditionally energy stable at the semi discrete level this scheme is further discretized in space using a second order finite difference method and implemented on a gpu for high performance computing the convergence rate in time is established using a mesh refinement test several numerical examples are presented to demonstrate the usefulness of the model and the effectiveness of the numerical scheme in simulating defect dynamics in flows of liquid crystals

a phase field model for the mixture of two incompressible fluids and its approximation by a fourier spectral method abstract a phase field model for the mixture of two incompressible fluids is presented in this paper the model is based on an energetic variational formulation it consists of a navier stokes system linear momentum equation coupled with a cahn hilliard equation phase field equation through an extra stress term and the transport term the extra stress represents the phase induced capillary effect for the mixture due to the surface tension a fourier spectral method for the numerical approximation of this system is proposed and analyzed numerical results illustrating the robustness and versatility of the model are presented

an efficient spectral projection method for the navier stokes equations in cylindrical geometries an efficient and accurate numerical scheme is presented for the three dimensional navier stokes equations in primitive variables in a cylinder the scheme is based on a spectral galerkin approximation for the space variables and a second order projection scheme for time the new spectral projection scheme is implemented to simulate unsteady incompressible flows in a cylinder

a high dimensional nonparametric multivariate test for mean vector this work is concerned with testing the population mean vector of nonnormal high dimensional multivariate data several tests for high dimensional mean vector based on modifying the classical hotelling t2 test have been proposed in the literature despite their usefulness they tend to have unsatisfactory power performance for heavy tailed multivariate data which frequently arise in genomics and quantitative finance this article proposes a novel high dimensional nonparametric test for the population mean vector for a general class of multivariate distributions with the aid of new tools in modern probability theory we proved that the limiting null distribution of the proposed test is normal under mild conditions when p is substantially larger than n we further study the local power of the proposed test and compare its relative efficiency with a modified hotelling t2 test for high dimensional data an interesting finding is that the newly proposed test can have even more substantial power gain with l

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

tests alternative to higher criticism for high dimensional means under sparsity and column wise dependence we consider two alternative tests to the higher criticism test of donoho and jin ann statist 32 2004 962 994 for high dimensional means under the sparsity of the nonzero means for sub gaussian distributed data with unknown column wise dependence the two alternative test statistics are constructed by first thresholding l 1 and l 2 statistics based on the sample means respectively followed by maximizing over a range of thresholding levels to make the tests adaptive to the unknown signal strength and sparsity the two alternative tests can attain the same detection boundary of the higher criticism test in ann statist 32 2004 962 994 which was established for uncorrelated gaussian data it is demonstrated that the maximal l 2 thresholding test is at least as powerful as the maximal l 1 thresholding test and both the maximal l 2 and l 1 thresholding tests are at least as powerful as the higher criticism test

spectral collocation methods and polar coordinate singularities this paper considers the numerical solution of elliptic differential equations on the unit disk using polar coordinates the disk is mapped onto a rectangle the resulting transformed problem is solved by a method related to collocation since the origin is a coordinate singularity some natural trial functions are singular there and a special technique is applied to use zero as a collocation point for poisson and helmholtz equations a fast algorithm with an operation count of o n2 log n is presented numerical results show the different stability and convergence properties of the algorithms

statistical uses of the spatial median the spatial median called the mediancentre in an earlier paper by j c gower is defined as the bivariate location measure to minimize the sum of absolute distances to observations its asymptotic efficiency relative to the sample mean vector with normal data is shown to exceed the usual univariate 2 ir 0 637 its estimating equations have an angular aspect and are used to develop angle tests which are analogues of sign tests in one dimension

efficient spectral galerkin methods iii polar and cylindrical geometries we present in this paper several extremely efficient and accurate spectral galerkin methods for second and fourth order equations in polar and cylindrical geometries these methods are based on appropriate variational formulations which incorporate naturally the pole condition s in particular the computational complexities of the chebyshev galerkin method in a disk and the chebyshev legendre galerkin method in a disk or a cylinder are quasi optimal optimal up to a logarithmic term as an indication of efficiency the cpu time for the poisson solver on a disk by our chebyshev galerkin method is only about 70 of the corresponding finite difference code in fishpack

multivariate sign based high dimensional tests for the two sample location problem abstractthis article concerns tests for the two sample location problem when data dimension is larger than the sample size existing multivariate sign based procedures are not robust against high dimensionality producing tests with type i error rates far away from nominal levels this is mainly due to the biases from estimating location parameters we propose a novel test to overcome this issue by using the leave one out idea the proposed test statistic is scalar invariant and thus is particularly useful when different components have different scales in high dimensional data asymptotic properties of the test statistic are studied compared with other existing approaches simulation studies show that the proposed method behaves well in terms of sizes and power supplementary materials for this article are available online

