simple finite element method in vorticity formulation for incompressible flows a very simple and efficient finite element method is introduced for two and three dimensional viscous incompressible flows using the vorticity formulation this method relies on recasting the traditional finite element method in the spirit of the high order accurate finite difference methods introduced by the authors in another work optimal accuracy of arbitrary order can be achieved using standard finite element or spectral elements the method is convectively stable and is particularly suited for moderate to high reynolds number flows

a characterization of markov equivalence classes for acyclic digraphs undirected graphs and acyclic digraphs adgs as well as their mutual extension to chain graphs are widely used to describe dependencies among variables in multivariate distributions in particular the likelihood functions of adg models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building bayesian networks for expert systems whereas the undirected graph associated with a dependence model is uniquely determined there may however be many adgs that determine the same dependence markov model thus the family of all adgs with a given set of vertices is naturally partitioned into markov equivalence classes each class being associated with a unique statistical model statistical procedures such as model selection or model averaging that fail to take into account these equivalence classes may incur substantial computational or other inefficiencies here it is shown that each markov equivalence class is uniquely determined by a single chain graph the essential graph that is itself simultaneously markov equivalent to all adgs in the equivalence class essential graphs are characterized a polynomial time algorithm for their construction is given and their applications to model selection and other statistical questions are described

time varying autoregressive moving average models for covariance estimation we consider large scale covariance estimation using a small number of samples in applications where there is a natural ordering between the random variables the two classical approaches to this problem rely on banded covariance and banded inverse covariance structures corresponding to time varying moving average ma and autoregressive ar models respectively motivated by this analogy to spectral estimation and the well known modeling power of autoregressive moving average arma processes we propose a novel time varying arma covariance structure similarly to known results in the context of ar and ma we address the completion of an arma covariance matrix from its main band and its estimation based on random samples finally we examine the advantages of our proposed methods using numerical experiments

tunneling through a singular potential barrier quantum tunneling of a nonrelativistic particle through a singular potential barrier v is studied on the line the hamiltonian is a self adjoint extension of the operator h1 d2 dx2 v x if h1 is essentially self adjoint on its natural domain the tunneling is forbidden for a class of potentials that includes all semiclassically impenetrable barriers if h1 is not essentially self adjoint the friedrichs extension of h1 yields no tunneling for another class of potentials which again includes the semiclassically impenetrable ones in general the occurrence of tunneling is not excluded and depends on the self adjoint extension we choose as the hamiltonian of our problem as an example we evaluate the transmission coefficient for all self adjoint extensions of the operator h1 referring to v x gx 2 with 0 g 3 4

discontinuous galerkin approximations of the stokes and navier stokes equations numerical schemes to compute approximate solutions of the evolutionary stokes and navier stokes equations are studied the schemes are discontinuous in time and conforming in space and of arbitrarily high order fully discrete error estimates are derived and dependence of the viscosity constant is carefully tracked it is shown that the errors are bounded by projection errors of the exact solution which exhibit optimal rates when the solutions are smooth

two level method based on finite element and crank nicolson extrapolation for the time dependent navier stokes equations a fully discrete two level finite element method the two level method is presented for solving the two dimensional time dependent navier stokes problem the method requires a crank nicolson extrapolation solution u h tau 0 p h tau 0 on a spatial time coarse grid j h tau 0 and a backward euler solution u h tau p h tau on a space time fine grid j h tau the error estimates of optimal order of the discrete solution for the two level method are derived compared with the standard crank nicolson extrapolation method the one level method based on a space time fine grid j h tau the two level method is of the error estimates of the same order as the one level method in the h1 norm for velocity and the l2 norm for pressure however the two level method involves much less work than the one level method

global identifiability of linear structural equation models structural equation models are multivariate statistical models that are defined by specifying noisy functional relationships among random variables we consider the classical case of linear relationships and additive gaussian noise terms we give a necessary and sufficient condition for global identifiability of the model in terms of a mixed graph encoding the linear structural equations and the correlation structure of the error terms global identifiability is understood to mean injectivity of the parametrization of the model and is fundamental in particular for applicability of standard statistical methodology

on a higher order accurate fully discrete galerkin approximation to the navier stokes equations we consider approximating the solution of the initial and boundary value problem for the navier stokes equations in bounded twoand three dimensional domains using a nonstandard galerkin finite element method for the space discretization and the third order accurate three step backward differentiation method coupled with extrapolation for the nonlinear terms for the time stepping the resulting scheme requires the solution of one linear system per time step plus the solution of five linear systems for the computation of the required initial conditions all these linear systems have the same matrix the resulting approximations of the velocity are shown to have optimal rate of convergence in l2 under suitable restrictions on the discretization parameters of the problem and the size of the solution in an appropriate function space

two level method and some a priori estimates in unsteady navier stokes calculations a two level method proposed for quasielliptic problems is adapted in this paper to the simulation of unsteady incompressible navier stokes flows the method requires a solution of a nonlinear problem on a coarse grid and a solution of linear symmetric problem on a fine grid the scaling between these two grids is superlinear approximation stability and convergence aspects of a fully discrete scheme are considered stability properties of the two level scheme are compared with those for a commonly used semi implicit scheme some new estimates are also proved for the latter

