semi global practical asymptotic stability and averaging abstract we prove a generalized liapunov theorem which guarantees practical asymptotic stability based on this theorem we show that if the averaged system x f av x corresponding to x f x t is globally asymptotically stable then starting from an arbitrarily large set of initial conditions the trajectories of x f x t e converge uniformly to an arbitrarily small residual set around the origin when e 0 is taken sufficiently small in other words the origin is semi globally practically asymptotically stable

an averaging theorem for time periodic degree zero homogeneous differential equations abstract this paper considers the stability of the differential equation dot x ex t x e x r n where x t x e is a time periodic degree zero homogeneous vector field and e 0 is a parameter it is shown that asymptotic stability of the time averaged equation implies asymptotic stability of the original system for e sufficiently small

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

covariance matrix selection and estimation via penalised normal likelihood summary we propose a nonparametric method for identifying parsimony and for producing a statistically efficient estimator of a large covariance matrix we reparameterise a covariance matrix through the modified cholesky decomposition of its inverse or the one step ahead predictive representation of the vector of responses and reduce the nonintuitive task of modelling covariance matrices to the familiar task of model selection and estimation for a sequence of regression models the cholesky factor containing these regression coefficients is likely to have many off diagonal elements that are zero or close to zero penalised normal likelihoods in this situation with l1 and l2 penalities are shown to be closely related to tibshirani s 1996 lasso approach and to ridge regression adding either penalty to the likelihood helps to produce more stable estimators by introducing shrinkage to the elements in the cholesky factor while because of its singularity the l1 penalty will set some elements to zero and produce interpretable models an algorithm is developed for computing the estimator and selecting the tuning parameter the proposed maximum penalised likelihood estimator is illustrated using simulation and a real dataset involving estimation of a 102 x 102 covariance matrix

pc algorithm for nonparanormal graphical models the pc algorithm uses conditional independence tests for model selection in graphical modeling with acyclic directed graphs in gaussian models tests of conditional independence are typically based on pearson correlations and high dimensional consistency results have been obtained for the pc algorithm in this setting analyzing the error propagation from marginal to partial correlations we prove that high dimensional consistency carries over to a broader class of gaussian copula or nonparanormal models when using rank based measures of correlation for graph sequences with bounded degree our consistency result is as strong as prior gaussian results in simulations the rank pc algorithm works as well as the pearson pc algorithm for normal data and considerably better for non normal data all the while incurring a negligible increase of computation time while our interest is in the pc algorithm the presented analysis of error propagation could be applied to other algorithms that test the vanishing of low order partial correlations

a pressure correction scheme for generalized form of energy stable open boundary conditions for incompressible flows we present a generalized form of open boundary conditions and an associated numerical algorithm for simulating incompressible flows involving open or outflow boundaries the generalized form represents a family of open boundary conditions which all ensure the energy stability of the system even in situations where strong vortices or backflows occur at the open outflow boundaries our numerical algorithm for treating these open boundary conditions is based on a rotational pressure correction type strategy with a formulation suitable for c 0 spectral element spatial discretizations we have introduced a discrete equation and associated boundary conditions for an auxiliary variable the algorithm contains constructions that prevent a numerical locking at the open outflow boundary in addition we have developed a scheme with a provable unconditional stability for a sub class of the open boundary conditions extensive numerical experiments have been presented to demonstrate the performance of our method for several flow problems involving open outflow boundaries we compare simulation results with the experimental data to demonstrate the accuracy of our algorithm long time simulations have been performed for a range of reynolds numbers at which strong vortices or backflows occur at the open outflow boundaries we show that the open boundary conditions and the numerical algorithm developed herein produce stable simulations in such situations