error analysis of a mixed finite element method for the molecular beam epitaxy model this paper investigates the error analysis of a mixed finite element method with crank nicolson time stepping for simulating the molecular beam epitaxy mbe model the fourth order differential equation of the mbe model is replaced by a system of equations consisting of one nonlinear parabolic equation and an elliptic equation then a mixed finite element method requiring only continuous elements is proposed to approximate the resulting system it is proved that the semidiscrete and fully discrete versions of the numerical schemes satisfy the nonlinearity energy stability property which is important in the numerical implementation moreover detailed analysis is provided to obtain the convergence rate numerical experiments are carried out to validate the theoretical results

periodic orbits for interval maps with sharp cusps abstract we study families of unimodal interval maps having a sharp cusp at a critical point we describe bifurcations and distribution of periodic orbits period adding for attracting periodic orbits is observed that is period k is followed by k 1 etc coexistence of attractors of successive periods as well as of an attracting fixed point and chaos occurs periodic orbits of various periods are grouped into bunches and occupy a relatively small part of the space

optimal rate convergence analysis of a second order scheme for a thin film model with slope selection abstract in this paper an energy stable second order mixed finite element scheme is proposed and analyzed for the thin film epitaxial growth model with slope selection we employ second order backward differentiation formula bdf scheme with a second order stabilized term which guarantees the long time energy stability to approximate the continuous model in terms of the convergence analysis the key difficulty to derive an optimal rate spatial estimate is associated with the appearance of the gradient operator in the nonlinear terms which may lead to a loss of optimal accuracy order to overcome this well known difficulty we make use of some auxiliary techniques over triangular elements and obtain an optimal convergence rate o h q 1 t 2 in comparison with o h q t 2 rate from a standard projection estimate furthermore we use an efficient preconditioned steepest descent psd solver for the numerical implementation a few numerical examples are presented to validate the stability and convergence

divergence free velocity fields in nonperiodic geometries the influence matrix method of enforcing incompressibility in pseudospectral simulations of fluid dynamics as described by kleiser and schumann for channel flow is generalized to other geometries a formalism of projection and matrix operators is introduced in which the influence matrix method is shown to be an application of the classic sherman morrison woodbury formula of numerical linear algebra special attention is paid to the tau correction applications to cartesian geometries illustrate the concepts and highlight the role of symmetry a coded implementation in a cylindrical geometry requiring special treatment of coordinate singularities is used to investigate properties of the influence matrix and to provide estimates of timings

numerical study of flows of two immiscible liquids at low reynolds number we review our recent results on fingering bamboo waves and drop breakup in low reynolds number flows composed of two viscous liquids under shear

comparing large covariance matrices under weak conditions on the dependence structure and its application to gene clustering comparing large covariance matrices has important applications in modern genomics where scientists are often interested in understanding whether relationships e g dependencies or co regulations among a large number of genes vary between different biological states we propose a computationally fast procedure for testing the equality of two large covariance matrices when the dimensions of the covariance matrices are much larger than the sample sizes a distinguishing feature of the new procedure is that it imposes no structural assumptions on the unknown covariance matrices hence the test is robust with respect to various complex dependence structures that frequently arise in genomics we prove that the proposed procedure is asymptotically valid under weak moment conditions as an interesting application we derive a new gene clustering algorithm which shares the same nice property of avoiding restrictive structural assumptions for high dimensional genomics data using an asthma gene expression dataset we illustrate how the new test helps compare the covariance matrices of the genes across different gene sets pathways between the disease group and the control group and how the gene clustering algorithm provides new insights on the way gene clustering patterns differ between the two groups the proposed methods have been implemented in an r package hdtest and are available on cran