higher order godunov methods for general systems of hyperbolic conservation laws abstract we describe an extension of higher order godunov methods to general systems of hyperbolic conservation laws this extension allow the method to be applied to problems that are not strictly hyperbolic and exhibit local linear degeneracies in the wave fields the method constructs an approximation of the riemann problem from local wave information a generalization of the engquist osher flux for systems is then used to compute a numerical flux based on this approximation this numerical flux replaces the godunov numerical flux in the algorithm thereby eliminating the need for a global riemann problem solution the additional modifications to the godunov methodology that are needed to treat loss of strict hyperbolicity are described in detail the method is applied to some simple model problems for which the global analytic structure is known the method is also applied to the black oil model for multiphase flow in petroleum reservoirs

an optimally accurate discrete regularization for second order timestepping methods for navier stokes equations abstract we propose a new optimally accurate numerical regularization stabilization for a family of second order timestepping methods for the navier stokes equations nse the method combines a linear treatment of the advection term together with stabilization terms that are proportional to discrete curvature of the solutions in both velocity and pressure we rigorously prove that the entire new family of methods are unconditionally stable and o t 2 accurate the idea of curvature stabilization is new to cfd and is intended as an improvement over the commonly used speed stabilization which is only first order accurate in time and can have an adverse effect on important flow quantities such as drag coefficients numerical examples verify the predicted convergence rate and show the stabilization term clearly improves the stability and accuracy of the tested flows

on the leray deconvolution model for the incompressible magnetohydrodynamics equations abstract we extend leray deconvolution modeling to the incompressible magnetohydrodynamics mhd the resulting model is shown to be well posed and have attractive limiting behavior both in its filtering radius and order of deconvolution additionally we present and study a numerical scheme for the model based on an extrapolated crank nicolson finite element method we show the numerical scheme is unconditionally stable preserves energy and cross helicity and optimally converges to the mhd solution numerical experiments are provided that verify convergence rates and test the scheme on benchmark problems of channel flow over a step and the orszag tang vortex problem

causal inference for complex longitudinal data the continuous case we extend robins theory of causal inference for complex longitudinal data to the case of continuously varying as opposed to discrete covariates and treatments in particular we establish versions of the key results of the discrete theory the g computation formula and a collection of powerful characterizations of the g null hypothesis of no treatment eect this is accomplished under natural continuity hypotheses concerning the conditional distributions of the outcome variable and of the covariates given the past we also show that our assumptions concerning counterfactual variables place no restriction on the joint distribution of the observed variables thus in a precise sense these assumptions are for free or if you prefer harmless

long term dissipativity of time stepping algorithms for an abstract evolution equation with applications to the incompressible mhd and navier stokes equations abstract this paper examines the long term dissipativity and unconditional non linear stability of time integration algorithms for the incompressible mhd equations the results apply as a particular case to the uncoupled incompressible navier stokes equations the analysis is performed in the setting defined by an abstract evolution equation possessing the same long term dissipative structure as the above physical systems a general class of time stepping schemes is analyzed in this context showing the non linear unconditional stability and long term dissipativity of particular members of this family of algorithms in particular algorithms showing these strong properties which additionally involve the solution of a linear problem at each time step are identified a particular linear algorithm is shown to exhibit optimal long term dissipative properties and to possess an algorithmic global attractor these abstract results apply rigorously to galerkin mixed finite element discretizations of the problem to this end it is shown that such galerkin type projection inherits the dissipative properties of the continuum problem computational aspects involved in the numerical implementation of these methods are addressed and illustrated in representative numerical simulations

graphical methods for efficient likelihood inference in gaussian covariance models in graphical modelling a bi directed graph encodes marginal independences among random variables that are identified with the vertices of the graph we show how to transform a bi directed graph into a maximal ancestral graph that i represents the same independence structure as the original bi directed graph and ii minimizes the number of arrowheads among all ancestral graphs satisfying i here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bi directed edges in gaussian models this construction can be used for more efficient iterative maximization of the likelihood function and to determine when maximum likelihood estimates are equal to empirical counterparts

estimation of a covariance matrix with zeros summary we consider estimation of the covariance matrix of a multivariate random vector under the constraint that certain covariances are zero we first present an algorithm which we call iterative conditional fitting for computing the maximum likelihood estimate of the constrained covariance matrix under the assumption of multivariate normality in contrast to previous approaches this algorithm has guaranteed convergence properties dropping the assumption of multivariate normality we show how to estimate the covariance matrix in an empirical likelihood approach these approaches are then compared via simulation and on an example of gene expression

quantum motion on a half line connected to a plane in this paper the free motion of a particle on a manifold that consists of a one dimensional and a two dimensional part connected in one point is discussed the class of admissible hamiltonians is found using the theory of self adjoint extensions particular attention is paid to those hamiltonians that allow the particle to pass through the point singularity the reflection coefficient and other quantities characterizing scattering on the connection point are calculated a possible application is also discussed