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

finite element discretization of the navier stokes equations with a free capillary surface the instationary navier stokes equations with a free capillary boundary are considered in 2 and 3 space dimensions a stable finite element discretization is presented the key idea is the treatment of the curvature terms by a variational formulation in the context of a discontinuous in time space time element discretization stability in weak energy norms can be proved numerical examples in 2 and 3 space dimensions are given

exponential stability and partial averaging the exponential stability of singularly perturbed time varying systems is investigated it turns out that under natural conditions exponential stability of an averaged system is equivalent to exponential stability of the perturbed system for small perturbation parameters explicit estimates for both the approximation of single trajectories and the order of the exponential decay are obtained the method of proof does not require smoothness of the averaged system

a sufficient condition for the uniform exponential stability of time varying systems with noise abstract the averaging method for nonlinear differential equations with a fast time variable is extended to systems with deterministic bounded noise the averaged system is used to present a sufficient condition for the uniform exponential stability of the original system the method of proof is based on a direct computation of the distance of trajectories of the averaged system to trajectories of the original system and manages without smoothness assumptions on the averaged system

exponential stability of nonlinear singularly perturbed differential equations nonlinear singularly perturbed differential equations with two natural time scales are under consideration it is shown that the exponential stability of both the boundary layer systems and the reduced system imply the exponential stability of the fully coupled system for small perturbation parameters the asymptotic behavior of gain and decay rates is investigated moreover it turns out that the achieved exponential stability is robust with respect to multivalued regular perturbations and small delays the method of proof does not rely on converse lyapunov theorems but on an appropriate version of tychonov s theorem and manages without the smoothness of the vector fields involved

robust gaussian graphical modeling via l1 penalization gaussian graphical models have been widely used as an effective method for studying the conditional independency structure among genes and for constructing genetic networks however gene expression data typically have heavier tails or more outlying observations than the standard gaussian distribution such outliers in gene expression data can lead to wrong inference on the dependency structure among the genes we propose a l1 penalized estimation procedure for the sparse gaussian graphical models that is robustified against possible outliers the likelihood function is weighted according to how the observation is deviated where the deviation of the observation is measured based on its own likelihood an efficient computational algorithm based on the coordinate gradient descent method is developed to obtain the minimizer of the negative penalized robustified likelihood where nonzero elements of the concentration matrix represents the graphical links among the genes after the graphical structure is obtained we re estimate the positive definite concentration matrix using an iterative proportional fitting algorithm through simulations we demonstrate that the proposed robust method performs much better than the graphical lasso for the gaussian graphical models in terms of both graph structure selection and estimation when outliers are present we apply the robust estimation procedure to an analysis of yeast gene expression data and show that the resulting graph has better biological interpretation than that obtained from the graphical lasso

the gauge uzawa finite element method part i the navier stokes equations the gauge uzawa fem is a new first order fully discrete projection method which combines advantages of both the gauge and uzawa methods within a variational framework a time step consists of a sequence of d 1 poisson problems d being the space dimension thereby avoiding both the incompressibility constraint as well as dealing with boundary tangential derivatives as in the gauge method this allows for a simple finite element discretization in space of any order in both two and three dimensions this first part introduces the method for the navier stokes equations of incompressible fluids and shows unconditional stability and error estimates for both velocity and pressure via a variational approach under realistic regularity assumptions several numerical experiments document performance of the gauge uzawa fem and compare it with other projection methods

error analysis of a fractional time stepping technique for incompressible flows with variable density in this paper we analyze the convergence properties of a new fractional time stepping technique for the solution of the variable density incompressible navier stokes equations the main feature of this method is that contrary to other existing algorithms the pressure is determined by just solving one poisson equation per time step first order error estimates are proved and stability of a formally second order variant of the method is established

robust and efficient estimation by minimising a density power divergence a minimum divergence estimation method is developed for robust parameter estimation the proposed approach uses new density based divergences which unlike existing methods of this type such as minimum hellinger distance estimation avoid the use of nonparametric density estimation and associated complications such as bandwidth selection the proposed class of density power divergences is indexed by a single parameter which controls the trade off between robustness and efficiency the methodology affords a robust extension of maximum likelihood estimation for which 0 choices of near zero afford considerable robustness while retaining efficiency close to that of maximum likelihood