accurate stable and efficient navier stokes solvers based on explicit treatment of the pressure term we present numerical schemes for the incompressible navier stokes equations based on a primitive variable formulation in which the incompressibility constraint has been replaced by a pressure poisson equation the pressure is treated explicitly in time completely decoupling the computation of the momentum and kinematic equations the result is a class of extremely efficient navier stokes solvers full time accuracy is achieved for all flow variables the key to the schemes is a neumann boundary condition for the pressure poisson equation which enforces the incompressibility condition for the velocity field irrespective of explicit or implicit time discretization of the viscous term in the momentum equation the explicit time discretization of the pressure term does not affect the time step constraint indeed we prove unconditional stability of the new formulation for the stokes equation with explicit treatment of the pressure term and first or second order implicit treatment of the viscous term systematic numerical experiments for the full navier stokes equations indicate that a second order implicit time discretization of the viscous term with the pressure and convective terms treated explicitly is stable under the standard cfl condition additionally various numerical examples are presented including both implicit and explicit time discretizations using spectral and finite difference spatial discretizations demonstrating the accuracy flexibility and efficiency of this class of schemes in particular a galerkin formulation is presented requiring only c0 elements to implement

numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method abstract the molecular beam epitaxial model is derived from the variation of a free energy that consists of either a fourth order ginzburg landau double well potential or a nonlinear logarithmic potential in terms of the gradient of a height function one challenge in solving the mbe model numerically is how to develop proper temporal discretization for the nonlinear terms in order to preserve energy stability at the time discrete level in this paper we resolve this issue by developing a first and second order time stepping scheme based on the invariant energy quadratization ieq method the novelty is that all nonlinear terms are treated semi explicitly and the resulted semi discrete equations form a linear system at each time step moreover the linear operator is symmetric positive definite and thus can be solved efficiently we then prove that all proposed schemes are unconditionally energy stable the semi discrete schemes are further discretized in space using finite difference methods and implemented on gpus for high performance computing various 2d and 3d numerical examples are presented to demonstrate stability and accuracy of the proposed schemes

an efficient moving mesh spectral method for the phase field model of two phase flows we develop in this paper a moving mesh spectral method for the phase field model of two phase flows with non periodic boundary conditions the method is based on a variational moving mesh pde for the phase function coupled with efficient semi implicit treatments for advancing the mesh function the phase function and the velocity and pressure in a decoupled manner ample numerical results are presented to demonstrate the accuracy and effectiveness of the moving mesh spectral method

entropy and rotation intervals for circle maps near saddle node bifurcations abstract consider a one parameter family of c r endomorphisms f lambda of the circle suppose that f 0 is on the boundary of an interval of phase locking and that f lambda generically unfolds f 0 so that f lambda exhibits nontrivial behavior for lambda 0 we show that the topological entropy and the width of the rotation interval of f lambda satisfy certain scaling laws as lambda searrow 0 and give estimates for each of these characteristics these scalings depend only on f 0 not on the particular family f lambda which unfolds it

efficient and accurate numerical schemes for a hydro dynamically coupled phase field diblock copolymer model in this paper we consider numerical approximations of a hydro dynamically coupled phase field diblock copolymer model in which the free energy contains a kinetic potential a gradient entropy a ginzburglandau double well potential and a long range nonlocal type potential we develop a set of second order time marching schemes for this system using the invariant energy quadratization approach for the double well potential the projection method for the navierstokes equation and a subtle implicit explicit treatment for the stress and convective term the resulting schemes are linear and lead to symmetric positive definite systems at each time step thus they can be efficiently solved we further prove that these schemes are unconditionally energy stable various numerical experiments are performed to validate the accuracy and energy stability of the proposed schemes

linear second order and unconditionally energy stable schemes for the viscous cahn hilliard equation with hyperbolic relaxation using the invariant energy quadratization method abstract in this paper we consider numerical approximations for the viscous cahn hilliard equation with hyperbolic relaxation this type of equations processes energy dissipative structure the main challenge in solving such a diffusive system numerically is how to develop high order temporal discretization for the hyperbolic and nonlinear terms allowing large time marching step while preserving the energy stability i e the energy dissipative structure at the time discrete level we resolve this issue by developing two second order time marching schemes using the recently developed invariant energy quadratization approach where all nonlinear terms are discretized semi explicitly in each time step one only needs to solve a symmetric positive definite spd linear system all the proposed schemes are rigorously proven to be unconditionally energy stable and the second order convergence in time has been verified by time step refinement tests numerically various 2d and 3d numerical simulations are presented to demonstrate the stability accuracy and efficiency of the proposed schemes

test for high dimensional regression coefficients using refitted cross validation variance estimation testing a hypothesis for high dimensional regression coefficients is of fundamental importance in the statistical theory and applications in this paper we develop a new test for the overall significance of coefficients in high dimensional linear regression models based on an estimated u statistics of order two with the aid of the martingale central limit theorem we prove that the asymptotic distributions of the proposed test are normal under two different distribution assumptions refitted cross validation rcv variance estimation is utilized to avoid the overestimation of the variance and enhance the empirical power we examine the finite sample performances of the proposed test via monte carlo simulations which show that the new test based on the rcv estimator achieves higher powers especially for the sparse cases we also demonstrate an application by an empirical analysis of a microarray data set on yorkshire gilts