complex potential description of the damped harmonic oscillator the multidimensional damped harmonic oscillator is treated by means of a non self adjoint hamiltonian with complex potential the propagator referring to the evolution semigroup is evaluated from the lie trotter formula the one dimensional case is discussed in detail with the following results a the nondamped limit gives the correct propagator including the maslov phase factor b for some initial conditions the classical limit of the solution can differ from the behavior of the classical damped oscillator the difference being negligible in the case of weak damping and c the point spectrum of the considered pseudo hamiltonian is found

markov equivalence for ancestral graphs ancestral graphs can encode conditional independence relations that arise in directed acyclic graph dag models with latent and selection variables however for any ancestral graph there may be several other graphs to which it is markov equivalent we state and prove conditions under which two maximal ancestral graphs are markov equivalent to each other thereby extending analogous results for dags given by other authors these conditions lead to an algorithm for determining markov equivalence that runs in time that is polynomial in the number of vertices in the graph

a finite element method for diffusion dominated unsteady viscous flows a general conforming finite element scheme for computing viscous flows is presented which is of second order accuracy in space and time viscous terms are treated implicitly and advection terms are treated explicitly in the time marching segment of the algorithm a method for solving the algebraic equations at each time step is given the method is demonstrated on two test problems one of them being a plane vortex flow for which asymptotic methods are used to obtain suitable numerical boundary conditions at each time step

bounds on natural direct effects in the presence of confounded intermediate variables in epidemiological studies we often want to learn about the direct effect of an exposure on an outcome i e the effect that is not relayed by a specific intermediate variable in the literature there are two common definitions of direct effects controlled and natural when the intermediate variable and the outcome have common causes neither the controlled nor the natural direct effect is identified cai et al biometrics 2007 64 3 695 701 derived bounds for the controlled direct effect under a set of hierarchical assumptions in this paper we derive bounds on the natural direct effect under the same assumptions copyright 2008 john wiley sons ltd

an energy stable scheme for incompressible navier stokes equations with periodically updated coefficient matrix abstract we present an energy stable scheme for simulating the incompressible navier stokes equations based on the generalized positive auxiliary variable gpav framework in the gpav reformulated system the original nonlinear term is replaced by a linear term plus a correction term where the correction term is put under control by an auxiliary variable the proposed scheme incorporates a pressure correction type strategy into the gpav procedure and it satisfies a discrete energy stability property the scheme entails the computation of two copies of the velocity and pressure within a time step by solving an individual de coupled linear equation for each of these field variables upon discretization the pressure linear system involves a constant coefficient matrix that can be pre computed while the velocity linear system involves a coefficient matrix that is updated periodically once every k 0 time steps in the current work where k 0 is a user specified integer the auxiliary variable being a scalar valued number is computed by a well defined explicit formula which guarantees the positivity of its computed values it is observed that the current method can produce accurate simulation results at large or fairly large time step sizes for the incompressible navier stokes equations the impact of the periodic coefficient matrix update on the overall cost of the method is observed to be small in typical numerical simulations several flow problems have been simulated to demonstrate the accuracy and performance of the method developed herein

a stabilized implicit fractional step method for the time dependent navier stokes equations using equal order pairs abstract a stabilized implicit fractional step method for numerical solutions of the time dependent navier stokes equations is presented in this paper the time advancement is decomposed into a sequence of two steps the first step has the structure of the linear elliptic problem the second step can be seen as the generalized stokes problem the two problems satisfy the full homogeneous dirichlet boundary conditions on the velocity on the other hand a locally stabilized term is added in the second step of the schemes it allows one to enhance the numerical stability and efficiency by using the equal order pairs convergence analysis and error estimates for the velocity and pressure of the schemes are established via the energy method some numerical experiments are also used to demonstrate the efficiency of this new method

stabilized finite element method based on the crank nicolson extrapolation scheme for the time dependent navier stokes equations this paper provides an error analysis for the crank nicolson extrapolation scheme of time discretization applied to the spatially discrete stabilized finite element approximation of the two dimensional time dependent navier stokes problem where the finite element space pair x h m h for the approximation u n h p n h of the velocity u and the pressure p is constructed by the low order finite element the q 1 p 0 quadrilateral element or the p 1 p 0 triangle element with mesh size h error estimates of the numerical solution u n h p n h to the exact solution u t n p t n with t n 0 t are derived

maximum likelihood analysis of linear mediation models with treatment mediator interaction this research concerns a mediation model where the mediator model is linear and the outcome model is also linear but with a treatment mediator interaction term and a residual correlated with the residual of the mediator model assuming the treatment is randomly assigned parameters in this mediation model are shown to be partially identifiable under the normality assumption on the residual of the mediator and the residual of the outcome explicit full information maximum likelihood estimates of model parameters are introduced given the correlation between the residual for the mediator and the residual for the outcome a consistent variance matrix of these estimates is derived currently the coefficients of this mediation model are estimated using the iterative feasible generalized least squares ifgls method that is originally developed for seemingly unrelated regressions surs we argue that this mediation model is not a system of surs while the ifgls estimates are consistent their variance matrix is not theoretical comparisons of the fimle variance matrix and the ifgls variance matrix are conducted our results are demonstrated by simulation studies and an empirical study the fimle method has been implemented in a freely available r package imediate