finite element approximations of the nonstationary navier stokes problem part iv error estimates for second order time discretization this paper provides an error analysis for the crank nicolson method of time discretization applied to spatially discrete galerkin approximations of the nonstationary navier stokes equations second order error estimates are proven locally in time under realistic assumptions about the regularity of the solution for approximations of an exponentially stable solution these local error estimates are extended uniformly in time as text t to infty

model selection and estimation in the gaussian graphical model summary we propose penalized likelihood methods for estimating the concentration matrix in the gaussian graphical model the methods lead to a sparse and shrinkage estimator of the concentration matrix that is positive definite and thus conduct model selection and estimation simultaneously the implementation of the methods is nontrivial because of the positive definite constraint on the concentration matrix but we show that the computation can be done effectively by taking advantage of the efficient maxdet algorithm developed in convex optimization we propose a bic type criterion for the selection of the tuning parameter in the penalized likelihood methods the connection between our methods and existing methods is illustrated simulations and real examples demonstrate the competitive performance of the new methods

brief on exponential stability of nonlinear time varying differential equations within the liapunov framework a sufficient condition for exponential stability of ordinary differential equations is proposed unlike with classical liapunov theory the time derivative of the liapunov function taken along solutions of the system may have positive and negative values verification of the conditions of the main theorem may be harder than in the classical case it is shown that the proposed conditions are useful for the investigation of the exponential stability of fast time varying systems this sets the stability study by means of averaging in a liapunov context in particular it is established that exponential stability of the averaged system implies exponential stability of the original fast time varying system a comparison of our work with results taken from the literature is included

analysis of algebraic chorin temam splitting for incompressible nse and comparison to yosida methods abstract algebraic splitting methods are a common approach to solving the saddle point linear systems that arise at each time step of an incompressible flow simulation there are two main classes of these methods those of yosida type and those of algebraic chorin temam act type with the yosida methods being predominantly used in practice we show herein through new analysis and extensive numerical testing that act methods that include the viscous term stiffness matrix in the modified a block are unconditionally stable and can be superior to yosida methods in a range of problems particular situations where the act type solvers are advantageous include problems where numerical stability is a concern as well as problems where strong enforcement of the divergence constraint is important

a constrained l1 minimization approach to sparse precision matrix estimation this article proposes a constrained l1 minimization method for estimating a sparse inverse covariance matrix based on a sample of n iid p variate random variables the resulting estimator is shown to have a number of desirable properties in particular the rate of convergence between the estimator and the true s sparse precision matrix under the spectral norm is when the population distribution has either exponential type tails or polynomial type tails we present convergence rates under the elementwise l norm and frobenius norm in addition we consider graphical model selection the procedure is easily implemented by linear programming numerical performance of the estimator is investigated using both simulated and real data in particular the procedure is applied to analyze a breast cancer dataset and is found to perform favorably compared with existing methods

a finite element pressure correction scheme for the navier stokes equations with traction boundary condition abstract we consider the popular pressure correction scheme for the solution of the time dependent navier stokes equations with traction boundary condition a finite element based method to improve the performance of the classical approach is proposed the improvement is achieved by modifying the traction boundary condition for the provisional velocity u n 1 in each time step the corresponding term consists of a simple boundary functional involving the normal derivative of the pressure correction that can be evaluated in a natural and easy way in the context of finite elements computational results show a significant improvement of the solution in particular for the pressure in the case of smooth domains

an overview of projection methods for incompressible flows a series of numerical issues related to the analysis and implementation of fractional step methods for incompressible flows are addressed in this paper these methods are often referred to in the literature as projection methods and can be classified into three classes namely the pressure correction methods the velocity correction methods and the consistent splitting methods for each class of schemes theoretical and numerical convergence results available in the literature are reviewed and open questions are discussed the essential results are summarized in a table which could serve as a useful reference to numerical analysts and practitioners published by elsevier b v msc 65n35 65n22 65f05 35j05

