a smooth transition model between kinetic and diffusion equations this paper presents a model which provides a smooth transition between a kinetic and a diffusion domain the idea is to use a buffer zone in which both diffusion and kinetic equations will be solved the solution of the original kinetic equation will be recovered as the sum of the solutions of these two equations we use an artificial connecting function which makes the equation on each domain degenerate at the end of the buffer zone thus no boundary condition is needed at the transition point this model avoids the delicate issue of finding the interface condition or iteration in a typical domain decomposition method that couples a kinetic equation with hydrodynamic equations a new asymptotic preserving method for this model is introduced and numerical examples are used to validate this new model and the new numerical method

an asymptotic preserving monte carlo method for the boltzmann equation abstract in this work we propose an asymptotic preserving monte carlo method for the boltzmann equation that is more efficient than the currently available monte carlo methods in the fluid dynamic regime this method is based on the successive penalty method 39 which is an improved bgk penalization method originally proposed by filbet and jin 16 here we introduce the monte carlo implementation of the method which despite its lower order accuracy is very efficient in higher dimensions or simulating some complicated chemical processes this method allows the time step independent of the mean free time which is prohibitively small in the fluid dynamic regime we study some basic properties of this method and compare it with some other asymptotic preserving monte carlo methods in terms of numerical performance in different regimes from rarefied to fluid dynamic regimes and their computational efficiency

logarithmically improved regularity criterion for the nematic liquid crystal flows in b 1 space abstract in this work we study the regularity criterion of the three dimensional nematic liquid crystal flows it is proved that if the vorticity satisfies 0 t t b 1 2 1 log e t b 1 d t where b 1 denotes the critical besov space then the solution u d becomes a regular solution on 0 t this result extends the recent regularity criterion obtained by fan and ozawa 2012 11

a one dimensional analysis for the probability of error of linear classifiers for normally distributed classes computing the probability of error is an important problem in evaluating classifiers when dealing with normally distributed classes this problem becomes intricate due to the fact that there is no closed form expression for integrating the probability density function in this paper we derive lower and upper bounds for the probability of error for a linear classifier where the random vectors representing the underlying classes obey the multivariate normal distribution the expression of the error is derived in the one dimensional space independently of the dimensionality of the original problem based on the two bounds we propose an approximating expression for the error of a generic linear classifier in particular we derive the corresponding bounds and the expression for approximating the error of fisher s classifier our empirical results on synthetic data including up to two hundred dimensional featured samples show that the computations for the error are extremely fast and quite accurate it differs from the actual error in at most e 0 0184340683 the scheme has also been successfully tested on real life data sets drawn from the uci machine learning repository

a novel series expansion for the multivariate normal probability integrals based on fourier series abstract in this article we derive a series expansion of the multivariate normal probability integrals based on fourier series the basic idea is to transform the limits of each integral from hi to to be from to by multiplying the integrand by a periodic square wave that approximates the domain of the integral this square wave is expressed by its fourier series expansion then a cholesky decomposition of the covariance matrix is applied to transform the integrand to a simple one that can be easily evaluated the resultant formula has a simple pattern that is expressed as multiple series expansion of trigonometric and exponential functions

the evaluation of general non centred orthant probabilities summary the evaluation of the cumulative distribution function of a multivariate normal distribution is considered the multivariate normal distribution can have any positive definite correlation matrix and any mean vector the approach taken has two stages in the first stage it is shown how non centred orthoscheme probabilities can be evaluated by using a recursive integration method in the second stage some ideas of schlafli and abrahamson are extended to show that any non centred orthant probability can be expressed as differences between at most m 1 non centred orthoscheme probabilities this approach allows an accurate evaluation of many multivariate normal probabilities which have important applications in statistical practice

an asymptotic preserving monte carlo method for the multispecies boltzmann equation an asymptotic preserving ap scheme is efficient in solving multiscale kinetic equations with a wide range of the knudsen number in this paper we generalize the asymptotic preserving monte carlo method ap dsmc developed in 25 to the multispecies boltzmann equation this method is based on the successive penalty method 26 originated from the bgk penalization based ap scheme developed in 7 for the multispecies boltzmann equation the penalizing maxwellian should use the unified maxwellian as suggested in 12 we give the details of ap dsmc for multispecies boltzmann equation show its ap property and verify through several numerical examples that the scheme can allow time step much larger than the mean free time thus making it much more efficient for flows with possibly small knudsen numbers than the classical dsmc