an incomplete data approach to the analysis of covariance structures in this paper linear structural equation models with latent variables are considered it is shown how many common models arise from incomplete observation of a relatively simple system subclasses of models with conditional independence interpretations are also discussed using an incomplete data point of view the relationships between the incomplete and complete data likelihoods assuming normality are highlighted for computing maximum likelihood estimates the em algorithm and alternatives are surveyed for the alternative algorithms simplified expressions for computing function values and derivatives are given likelihood ratio tests based on complete and incomplete data are related and an example on using their relationship to improve the fit of a model is given

on the convergence of a time discretization scheme for the navier stokes equations version linearisee du schema d euler implicite utilisee pour l approximation des solutions des equations de navier stokes dans un domaine bidimensionnel

galerkin runge kutta discretizations for semilinear parabolic equations a new class of fully discrete galerkin runge kutta methods is constructed and analyzed for semilinear parabolic initial boundary value problems unlike any classical counterpart this class offers arbitrarily high order convergence without suffering from what has been called order reduction in support of this claim error estimates are proved and computational results are presented furthermore it is noted that special runge kutta methods allow computations to be performed in parallel so that the final execution time can be reduced to that of a low order method

stability and convergence of iterative methods related to viscosities for the 2d 3d steady navier stokes equations abstract some finite element iterative methods related to viscosities are designed to solve numerically the steady 2d 3d navier stokes equations the two level finite element iterative methods are designed to solve numerically the steady 2d 3d navier stokes equations for a large viscosity such that a strong uniqueness condition holds the two level finite element iterative methods consist of using the stokes newton and oseen iterations of m times on a coarse mesh with mesh size h and computing the stokes newton and oseen correction of one time on a fine grid with mesh size h h moreover the one level oseen finite element iterative method based on a fine mesh with a small mesh size is designed to solve numerically the steady 2d 3d navier stokes equations for small viscosity such that a weak uniqueness condition holds the uniform stability and convergence of these methods with respect to and mesh sizes h and h and iterative times m are provided

unconditional convergence and optimal l2 error estimates of the crank nicolson extrapolation fem for the nonstationary navier stokes equations abstract in this paper we study stability and convergence of fully discrete finite element method on large timestep which used crank nicolson extrapolation scheme for the nonstationary navier stokes equations this approach bases on a finite element approximation for the space discretization and the crank nicolson extrapolation scheme for the time discretization it reduces nonlinear equations to linear equations thus can greatly increase the computational efficiency we prove that this method is unconditionally stable and unconditionally convergent moreover taking the negative norm technique we derive the l 2 h 1 unconditionally optimal error estimates for the velocity and the l 2 unconditionally optimal error estimate for the pressure also numerical simulations on unconditional l 2 stability and convergent rates of this method are shown

an unconditionally stable rotational velocity correction scheme for incompressible flows we present an unconditionally stable splitting scheme for incompressible navier stokes equations based on the rotational velocity correction formulation the main advantages of the scheme are i it allows the use of time step sizes considerably larger than the widely used semi implicit type schemes the time step size is only constrained by accuracy ii it does not require the velocity and pressure approximation spaces to satisfy the usual inf sup condition in particular the equal order finite element spectral element approximation spaces can be used iii it only requires solving a pressure poisson equation and a linear convection diffusion equation at each time step numerical tests indicate that the computational cost of the new scheme for each time step under identical time step sizes is even less expensive than the semi implicit scheme with low element orders therefore the total computational cost of the new scheme can be significantly less than the usual semi implicit scheme

sharp sensitivity bounds for mediation under unmeasured mediator outcome confounding it is often of interest to decompose the total effect of an exposure into a component that acts on the outcome through some mediator and a component that acts independently through other pathways said another way we are interested in the direct and indirect effects of the exposure on the outcome even if the exposure is randomly assigned it is often infeasible to randomize the mediator leaving the mediator outcome confounding not fully controlled we develop a sensitivity analysis technique that can bound the direct and indirect effects without parametric assumptions about the unmeasured mediator outcome confounding

the euler implicit explicit scheme for the 2d time dependent navier stokes equations with smooth or non smooth initial data this paper considers the stability and convergence results for the euler implicit explicit scheme applied to the spatially discretized two dimensional 2d time dependent navier stokes equations a galerkin finite element spatial discretization is assumed and the temporal treatment is implicit explict scheme which is implicit for the linear terms and explicit for the nonlinear term here the stability condition depends on the smoothness of the initial data u 0 h i e the time step condition is c 0 in the case of a 2 log h c 0 in the case of a 1 and th 2 c 0 in the case of a 0 for mesh size h and some positive constant c 0 we provide the h 2 stability of the scheme under the stability condition with a 0 1 2 and obtain the optimal h 1 l 2 error estimate of the numerical velocity and the optimal l 2 error estimate of the numerical pressure under the stability condition with a 1 2

