analysis of asymptotic preserving dg imex schemes for linear kinetic transport equations in a diffusive scaling in this paper some theoretical aspects will be addressed for the asymptotic preserving discontinuous galerkin implicit explicit dg imex schemes recently proposed in j jang f li j m qiu and t xiong high order asymptotic preserving dg imex schemes for discrete velocity kinetic equations in a diffusive scaling http arxiv org abs 1306 0227 2013 submitted for kinetic transport equations under a diffusive scaling we will focus on the methods that are based on discontinuous galerkin dg spatial discretizations with the p k polynomial space and a first order implicit explicit imex temporal discretization and apply them to two linear models the telegraph equation and the one group transport equation in slab geometry in particular we will establish uniform numerical stability with respect to knudsen number varepsilon using energy methods as well as error estimates for any given varepsilon when varepsilon rightarrow 0 a rigorous asymptotic analysis of the schemes is also obta

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

the radiative transfer equations diffusion approximation under accretiveness and compactness assumptions abstract when the mean free path of photons is small the radiative transfer equations may be approximated by a nonlinear angular diffusion equation in order to find the diffusion equation one uses accretiveness assumptions but theses assumptions impose some conditions on the opacity to overcome this inconvenient we propose to study the solution of the radiative transfer equations by compactness techniques

on fluid limit for the semiconductors boltzmann equation abstract this paper is devoted to the derivation of non linear drift diffusion equations from the semiconductor boltzmann equation collisions are taken into account through the non linear pauli operator but we do not assume relation on the cross section such as the so called detailed balance principle in turn equilibrium states are implicitly defined this article follows and completes the contribution of mellet monatsh math 134 4 2002 305 329 where the electric field is given and does not depend on time here we treat the self consistent problem the electric potential satisfying the poisson equation by means of a hilbert expansion we shall formally derive the asymptotic model in the general case we shall then rigorously prove the convergence in the one dimensional case by using a modified hilbert expansion

regularity of the moments of the solution of a transport equation let u u x v satisfy the transport equation u v xu f x rn v rnwhere f belongs to some space of type lp dx d v where is a positive bounded measure on rn we study the resulting regularity of the moment u x v d v in terms of fractional sobolev spaces for example counter examples are given in order to test the optimality of our results

two dimensional approximations of three dimensional eigenvalue problems in plate theory the eigenvalues and eigenfunctions corresponding to the three dimensional equations for the linear elastic equilibrium of a clamped plate of thickness 2 are shown to converge in a specific sense to the eigenvalues and eigenfunctions of the well known two dimensional biharmonic operator of plate theory as approaches zero in the process it is found in particular that the displacements and stresses are indeed of the specific forms usually assumed a priori in the literature it is also shown that the limit eigenvalues and eigenfunctions can be equivalently characterized as the leading terms in an asymptotic expansion of the three dimensional solutions in terms of powers of the method presented here applies equally well to the stationary problem of linear plate theory as shown elsewhere by p destuynder

on a system of nonlinear boltzmann equations of semiconductor physics a system of kinetic equations of semiconductors is studied these are vlasov poisson equations with additional nonlinear boltzmann operators first h theorems for these integral operators are given and their null space is determined then the existence and uniqueness of smooth solutions is proven the proof relies on the construction of a sequence of solutions for a suitable linearized system which converges strongly

large scale dynamics of the persistent turning walker model of fish behavior this paper considers a new model of individual displacement based on fish motion the so called persistent turning walker ptw model which involves an ornstein uhlenbeck process on the curvature of the particle trajectory the goal is to show that its large time and space scale dynamics is of diffusive type and to provide an analytic expression of the diffusion coefficient two methods are investigated in the first one we compute the large time asymptotics of the variance of the individual stochastic trajectories the second method is based on a diffusion approximation of the kinetic formulation of these stochastic trajectories the kinetic model is a fokker planck type equation posed in an extended phase space involving the curvature among the kinetic variables we show that both methods lead to the same value of the diffusion constant we present some numerical simulations to illustrate the theoretical results

variance component testing in generalised linear models with random effects summary there is considerable interest in testing for overdispersion correlation and heterogeneity across groups in biomedical studies in this paper we cast the problem in the framework of generalised linear models with random effects we propose a global score test for the null hypothesis that all the variance components are zero this test is a locally asymptotically most stringent test and is robust in the special sense that the test does not require specifying the joint distribution of the random effects we also propose individual score tests and their approximations for testing the variance components separately both tests can be easily implemented using existing statistical software we illustrate these tests with an application to the study of heterogeneity of mating success across males and females in an experiment on salamander matings and evaluate their performance through simulation

restricted p isometry properties of nonconvex matrix recovery recently a nonconvex relaxation of low rank matrix recovery lmr called the schatten p quasi norm minimization 0 p 1 was introduced instead of the previous nuclear norm minimization in order to approximate the problem of lmr closer in this paper we introduce a notion of the restricted p isometry constants 0 p 1 and derive a p rip condition for exact reconstruction of lmr via schatten p quasi norm minimization in particular we determine how many random gaussian measurements are needed for the p rip condition to hold with high probability which gives a theoretical result that it needs fewer measurements with small p for exact recovery via schatten p quasi norm minimization than when p 1