an implicit monte carlo method for rarefied gas dynamics for the space homogeneous boltzmann equation we formulate a hybrid monte carlo method that is robust in the fluid dynamic limit this method is based on an analytic representation of the solution over a single time step and involves implicit time differencing derived from a suitable power series expansion of the solution a generalized wild expansion a class of implicit yet explicitly implementable numerical schemes is obtained by substituting a maxwellian distribution in place of the high order terms in the expansion the numerical solution is represented as a convex combination of a non equilibrium particle distribution and a maxwellian the hybrid distribution is then evolved by monte carlo using the implicit formulation for the time evolution computational simulations of spatially homogeneous problems by our method are presented here for the kac model and for the variable hard sphere model including maxwell molecules comparison to exact solutions and to direct simulation monte carlo dsmc computations shows the robustness and the efficiency of the new method

nonparabolic dissipative systems modeling the flow of liquid crystals we study a simplified system which retains most of the interesting mathematical properties of the original ericksen leslie equations for the flow of liquid crystals this is a coupled nonparabolic dissipative dynamic system we derive several energy laws which enable us to prove the global existence of the weak solutions and the classical solutions we also discuss uniqueness and some stability properties of the system 1995 john wiley sons inc

nonlinear theory of defects in nematic liquid crystals phase transition and flow phenomena this is a summary of some recent mathematical advances in the theory of defects in nematic liquid crystals it also includes some new results concerning disclinations line defects which have not been published elsewhere

a fast iterative scheme for the linearized boltzmann equation iterative schemes to find steady state solutions to the boltzmann equation is efficient for highly rarefied gas flows but can be very slow to converge in the near continuum flow regime in this paper a synthetic iterative scheme is developed to speed up the solution of the linearized boltzmann equation by penalizing the collision operator l into the form l l n delta h n delta h where delta is the gas rarefaction parameter h is the velocity distribution function and n is a tuning parameter controlling the convergence rate the velocity distribution function is first solved by the conventional iterative scheme then it is corrected such that the macroscopic flow velocity is governed by a diffusion type equation that is asymptotic preserving into the navier stokes limit the efficiency of this new scheme is assessed by calculating the eigenvalue of the iteration as well as solving for poiseuille and thermal transpiration flows we find that the fastest convergence of our synthetic scheme for the linearized boltzmann equation is achieved when n delta is close to the average collision frequency the synthetic iterative scheme is significantly faster than the conventional iterative scheme in both the transition and the near continuum gas flow regimes moreover due to its asymptotic preserving properties the synthetic iterative scheme does not need high spatial resolution in the near continuum flow regime which makes it even faster than the conventional iterative scheme using this synthetic scheme with the fast spectral approximation of the linearized boltzmann collision operator poiseuille and thermal transpiration flows between two parallel plates through channels of circular rectangular cross sections and various porous media are calculated over the whole range of gas rarefaction finally the flow of a ne ar gas mixture is solved based on the linearized boltzmann equation with the lennard jones intermolecular potential for the first time and the difference between these results and those using the hard sphere potential is discussed

a new reconstruction of multivariate normal orthant probabilities a new method is introduced for geometrically reconstructing orthant probabilities for non singular multivariate normal distributions orthant probabilities are expressed in terms of those for auto regressive sequences and an efficient method is developed for numerical approximation of the latter the approach allows more efficient accurate evaluation of the multivariate normal cumulative distribution function than previously for many situations where the original distribution arises from a graphical model an implementation is available as a package for the statistical software r and an application is given to multivariate probit models

conjugate bayes for probit regression via unified skew normal distributions regression models for dichotomous data are ubiquitous in statistics besides being useful for inference on binary responses these methods serve also as building blocks in more complex formulations such as density regression nonparametric classification and graphical models within the bayesian framework inference proceeds by updating the priors for the coefficients typically set to be gaussians with the likelihood induced by probit or logit regressions for the responses in this updating the apparent absence of a tractable posterior has motivated a variety of computational methods including markov chain monte carlo routines and algorithms which approximate the posterior despite being routinely implemented markov chain monte carlo strategies face mixing or time inefficiency issues in large p and small n studies whereas approximate routines fail to capture the skewness typically observed in the posterior this article proves that the posterior distribution for the probit coefficients has a unified skew normal kernel under gaussian priors such a novel result allows efficient bayesian inference for a wide class of applications especially in large p and small to moderate n studies where state of the art computational methods face notable issues these advances are outlined in a genetic study and further motivate the development of a wider class of conjugate priors for probit models along with methods to obtain independent and identically distributed samples from the unified skew normal posterior