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

identification and efficient estimation of the natural direct effect among the untreated the natural direct effect nde or the effect of an exposure on an outcome if an intermediate variable was set to the level it would have been in the absence of the exposure is often of interest to investigators in general the statistical parameter associated with the nde is difficult to estimate in the non parametric model particularly when the intermediate variable is continuous or high dimensional in this paper we introduce a new causal parameter called the natural direct effect among the untreated discus identifiability assumptions propose a sensitivity analysis for some of the assumptions and show that this new parameter is equivalent to the nde in a randomized controlled trial we also present a targeted minimum loss estimator tmle a locally efficient double robust substitution estimator for the statistical parameter associated with this causal parameter the tmle can be applied to problems with continuous and high dimensional intermediate variables and can be used to estimate the nde in a randomized controlled trial with such data additionally we define and discuss the estimation of three related causal parameters the natural direct effect among the treated the indirect effect among the untreated and the indirect effect among the treated

convergence analysis for a higher order scheme for the time dependent navier stokes equations abstract in this paper we discuss a third order fully discrete scheme for the time dependent navier stokes equations we prove that the third order temporal discrete scheme proposed by baker et al in 1 is convergent under a weaker condition some numerical experiments are shown to confirm the efficiency of the higher order scheme

convergence of three iterative methods based on the finite element discretization for the stationary navier stokes equations abstract this paper considers three iterative methods for solving the stationary navier stokes equations iterative method i consists in solving the stationary stokes equations iterative method ii consists in solving the stationary linearized navier stokes equations and iterative method iii consists in solving the stationary oseen equations under the finite element discretization respectively at each iterative step also we discuss the stability and convergence of three iterative methods the iterative methods i and ii are stability and convergence under the strong uniqueness conditions where the iterative method ii is the second order convergence furthermore the iterative method iii is uncondition stability and convergence under the uniqueness condition finally some numerical tests show that the efficiency of the theoretical analysis

some iterative finite element methods for steady navier stokes equations with different viscosities two level iterative finite element methods are designed to solve numerically the steady 2d 3d navier stokes equations for a large viscosity n such that a strong uniqueness condition holds moreover the one level oseen iterative finite element method based on a fine mesh with small mesh size h is designed to solve numerically the steady 2d 3d navier stokes equations for small viscosity n such that a weak uniqueness condition holds meanwhile the numerical investigations are provided to show that the proposed methods are efficient for solving the 2d 3d steady navier stokes equations for different viscosities

euler implicit explicit iterative scheme for the stationary navier stokes equations in this paper a new uniqueness assumption a2 of the solution for the stationary navier stokes equations is presented under assumption a2 the exponential stability of the solution bar u bar p for the stationary navier stokes equations is proven moreover the euler implicit explicit scheme based on the mixed finite element is applied to solve the stationary navier stokes equations finally the almost unconditionally stability is proven and the optimal error estimates uniform in time are provided for the scheme

numerical approximation of incompressible navier stokes equations based on an auxiliary energy variable abstract we present a numerical scheme for approximating the incompressible navier stokes equations based on an auxiliary variable associated with the total system energy by introducing a dynamic equation for the auxiliary variable and reformulating the navier stokes equations into an equivalent system the scheme satisfies a discrete energy stability property in terms of a modified energy and it allows for an efficient solution algorithm and implementation within each time step the algorithm involves the computations of two pressure fields and two velocity fields by solving several de coupled individual linear algebraic systems with constant coefficient matrices together with the solution of a nonlinear algebraic equation about a scalar number involving a negligible cost a number of numerical experiments are presented to demonstrate the accuracy and the performance of the presented algorithm

a crank nicolson leapfrog stabilization we propose and analyze a linear stabilization of the crank nicolson leapfrog cnlf method that removes all time step cfl conditions for stability and controls the unstable mode it also increases the spd part of the linear system to be solved at each time step while increasing solution accuracy we give a proof of unconditional stability of the method as well as a proof of unconditional asymptotic stability of both the stable and unstable modes we illustrate two applications of the method uncoupling groundwater surface water flows and stokes flow plus a coriolis term

graphical gaussian modelling of multivariate time series with latent variable in time series analysis inference about causeeffect relationships among multiple times series is commonly based on the concept of granger causality which exploits temporal structure to achieve causal ordering of dependent variables one major problem in the application of granger causality for the identification of causal relationships is the possible presence of latent variables that affect the measured components and thus lead to so called spurious causalities in this paper we describe a new graphical approach for modelling the dependence structure of multivariate stationary time series that are affected by latent variables to this end we introduce dynamic maximal ancestral graphs dmags in which each time series is represented by a single vertex for gaussian processes this approach leads to vector autoregressive models with errors that are not independent but correlated according to the dashed edges in the graph we discuss identifiability of the parameters and show that these models can be viewed as graphical arma models that satisfy the granger causality restrictions encoded by the associated dynamic maximal ancestral graph

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

group theoretical approach to reflectionless potentials we examine the general form of potentials with zero reflection coefficient in one dimensional hamiltonians connected with casimir invariants of non compact groups

the stabilized extrapolated trapezoidal finite element method for the navier stokes equations abstract we consider a fully discrete stabilized finite element method for the navier stokes equations which is unconditionally stable and has second order temporal accuracy of o k 2 hk spatial error the method involves a simple artificial viscosity stabilization of the linear system for the approximation of the new time level connected to anti diffusion of its effects at the old time level lowering the cell reynolds number of the oseen problem to o 1 the method requires only the solution of one linear system arising from an oseen problem per time step the effectiveness of the method is illustrated in several numerical experiments