a multivariate two sample mean test for small sample size and missing data we develop a new statistic for testing the equality of two multivariate mean vectors a scaled chi squared distribution is proposed as an approximating null distribution because the test statistic is based on componentwise statistics it has the advantage over hotelling s t2 test of being applicable to the case where the dimension of an observation exceeds the number of observations an appealing feature of the new test is its ability to handle missing data by relying on only componentwise sample moments monte carlo studies indicate good power compared to hotelling s t2 and a recently proposed test by srivastava 2004 technical report university of toronto the test is applied to drug discovery data

initial and boundary conditions for diffusive linear transport problems the initial and boundary layer analyses of an asymptotic expansion that yields a diffusive description of linear particle transport are carried out in some generality this yields initial and boundary conditions that apply to the diffusion equation previously reported in the literature effects treated include boundary curvature the variation of the transport boundary condition along the bounding surface and spatial and temporal variations of the interaction coefficients cross sections in the initial and boundary layers

on the theorem of uniform recovery of random sampling matrices we consider two theorems from the theory of compressive sensing mainly a theorem concerning uniform recovery of random sampling matrices where the number of samples needed in order to recover an s sparse signal from linear measurements with high probability is known to be m s ln s 3 lnn we present new and improved constants together with what we consider to be a more explicit proof a proof that also allows for a slightly larger class of m n matrices by considering what is called effective sparsity we also present a condition on the so called restricted isometry constants s ensuring sparse recovery via l 1 minimization we show that is sufficient and that this can be improved further to almost allow for a sufficient condition of the type

a new asymptotic preserving scheme based on micro macro formulation for linear kinetic equations in the diffusion limit we propose a new numerical scheme for linear transport equations it is based on a decomposition of the distribution function into equilibrium and nonequilibrium parts we also use a projection technique that allows us to reformulate the kinetic equation into a coupled system of an evolution equation for the macroscopic density and a kinetic equation for the nonequilibrium part by using a suitable time semi implicit discretization our scheme is able to accurately approximate the solution in both kinetic and diffusion regimes it is asymptotic preserving in the following sense when the mean free path of the particles is small our scheme is asymptotically equivalent to a standard numerical scheme for the limit diffusion model a uniform stability property is proved for the simple telegraph model various boundary conditions are studied our method is validated in one dimensional cases by several numerical tests and comparisons with previous asymptotic preserving schemes

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

control of modeling errors in the coupling of linear transport and diffusion models we consider in this paper the initial boundary value problem for the 1d neutron transport equation with isotropic scattering set in some bounded interval with inflow boundary conditions the usual parabolic scaling yields the diffusive limit a surrogate model coupling transport and diffusion equations is then introduced in order to accurately assess the value of specific quantities of interest the control of the quality of the computation with respect to such a quantity of interest is performed by means of an approach for modeling error estimation combined with an adaptive strategy to control the surrogate model if necessary

analysis of an asymptotic preserving scheme for linear kinetic equations in the diffusion limit we present a mathematical analysis of the asymptotic preserving scheme proposed in m lemou and l mieussens siam j sci comput 31 2008 pp 334 368 for linear transport equations in kinetic and diffusive regimes we prove that the scheme is uniformly stable and accurate with respect to the mean free path of the particles this property is satisfied under an explicitly given cfl condition this condition tends to a parabolic cfl condition for small mean free paths and is close to a convection cfl condition for large mean free paths our analysis is based on very simple energy estimates

diffusion approximation and computation of the critical size this paper is devoted to the mathematical definition of the extrapolation length which appears in the diffusion approximation to obtain this result we describe the spectral properties of the transport equation and we show how the diffusion approximation is related to the computation of the critical size the paper also contains some simple numerical examples and some new results for the milne problem introduction the computation of the critical size and the diffusion approximation for the transport equation have been closely related and this is due to the following facts first the computation of the critical size is much easier for the diffusion approximation than for the original transport equation second for the critical size one can consider a host media x nxq with x0 given and 17 a positive number the problem of the critical size is then reduced to the computation of the parameter zj it turns out that when the transport operator is almost conservative the critical value of the parameter 17 is large and it is exactly for this range of value that the diffusion approximation is accurate on the other hand the physical boundary condition for the diffusion approxi mation is of the form

the brownian motion as the limit of a deterministic system of hard spheres we provide a rigorous derivation of the brownian motion as the limit of a deterministic system of hard spheres as the number of particles n goes to infinity and their diameter varepsilon simultaneously goes to 0 in the fast relaxation limit alpha n varepsilon d 1 rightarrow infty with a suitable diffusive scaling of the observation time as suggested by hilbert in his sixth problem we rely on a kinetic formulation as an intermediate level of description between the microscopic and the fluid descriptions we use indeed the linear boltzmann equation to describe one tagged particle in a gas close to global equilibrium our proof is based on the fundamental ideas of lanford the main novelty here is the detailed study of the branching process leading to explicit estimates on pathological collision trees