efficient asymptotic preserving ap schemes for some multiscale kinetic equations many kinetic models of the boltzmann equation have a diffusive scaling that leads to the navier stokes type parabolic equations as the small scaling parameter approaches zero in practical applications it is desirable to develop a class of numerical schemes that can work uniformly with respect to this relaxation parameter from the rarefied kinetic regimes to the hydrodynamic diffusive regimes an earlier approach in s jin l pareschi and g toscani siam j numer anal 35 1998 pp 2405 2439 reformulates such systems into the common hyperbolic relaxation system by jin and xin for hyperbolic conservation laws used to construct the relaxation schemes and then uses a multistep time splitting method to solve the relaxation system here we observe that the combination of the two time split steps may yield hyperbolic parabolic systems that are more advantageous in both stability and efficiency for numerical computations we show that such an approach yields a class of asymptotic preserving ap schemes which are suitable for the computation of multiscale kinetic problems we use the goldstein taylor and carleman models to illustrate this approach

the normal law under linear restrictions simulation and estimation via minimax tilting summary simulation from the truncated multivariate normal distribution in high dimensions is a recurrent problem in statistical computing and is typically only feasible by using approximate markov chain monte carlo sampling we propose a minimax tilting method for exact independently and identically distributed data simulation from the truncated multivariate normal distribution the new methodology provides both a method for simulation and an efficient estimator to hitherto intractable gaussian integrals we prove that the estimator has a rare vanishing relative error asymptotic property numerical experiments suggest that the scheme proposed is accurate in a wide range of set ups for which competing estimation schemes fail we give an application to exact independently and identically distributed data simulation from the bayesian posterior of the probit regression model

regression to the mean for nonnormal populations summary regression to the mean is the term used to describe the effect by which individuals selected on the basis of an initial measurement which is particularly high or low compared with other members of the population have upon remeasurement values nearer to the population mean for the normal distribution simple expressions for estimates of the magnitude of this effect are available many biological variables however are not normally distributed estimates for regression to the mean for nonnormal data are given based on edgeworth series and saddlepoint approximations

a class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources in this paper we propose a general time discrete framework to design asymptotic preserving schemes for initial value problem of the boltzmann kinetic and related equations numerically solving these equations are challenging due to the nonlinear stiff collision source terms induced by small mean free or relaxation time we propose to penalize the nonlinear collision term by a bgk type relaxation term which can be solved explicitly even if discretized implicitly in time moreover the bgk type relaxation operator helps to drive the density distribution toward the local maxwellian thus naturally imposes an asymptotic preserving scheme in the euler limit the scheme so designed does not need any nonlinear iterative solver or the use of wild sum it is uniformly stable in terms of the possibly small knudsen number and can capture the macroscopic fluid dynamic euler limit even if the small scale determined by the knudsen number is not numerically resolved it is also consistent to the compressible navier stokes equations if the viscosity and heat conductivity are numerically resolved the method is applicable to many other related problems such as hyperbolic systems with stiff relaxation and high order parabolic equations

gaussian process single index models as emulators for computer experiments a single index model sim provides for parsimonious multidimensional nonlinear regression by combining parametric linear projection with univariate nonparametric nonlinear regression models we show that a particular gaussian process gp formulation is simple to work with and ideal as an emulator for some types of computer experiment as it can outperform the canonical separable gp regression model commonly used in this setting our contribution focuses on drastically simplifying reinterpreting and then generalizing a recently proposed fully bayesian gp sim combination favorable performance is illustrated on synthetic data and a real data computer experiment two r packages both released on cran have been augmented to facilitate inference under our proposed model s

a further note on the regularity criterion for the 3d nematic liquid crystal flows we give a further investigation on the regularity criteria for the 3d nematic liquid crystal flows it is show that the obtained anisotropic integrability regularity criterion to the 3d nematic liquid crystal flows involving i u j i j 1 2 3 is an improvement of qian 2016 besides we also prove the solution is regular if some components of the velocity field and the vectorial field modeling the orientation of the crystal molecules satisfy the prodi serrin type conditions