joint response graphs and separation induced by triangular systems summary we consider joint probability distributions generated recursively in terms of univariate conditional distributions satisfying conditional independence restrictions the independences are captured by missing edges in a directed graph a matrix form of such a graph called the generating edge matrix is triangular so the distributions that are generated over such graphs are called triangular systems we study consequences of triangular systems after grouping or reordering of the variables for analyses as chain graph models i e for alternative recursive factorizations of the given density using joint conditional distributions for this we introduce families of linear triangular equations which do not require assumptions of distributional form the strength of the associations that are implied by such linear families for chain graph models is derived the edge matrices of chain graphs that are implied by any triangular system are obtained by appropriately transforming the generating edge matrix it is shown how induced independences and dependences can be studied by graphs by edge matrix calculations and via the properties of densities some ways of using the results are illustrated

a family of second order time stepping methods for the darcy brinkman equations abstract this study presents an efficient accurate effective and unconditionally stable time stepping scheme for the darcy brinkman equations in double diffusive convection the stabilization within the proposed method uses the idea of stabilizing the curvature for velocity temperature and concentration equations accuracy in time is proven and the convergence results for the fully discrete solutions of problem variables are given several numerical examples including a convergence study are provided that support the derived theoretical results and demonstrate the efficiency and the accuracy of the method

generalized cornfield conditions for the risk difference a central question in causal inference with observational studies is the sensitivity of conclusions to unmeasured confounding the classical cornfield condition allows us to assess whether an unmeasured binary confounder can explain away the observed relative risk of the exposure on the outcome it states that for an unmeasured confounder to explain away an observed relative risk the association between the unmeasured confounder and the exposure and the association between the unmeasured confounder and the outcome must both be larger than the observed relative risk in this paper we extend the classical cornfield condition in three directions first we consider analogous conditions for the risk difference and allow for a categorical not just a binary unmeasured confounder second we provide more stringent thresholds that the maximum of the above mentioned associations must satisfy rather than weaker conditions that both must satisfy third we show that all the earlier results on cornfield conditions hold under weaker assumptions than previously used we illustrate the potential applications by real examples where our new conditions give more information than the classical ones

compactness properties of the dg and cg time stepping schemes for parabolic equations for a broad class of parabolic equations it is shown that numerical solutions computed using the discontinuous galerkin or the continuous galerkin time stepping schemes of arbitrary order will inherit the compactness properties of the underlying equation convergence of numerical schemes for a phase field approximation of the flow of two fluids with surface tension is presented to illustrate these results

identifying direct and indirect effects in a non counterfactual framework summary identifying direct and indirect effects is a common problem in the social science and medical literature and can be described as follows a treatment is administered and a response is recorded however another variable mediates the effect of the treatment on the response in some way channelling a part of the treatment effect the question is how to extricate the direct and channelled indirect effects from one another when it is not possible to intervene on the mediating variable the aim of the paper is to tackle this problem by using a model for direct and indirect effects based on the decision theoretic framework for causal inference

a two level method with backtracking for the navier stokes equations we consider a two level method for resolving the nonlinearity in finite element approximations of the equilibrium navier stokes equations the method yields l2 and h1 optimal velocity approximations and an l2 optimal pressure approximation the two level method involves solving one small nonlinear coarse mesh system one oseen problem hence linear with positive definite symmetric part on the fine mesh and one linear correction problem on the coarse mesh the algorithm we study produces an approximate solution with the optimal asymptotic in h accuracy for any fixed reynolds number we do not consider the behavior of the error for fixed h as re rightarrow infty i e for flows in transition to turbulence

stability of implicit explicit backward difference formulas for nonlinear parabolic equations we analyze stability properties of backward difference formulas bdf methods of order up to 5 for linear parabolic equations as well as of implicit explicit bdf methods for nonlinear parabolic equations by energy techniques time dependent norms also play a key role in the analysis

computing maximum likelihood estimates in recursive linear models with correlated errors in recursive linear models the multivariate normal joint distribution of all variables exhibits a dependence structure induced by a recursive or acyclic system of linear structural equations these linear models have a long tradition and appear in seemingly unrelated regressions structural equation modelling and approaches to causal inference they are also related to gaussian graphical models via a classical representation known as a path diagram despite the models long history a number of problems remain open in this paper we address the problem of computing maximum likelihood estimates in the subclass of bow free recursive linear models the term bow free refers to the condition that the errors for variables i and j be uncorrelated if variable i occurs in the structural equation for variable j we introduce a new algorithm termed residual iterative conditional fitting ricf that can be implemented using only least squares computations in contrast to existing algorithms ricf has clear convergence properties and yields exact maximum likelihood estimates after the first iteration whenever the mle is available in closed form

unconditional optimal error estimates for bdf2 fem for a nonlinear schrodinger equation abstract this paper analyzes unconditional optimal error estimates for a 2 step backward differentiation formula bdf2 method for a nonlinear schrodinger equation in the analysis we split an error estimate into two parts one from the spatial discretization and the other from the temporal discretization we present the boundedness of the solution of the time discrete system in the certain strong norms and the error estimates for time discretization by these boundedness and temporal error estimates we obtain the l 2 error estimates without any conditions on a time step size numerical experiments are provided to validate our analysis and check the efficiency of our method