multivariate sign based high dimensional tests for sphericity this article concerns tests for sphericity in cases where the data dimension is larger than the sample size the existing multivariate sign based procedure hallin paindaveine 2006 for sphericity is not robust with respect to high dimensionality producing tests with type i error rates that are much larger than the nominal levels this is mainly due to bias from estimating the location parameter we develop a correction that makes the existing test statistic robust with respect to high dimensionality and show that the proposed test statistic is asymptotically normal under elliptical distributions the proposed method allows the dimensionality to increase as the square of the sample size simulations demonstrate that it has good size and power in a wide range of settings

error analysis of full discrete invariant energy quadratization schemes for the cahn hilliard type equation abstract in this paper we present the error analysis for a fully discrete scheme of the cahn hilliard type equation along with numerical verifications the numerical schedule is developed by first transforming the cahn hilliard type equation into an equivalent form using the invariant energy quadratization ieq technique then the equivalent form is discretized by using the linear implicit crank nicolson method for the time variable and the fourier pseudo spectral method for the spatial variables the resulted full discrete scheme is linear and unconditionally energy stable which makes it easy to implement by constructing an appropriate interpolation equation the uniform boundedness of the numerical solution is obtained theoretically then we prove that the numerical solutions converge with the order o t 2 h m where t is the temporal step and h is the spatial step with m the regularity of the exact solution several numerical examples are presented to confirm the theoretical results and demonstrate the effectiveness of the presented linear scheme the numerical strategies and analytical tools in this paper could be readily applied to study other phase field models or gradient flow problems

decoupled energy stable schemes for phase field models of two phase complex fluids we consider in this paper numerical approximations of phase field models for two phase complex fluids we first reformulate the phase field models derived from an energetic variational formulation into a form which is suitable for numerical approximation and establish their energy laws then we construct two classes stabilized and convex splitting of decoupled time discretization schemes for the coupled nonlinear systems these schemes are unconditionally energy stable and lead to decoupled elliptic equations to solve at each time step furthermore these elliptic equations are linear for the stabilized version stability analysis and ample numerical simulations are presented

linear first and second order unconditionally energy stable numerical schemes for the phase field model of homopolymer blends abstract in this paper we develop a series of efficient numerical schemes to solve the phase field model for homopolymer blends the governing system is derived from the energetic variational approach of a total free energy that consists of a nonlinear logarithmic flory huggins potential and a gradient entropy with a concentration dependent de gennes type coefficient the main challenging issue to solve this kind of models numerically is about the time marching problem i e how to develop suitable temporal discretizations for the nonlinear terms in order to preserve the energy stability at the discrete level we solve this issue in this paper by developing the first and second order temporal approximation schemes based on the invariant energy quadratization method where all nonlinear terms are treated semi explicitly consequently the resulting numerical schemes lead to a symmetric positive definite linear system to be solved at each time step the unconditional energy stabilities are further proved various numerical simulations of 2d and 3d are presented to demonstrate the stability and the accuracy of the proposed schemes

at the other side of a saddle node we describe phenomena occurring just before a saddle node bifurcation for one parameter families of interval maps in particular as a parameter approaches the bifurcation value attracting periodic orbits of periodsk k 1 k 2 k 3 can appear we make a detailed study of a family of cusp shaped maps where this phenomenon occurs in a pure form

a finite difference scheme for three dimensional incompressible flows in cylindrical coordinates a finite difference scheme for the direct simulation of the incompressible time dependent three dimensional navier stokes equations in cylindrical coordinates is presented the equations in primitive variables vr v vzandp are solved by a fractional step method together with an approximate factorization technique cylindrical coordinates are singular at the axis the introduction of the radial fluxqr r vron a staggered grid simplifies the treatment of the region atr 0 the method is tested by comparing the evolution of a free vortex ring and its collision with a wall with the theory experiments and other numerical results the formation of a tripolar vortex where the highest vorticity is atr 0 is also considered finally to emphasize the accurate treatment near the axis the motion of a lamb dipole crossing the origin is simulated