a successive penalty based asymptotic preserving scheme for kinetic equations we propose an asymptotic preserving ap scheme for kinetic equations that is efficient also in the hydrodynamic regimes this scheme is based on the bhantnagar gross krook bgk penalty method introduced by filbet and jin j comput phys 229 2010 pp 7625 7648 but uses the penalization successively to achieve the desired asymptotic property this method possesses a stronger ap property than the original method of filbet and jin with the additional feature of being positivity preserving when applied on the boltzmann equation it is also general enough to be applicable to several important classes of kinetic equations including the boltzmann equation and the landau equation numerical experiments verify these properties

exponential runge kutta methods for stiff kinetic equations we introduce a class of exponential runge kutta integration methods for kinetic equations the methods are based on a decomposition of the collision operator into an equilibrium and a nonequilibrium part and are exact for relaxation operators of bgk type for boltzmann type kinetic equations they work uniformly for a wide range of relaxation times and avoid the solution of nonlinear systems of equations even in stiff regimes we give sufficient conditions in order that such methods are unconditionally asymptotically stable and asymptotic preserving such stability properties are essential to guarantee the correct asymptotic behavior for small relaxation times the methods also offer favorable properties such as nonnegativity of the solution and entropy inequality for this reason as we will show the methods are suitable both for deterministic as well as probabilistic numerical techniques

time relaxed monte carlo methods for the boltzmann equation a new family of monte carlo schemes is introduced for the numerical solution of the boltzmann equation of rarefied gas dynamics the schemes are inspired by the wild sum expansion of the solution of the boltzmann equation for maxwellian molecules and consist of a novel time discretization of the equation in particular high order terms in the expansion are replaced by the equilibrium maxwellian distribution the two main features of the schemes are high order accuracy in time and asymptotic preservation the first property allows to recover accurate solutions with time steps larger than those required by direct simulation monte carlo dsmc while the latter guarantees that for the vanishing knudsen number the numerical solution relaxes to the local maxwellian conservation of mass momentum and energy are preserved by the scheme numerical results on several space homogeneous problems show the improvement of the new schemes over standard dsmc applications to a one dimensional shock wave problem are also presented

logarithmically improved regularity criteria for the navier stokes equations in multiplier spaces abstract the aim of this paper is to establish some logarithmically improved regularity criteria in term of the multiplier spaces to the navier stokes equations

relaxation schemes for nonlinear kinetic equations a class of numerical schemes for nonlinear kinetic equations of boltzmann type is described following wild s approach the solution is represented as a power series with parameter depending exponentially on the knudsen number this permits us to derive accurate and stable time discretizations for all ranges of the mean free path these schemes preserve the main physical properties positivity conservation of mass momentum and energy moreover for some particular models the entropy property is also shown to hold

remarks on the regularity criterion for the nematic liquid crystal flows in r 3 in this paper we investigate the regularity criteria for the 3d nematic liquid crystal flows in terms of the components of the velocity u or the gradient of the velocity field u more precisely the anisotropic integrability regularity criterion to the 3d nematic liquid crystal flows involving i u i 1 2 3 is obtained firstly and then it is proved that the solution is regular if u3 3 or 3u3 3 satisfies the prodi serrin type conditions

existence and uniqueness of mild solutions to the incompressible nematic liquid crystal flow abstract in this paper we consider the incompressible nematic liquid crystal flow in the whole space and obtain the existence of global mild solutions with small initial data in the scaling invariant space the main tool we have used is the implicit function theorem furthermore we establish the asymptotic stability of solutions when the time goes to infinity and in view of our construction of solutions in the weak l p spaces we finally derive the existence of self similar solutions provided the initial data are small homogeneous functions

a smooth transition model between kinetic and hydrodynamic equations this paper presents a model which provides a smooth transition between a kinetic and a hydrodynamic domain the idea is to use a buffer zone in which both hydrodynamics and kinetic equations will be solved the solution of the original kinetic equation will be recovered as the sum of the solutions of these two equations we use an artificial connecting function which makes the equation on each domain degenerate at the end of the buffer zone thus no boundary condition is needed at the transition point consequently this model avoids the delicate issue of finding the interface condition in a typical domain decomposition method that couples a kinetic equation with hydrodynamic equations a simple kinetic scheme is developed to discretize our model and numerical examples are used to validate the method