stable and time dissipative finite element methods for the incompressible navier stokes equations in advection dominated flows we examine a new galerkin method with scaled bubble functions which replicates the exact artificial diffusion methods in the case of 1 d scalar advection diffusion and that leads to non oscillatory solutions as the streamline upwinding algorithms for 2 d scalar advection diffusion and incompressible navier stokes this method retains the satisfaction of the babuska brezzi condition and thus leads to optimal performance in the incompressible limit this method when combined with the recently proposed linear unconditionally stable algorithms of simo and armero yields a method for solution of the incompressible navier stokes equations ideal for either diffusive or advection dominated flows examples from scalar advection diffusion and the solution of the incompressible navier stokes equations are presented

global well posedness of the velocity vorticity voigt model of the 3d navier stokes equations abstract the velocity vorticity formulation of the 3d navier stokes equations was recently found to give excellent numerical results for flows with strong rotation in this work we propose a new regularization of the 3d navier stokes equations which we call the 3d velocity vorticity voigt vvv model with a voigt regularization term added to momentum equation in velocity vorticity form but with no regularizing term in the vorticity equation we prove global well posedness and regularity of this model under periodic boundary conditions we prove convergence of the model s velocity and vorticity to their counterparts in the 3d navier stokes equations as the voigt modeling parameter tends to zero we prove that the curl of the model s velocity converges to the model vorticity which is solved for directly as the voigt modeling parameter tends to zero finally we provide a criterion for finite time blow up of the 3d navier stokes equations based on this inviscid regularization

connection conditions and the spectral family under singular potentials to describe a quantum system whose potential is divergent at one point one must provide proper connection conditions for the wavefunctions at the singularity generalizing the scheme used for point interactions in one dimension we present a set of connection conditions which are well defined even if the wavefunctions and or their derivatives are divergent at the singularity our generalized scheme covers the entire u 2 family of quantizations self adjoint hamiltonians admitted for the singular system we use this scheme to examine the spectra of the coulomb potential v x e2 x and the harmonic oscillator with square inverse potential v x m 2 2 x2 g x2 and thereby provide a general perspective for these models which have previously been treated with restrictive connection conditions resulting in conflicting spectra we further show that for any parity invariant singular potential v x v x the spectrum is determined solely by the eigenvalues of the characteristic matrix u u 2

ancestral graph markov models this paper introduces a class of graphical independence models that is closed under marginalization and conditioning but that contains all dag independence models this class of graphs called maximal ancestral graphs has two attractive features there is at most one edge between each pair of vertices every missing edge corresponds to an independence relation these features lead to a simple parameterization of the corresponding set of distributions in the gaussian case

multimodality of the likelihood in the bivariate seemingly unrelated regressions model summary we analyse the simplest two equation seemingly unrelated regressions model and demonstrate that its likelihood may have up to five stationary points and thus there may be up to three local modes consequently the estimates obtained via iterative estimation methods may depend on starting values we further show that the probability of multi modality vanishes asymptotically monte carlo simulations suggest that multimodality rarely occurs if the seemingly unrelated regressions model is true but can become more frequent if the model is misspecified the existence of multimodality in the likelihood for seemingly unrelated regressions models contradicts several claims in the literature

a connection between scott vogelius and grad div stabilized taylor hood fe approximations of the navier stokes equations this article studies two methods for obtaining excellent mass conservation in finite element computations of the navier stokes equations using continuous velocity fields with a particular mesh construction the scott vogelius element pair has recently been shown to be inf sup stable and have optimal approximation properties while also providing pointwise mass conservation we present herein the first numerical tests of this element pair for the time dependent navier stokes equations we also prove that the limit of the grad div stabilized taylor hood solutions to the navier stokes problem converges to the scott vogelius solution as the stabilization parameter tends to infinity that is we provide theoretical justification that choosing the grad div parameter large does not destroy the solution numerical tests are provided which verify the theory and show how both scott vogelius and grad div stabilized taylor hood with large stabilization parameter elements can provide accurate results with excellent mass conservation for navier stokes approximations

stability and convergence of the crank nicolson adams bashforth scheme for the time dependent navier stokes equations in this paper we study the stability and convergence of the crank nicolson adams bashforth scheme for the two dimensional nonstationary navier stokes equations a finite element method is applied for the spatial approximation of the velocity and pressure the time discretization is based on the crank nicolson scheme for the linear term and the explicit adams bashforth scheme for the nonlinear term moreover we present optimal error estimates and prove that the scheme is almost unconditionally stable and convergent i e stable and convergent when the time step is less than or equal to a constant

a gpav based unconditionally energy stable scheme for incompressible flows with outflow open boundaries abstract we present an unconditionally energy stable scheme for approximating the incompressible navier stokes equations on domains with outflow open boundaries the scheme combines the generalized positive auxiliary variable gpav approach and a rotational velocity correction type strategy and the adoption of the auxiliary variable simplifies the numerical treatment for the open boundary conditions the discrete energy stability of the proposed scheme has been proven irrespective of the time step sizes within each time step the scheme entails the computation of two velocity fields and two pressure fields by solving an individual de coupled helmholtz including poisson type equation with a constant pre computable coefficient matrix for each of these field variables the auxiliary variable being a scalar number is given by a well defined explicit formula within a time step which ensures the positivity of its computed values extensive numerical experiments with several flows involving outflow open boundaries in regimes where the backflow instability becomes severe have been presented to test the performance of the proposed method and to demonstrate its stability at large time step sizes