pole condition for singular problems the pseudospectral approximation abstract this paper deals with the pseudospectral solution of differential equations with coordinate singularities such as those which describe situations in spherical or cylindrical geometries we use the differential equation together with a smoothness assumption on the solution to construct pole conditions the pole conditions which are straightforward and easily implemented serve as numerical boundary conditions at the coordinate singularity standard pseudospectral methods including fast transformation techniques can then be applied to the singular problem the method is illustrated using the eigenvalue problem of bessel s equation and a poisson equation on the unit disk numerical results show that spectral convergence is achieved

a pseudospectral method for solution of the three dimensional incompressible navier stokes equations abstract a new chebyshev pseudospectral technique based on the projection method that was previously applied by the authors to the solution of two dimensional incompressible navier stokes equations in primitive variables for nonperiodic boundary conditions is extended to solve the three dimensional navier stokes equations the crucial point of the method is the requirement that the continuity equation be satisfied everywhere in the domain on the boundaries as well as in the interior the key feature of the work presented in this paper is that the computer storage requirements of the full matrix inversion resulting from direct solution of the pressure poisson equation in three dimensions is greatly reduced by considering an eigenfunction decomposition the method was tested on a two dimensional driven cavity flow and the results were compared with those of the most accurate finite difference calculation the three dimensional driven cavity flow was then calculated at the same reynolds numbers as the two dimensional cases i e re 100 400 and 1000 in the calculated results three dimensional boundary effects were observed in all cases and became more apparent with increasing reynolds number

semiparametrically efficient inference based on signed ranks in symmetric independent component models we consider semiparametric location scatter models for which the p variate observation is obtained as x lambda z mu where mu is a p vector lambda is a full rank p p matrix and the unobserved random p vector z has marginals that are centered and mutually independent but are otherwise unspecified as in blind source separation and independent component analysis ica the parameter of interest throughout the paper is lambda on the basis of n i i d copies of x we develop under a symmetry assumption on z signed rank one sample testing and estimation procedures for lambda we exploit the uniform local and asymptotic normality ulan of the model to define signed rank procedures that are semiparametrically efficient under correctly specified densities yet as usual in rank based inference the proposed procedures remain valid correct asymptotic size under the null for hypothesis testing and root n consistency for point estimation under a very broad range of densities we derive the asymptotic properties of the proposed procedures and investigate their finite sample behavior through simulations

two sample test of high dimensional means under dependence summary the paper considers in the high dimensional setting a canonical testing problem in multivariate analysis namely testing the equality of two mean vectors we introduce a new test statistic that is based on a linear transformation of the data by the precision matrix which incorporates the correlations between the variables the limiting null distribution of the test statistic and the power of the test are analysed it is shown that the test is particularly powerful against sparse alternatives and enjoys certain optimality a simulation study is carried out to examine the numerical performance of the test and to compare it with other tests given in the literature the results show that the test proposed significantly outperforms those tests in a range of settings

bifurcation analysis of a system of coupled chemical oscillators bromate chlorite iodine abstract we examine the dynamical behavior of two chemical oscillators bromate iodide and chlorite iodide as well as the coupled system bromate chlorite iodide study of the period and amplitude of oscillation at the bifurcation points yields phase portraits for the uncoupled systems these are combined to give insights into the nature of the higher dimensional coupled system which exhibits a wide variety of dynamical phenomena including multiple stationary states birhythmicity and chaos analysis of the topology of the phase portraits and of the experimentally determined return map suggests that the coupled system may possess a lorenz type attractor

on the efficiency of multivariate spatial sign and rank tests conditionally and asymptotically distribution free rotation but not scale invariant competitors of the well known hotelling s t 2 tests in this paper efficiency results are found in the one sample location case but they natu rally hold in the multivariate two sample multisample and general linear model case the lack of means of fully ordering or ranking observations in the multivariate setting seems at first sight to be an obstacle to generalizing the concepts of sign and rank and signed rank however these concepts can be defined also by using l criterion functions as follows consider k 1 variate observations y y utilizing the euclidean distance y s 1 n 2 2 1r2 y q qy the three objective functions 1 k t y s s y y i i i i i