partial correlation estimation by joint sparse regression models this article features online supplementary material in this article we propose a computationally efficient approach space sparse partial correlation estimation for selecting nonzero partial correlations under the high dimension low sample size setting this method assumes the overall sparsity of the partial correlation matrix and employs sparse regression techniques for model fitting we illustrate the performance of space by extensive simulation studies it is shown that space performs well in both nonzero partial correlation selection and the identification of hub variables and also outperforms two existing methods we then apply space to a microarray breast cancer dataset and identify a set of hub genes that may provide important insights on genetic regulatory networks finally we prove that under a set of suitable assumptions the proposed procedure is asymptotically consistent in terms of model selection and parameter estimation

sparsistency and rates of convergence in large covariance matrix estimation this paper studies the sparsistency and rates of convergence for estimating sparse covariance and precision matrices based on penalized likelihood with nonconvex penalty functions here sparsistency refers to the property that all parameters that are zero are actually estimated as zero with probability tending to one depending on the case of applications sparsity priori may occur on the covariance matrix its inverse or its cholesky decomposition we study these three sparsity exploration problems under a unified framework with a general penalty function we show that the rates of convergence for these problems under the frobenius norm are of order s n log p n n 1 2 where s n is the number of nonzero elements p n is the size of the covariance matrix and n is the sample size this explicitly spells out the contribution of high dimensionality is merely of a logarithmic factor the conditions on the rate with which the tuning parameter n goes to 0 have been made explicit and compared under different penalties as a result for the l 1 penalty to guarantee the sparsistency and optimal rate of convergence the number of nonzero elements should be small sn o pn at most among o pn2 parameters for estimating sparse covariance or correlation matrix sparse precision or inverse correlation matrix or sparse cholesky factor where sn is the number of the nonzero elements on the off diagonal entries on the other hand using the scad or hard thresholding penalty functions there is no such a restriction

error analysis of pressure correction schemes for the time dependent stokes equations with open boundary conditions the incompressible stokes equations with prescribed normal stress open boundary conditions on part of the boundary are considered it is shown that the standard pressure correction method is not suitable for approximating the stokes equations with open boundary conditions whereas the rotational pressure correction method yields reasonably good error estimates these results appear to be the first ever published for splitting schemes with open boundary conditions numerical results in agreement with the error estimates are presented

gaussian markov distributions over finite graphs on s interesse aux modeles de selection de covariance introduites par dempster 1972 et etudies par wermuth 1976

stability analysis of pressure correction schemes for the navier stokes equations with traction boundary conditions abstract we present a stability analysis for two different rotational pressure correction schemes with open and traction boundary conditions first we provide a stability analysis for a rotational version of the grad div stabilized scheme of bonito et al 2015 this scheme turns out to be unconditionally stable provided the stabilization parameter is suitably chosen we also establish a conditional stability result for the boundary correction scheme presented in bansch 2014 these results are shown by employing the equivalence between stabilized gauge uzawa methods and rotational pressure correction schemes with traction boundary conditions

nonparametric estimation of large covariance matrices of longitudinal data summary estimation of an unstructured covariance matrix is difficult because of its positivedefiniteness constraint this obstacle is removed by regressing each variable on its predecessors so that estimation of a covariance matrix is shown to be equivalent to that of estimating a sequence of varying coefficient and varying order regression models our framework is similar to the use of increasing order autoregressive models in approximating the covariance matrix or the spectrum of a stationary time series as an illustration we adopt fan zhang s 2000 two step estimation of functional linear models and propose nonparametric estimators of covariance matrices which are guaranteed to be positive definite for parsimony a suitable order for the sequence of auto regression models is found using penalised likelihood criteria like aic and bic some asymptotic results for the local polynomial estimators of components of a covariance matrix are established two longitudinal datasets are analysed to illustrate the methodology a simulation study reveals the advantage of the nonparametric covariance estimator over the sample covariance matrix for large covariance matrices