stability and convergence of the numerical solution for the species equation of a model for pemfcs abstract many mathematical models for fuel cells are presented in the literature however few studies of numerical analysis of these models are found the novelty of this work is the proof of stability and convergence of the equation of chemical species providing greater reliability to the mathematical model in this work a mathematical model is proposed that calculates the flow and concentration of species for proton exchange membrane fuel cells using the finite element method for spatial discretization and the crank nicolson method for temporal discretization the model takes into account the losses overpotentials at the anode and at the cathode the numerical results confirm the proof of convergence which is demonstrated in this work finally a numerical result is shown for the fuel cell power relative to current density and the result has good relation with the experimental data

what can we learn from the path equations identifiability constraints equivalence procedures are given for determining identified parameters finding constraints on the covariances and checking equivalence in acyclic recursive linear path models with correlated error terms disturbances by inspection of the path equations aided by simple recursions this provides a useful and general alternative to the employment of directed acyclic graph theory for such purposes

on the causal interpretation of acyclic mixed graphs under multivariate normality abstract in multivariate statistics acyclic mixed graphs with directed and bidirected edges are widely used for compact representation of dependence structures that can arise in the presence of hidden i e latent or unobserved variables indeed under multivariate normality every mixed graph corresponds to a set of covariance matrices that contains as a full dimensional subset the covariance matrices associated with a causally interpretable acyclic digraph this digraph generally has some of its nodes corresponding to hidden variables we seek to clarify for which mixed graphs there exists an acyclic digraph whose hidden variable model coincides with the mixed graph model restricting to the tractable setting of chain graphs and multivariate normality we show that decomposability of the bidirected part of the chain graph is necessary and sufficient for equality between the mixed graph model and some hidden variable model given by an acyclic digraph

likelihood ratio tests and singularities many statistical hypotheses can be formulated in terms of polynomial equalities and inequalities in the unknown parameters and thus correspond to semi algebraic subsets of the parameter space we consider large sample asymptotics for the likelihood ratio test of such hypotheses in models that satisfy standard probabilistic regularity conditions we show that the assumptions of chernoff s theorem hold for semi algebraic sets such that the asymptotics are determined by the tangent cone at the true parameter point at boundary points or singularities the tangent cone need not be a linear space and limiting distributions other than chi square distributions may arise while boundary points often lead to mixtures of chi square distributions singularities give rise to nonstandard limits we demonstrate that minima of chi square random variables are important for locally identifiable models and in a study of the factor analysis model with one factor we reveal connections to eigenvalues of wishart matrices

semiparametric theory for causal mediation analysis efficiency bounds multiple robustness and sensitivity analysis whilst estimation of the marginal total causal effect of a point exposure on an outcome is arguably the most common objective of experimental and observational studies in the health and social sciences in recent years investigators have also become increasingly interested in mediation analysis specifically upon evaluating the total effect of the exposure investigators routinely wish to make inferences about the direct or indirect pathways of the effect of the exposure not through or through a mediator variable that occurs subsequently to the exposure and prior to the outcome although powerful semiparametric methodologies have been developed to analyze observational studies that produce double robust and highly efficient estimates of the marginal total causal effect similar methods for mediation analysis are currently lacking thus this paper develops a general semiparametric framework for obtaining inferences about so called marginal natural direct and indirect causal effects while appropriately accounting for a large number of pre exposure confounding factors for the exposure and the mediator variables our analytic framework is particularly appealing because it gives new insights on issues of efficiency and robustness in the context of mediation analysis in particular we propose new multiply robust locally efficient estimators of the marginal natural indirect and direct causal effects and develop a novel double robust sensitivity analysis framework for the assumption of ignorability of the mediator variable

numerics and hydrodynamic stability toward error control in computational fluid dynamics we critically review the available error analysis in computational fluid dynamics cfd and come to the conclusion that the existing error estimates are meaningless in most cases of interest we propose a new approach to error analysis in cfd aiming at reliable and efficient adaptive quantitative error control this is based on a precise analysis of hydrodynamic stability coupled with galerkin orthogonality we prove a priori and a posteriori type error estimates in a model case for pipe flow formulate corresponding adaptive algorithms and discuss the potential of this approach for adaptive error control in cfd

numerical approximation of the voigt regularization for incompressible navier stokes and magnetohydrodynamic flows we study voigt regularizations for the navier stokes equations nses and magnetohydrodynamic mhd equations in the presence of physical boundary conditions in particular we develop the first finite element numerical algorithms for these systems prove stability and convergence of the algorithms and test them computationally on problems of practical interest it is found that unconditionally stable implementations of the voigt regularization can be made from a simple change to existing nse and mhd codes and moreover optimal convergence of the developed algorithms solutions to physical solutions can be obtained if lower order mixed finite elements are used finally we show that for several benchmark problems the voigt regularization on a coarse mesh produces good approximations to nse and mhd systems that is the voigt regularization provides accurate reduced order models for nse and mhd flows