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

estimating sufficient reductions of the predictors in abundant high dimensional regressions we study the asymptotic behavior of a class of methods for sufficient dimension reduction in high dimension regressions as the sample size and number of predictors grow in various alignments it is demonstrated that these methods are consistent in a variety of settings particularly in abundant regressions where most predictors contribute some information on the response and oracle rates are possible simulation results are presented to support the theoretical conclusion

semiparametrically efficient rank based inference for shape i optimal rank based tests for sphericity we propose a class of rank based procedures for testing that the shape matrix v of an elliptical distribution with unspecified center of symmetry scale and radial density has some fixed value v 0 this includes for v 0 i k the problem of testing for sphericity as an important particular case the proposed tests are invariant under translations monotone radial transformations rotations and reflections with respect to the estimated center of symmetry they are valid without any moment assumption for adequately chosen scores they are locally asymptotically maximin in the le cam sense at given radial densities they are strictly distribution free when the center of symmetry is specified and asymptotically so when it must be estimated the multivariate ranks used throughout are those of the distances in the metric associated with the null value v 0 of the shape matrix between the observations and the estimated center of the distribution local powers against elliptical alternatives and asymptotic relative efficiencies ares are derived with respect to the adjusted mauchly test a modified version of the gaussian likelihood ratio procedure proposed by muirhead and waternaux biometrika 67 1980 31 43 or equivalently with respect to an extension of the test for sphericity introduced by john biometrika 59 1972 169 173 for gaussian scores these ares are uniformly larger than one irrespective of the actual radial density necessary and or sufficient conditions for consistency under nonlocal possibly nonelliptical alternatives are given finite sample performance is investigated via a monte carlo study

a two sample test for high dimensional data with applications to gene set testing we proposed a two sample test for means of high dimensional data when the data dimension is much larger than the sample size the classical hotelling s t 2 test does not work for this large p small n situation the proposed test does not require explicit conditions on the relationship between the data dimension and sample size this offers much flexibility in analyzing high dimensional data an application of the proposed test is in testing significance for sets of genes which we demonstrate in an empirical study on a leukemia data set

a phase field approach in the numerical study of the elastic bending energy for vesicle membranes in this paper we compute the equilibrium configurations of a vesicle membrane under elastic bending energy with prescribed volume and surface area a variational phase field method is developed for such a problem discrete finite difference approximations and numerical simulations are carried out in the axial symmetrical cases different energetic bifurcation phenomena are discussed

energy law preserving c0 finite element schemes for phase field models in two phase flow computations we use the idea in 33 to develop the energy law preserving method and compute the diffusive interface phase field models of allen cahn and cahn hilliard type respectively governing the motion of two phase incompressible flows we discretize these two models using a c 0 finite element in space and a modified midpoint scheme in time to increase the stability in the pressure variable we treat the divergence free condition by a penalty formulation under which the discrete energy law can still be derived for these diffusive interface models through an example we demonstrate that the energy law preserving method is beneficial for computing these multi phase flow models we also demonstrate that when applying the energy law preserving method to the model of cahn hilliard type un physical interfacial oscillations may occur we examine the source of such oscillations and a remedy is presented to eliminate the oscillations a few two phase incompressible flow examples are computed to show the good performance of our method

two phase fluid simulation using a diffuse interface model with peng robinson equation of state in this paper two phase fluid systems are simulated using a diffusive interface model with the peng robinson equation of state eos a widely used realistic eos for hydrocarbon fluid in the petroleum industry we first utilize the gradient theory of thermodynamics and variational calculus to derive a generalized chemical equilibrium equation which is mathematically a second order elliptic partial differential equation pde in molar density with a strongly nonlinear source term to solve this pde we convert it to a time dependent parabolic pde with the main interest in its final steady state solution a lagrange multiplier is used to enforce mass conservation the parabolic pde is then solved by mixed finite element methods with a semi implicit time marching scheme convex splitting of the energy functional is proposed to construct this time marching scheme where the volume exclusion effect of an eos is treated implicitly while the pairwise attraction effect of eos is calculated explicitly this sch

empirical likelihood test for a large dimensional mean vector summarythis paper is concerned with empirical likelihood inference on the population mean when the dimension p and the sample size n satisfy p n rightarrow c in 1 infty as shown in tsao 2004 the empirical likelihood method fails with high probability when p n 1 2 because the convex hull of the n observations in mathbb r p becomes too small to cover the true mean value moreover when p n the sample covariance matrix becomes singular and this results in the breakdown of the first sandwich approximation for the log empirical likelihood ratio to deal with these two challenges we propose a new strategy of adding two artificial data points to the observed data we establish the asymptotic normality of the proposed empirical likelihood ratio test the proposed test statistic does not involve the inverse of the sample covariance matrix furthermore its form is explicit so the test can easily be carried out with low computational cost our numerical comparison shows that the proposed test outperforms some existing tests for high dimensional mean vectors in terms of power we also illustrate the proposed procedure with an empirical analysis of stock data