semiparametric regression for clustered data we consider estimation in a semiparametric partially generalised linear model for clustered data using estimating equations a marginal model is assumed where the mean of the outcome variable depends on some covariates parametrically and a cluster level covariate nonparametrically a profile kernel method allowing for working correlation matrices is developed we show that the nonparametric part of the model can be estimated using standard nonparametric methods including smoothing parameter estimation and the parametric part of the model can be estimated in a profile fashion the asymptotic distributions of the parameter estimators are derived and the optimal estimators of both the nonparametric and parametric parts are shown to be obtained when the working correlation matrix equals the actual correlation matrix the asymptotic covariance matrix of the parameter estimator is consistently estimated by the sandwich estimator we show that the semiparametric efficient score takes on a simple form and our profile kernel method is semiparametric efficient the results for the case where the nonparametric part of the model is an observation level covariate are noted to be dramatically different

the milne problem for the radiative transfer equations with frequency dependence we study the following stationary frequency dependent transport equation 28xf a t f bv t o x o m o u e 1 1 tr t bv t fjdm 8 0 r x 1 1 2 f 2 w 2 0 u e j0 1 where bv is the well known planck function appearing in astrophysics we are able to describe the asymptotic behavior of f and t for x large when a t is of the special form a t tr m k t our method relies mainly on the monotonicity of the nonlinearity the proof does not use any linearization of the equation in particular no smallness assumption on the data yp in any sense is required resume nous etudions l equation de transport stationnaire avec dependance en frequence 2dxf a t f bv t o x o m o u e 1 1 tr t bv t fjdm 8 0 r x 1 1 2 8 2 w 2 o u e 0 1 lorsque a t est de la forme particuliere a t a m k t nous savons decrire le comportement asymptotique de f et t pour x grand notre methode repose principalement sur la monotonie de la non linearite la preuve n utilise aucune linearisation de l equation en particulier nous n avons besoin d aucune hypothese de petitesse d aucune sorte sur la donnee y

asymptotic analysis and coupling conditions for kinetic and hydrodynamic equations abstract this article is concerned with domain decomposition for kinetic and hydrodynamic equations coupling conditions at the interface between the equations are developed and investigated in particular for nonequilibrium situations at the interface new coupling conditions are developed by considering interface layers this leads to kinetic half space problems a fast procedure to solve these problems is given

the effects of mixture distribution misspecification when fitting mixed effects logistic models summary mixed effects logistic models are often used to analyze binary response data which have been gathered in clusters or groups responses are assumed to follow a logistic model within clusters with an intercept which varies across clusters according to a specified probability distribution g in this paper we examine the performance of mixed effects logistic regression analysis when a main component of the model the mixture distribution is misspecified we show that when the mixture distribution is misspecified estimates of model parameters including the effects of covariates typically are asymptotically biased i e inconsistent however we present some approximations which suggest that the magnitude of the bias in the estimated covariate effects is typically small these findings are corroborated by a set of simulations which also suggest that valid variance estimates of estimated covariate effects can be obtained when the mixture distribution is misspecified

moment methods for the semiconductor boltzmann equation on bounded position domains galerkin methods for the semiconductor boltzmann equation based on moment expansions for a discretization in the velocity direction are studied for the moment equations boundary conditions are proposed which are analogues to inflow and respectively reflecting boundary conditions for the boltzmann equation stability and an error estimate are proved for an expansion in terms of hermite polynomials finally an adaptive numerical implementation is introduced and results of numerical experiments are presented

the nonaccretive radiative transfer equations existence of solutions and rosseland approximation we present an existence theory and an asymptotic analysis for the radiative transfer equations ue t xuee ue e2 ue ue 0 inx 1 ue x sn k ue t 0 u0 where u u t x t r x x rn 1 sn and ue t x 1 sn ue t x d we prove that even if has a singularity 0 1 has a solution ue l r x sn as e 0 we show that ue converges pointwise to a function u l r x solution of the degenerate parabolic equation u t f u 0 in x u x k u t 0 u0 this is achieved without any monotonicity assumption on and therefore one cannot use the theory of nonlinear contraction semigroups

misspecified maximum likelihood estimates and generalised linear mixed models summary we investigate the impact of model violations on the estimate of a regression coefficient in a generalised linear mixed model specifically we evaluate the asymptotic relative bias that results from incorrect assumptions regarding the random effects we compare the impact of model violation for two parameterisations of the regression model substantial bias in the conditionally specified regression point estimators can result from using a simple random intercepts model when either the random effects distribution depends on measured covariates or there are autoregressive random effects a marginally specified regression structure that is estimated using maximum likelihood is much less susceptible to bias resulting from random effects model misspecification

new models for the solution of intermediate regimes in transport theory and radiative transfer existence theory positivity asymptotic analysis and approximations in this paper we propose derive and establish the mathematical foundations of new models for the solution of intermediate regimes in transport theory and radiative transfer these new models consist of coupling the transport equations with their diffusion approximations our mathematical theory includes the existence theory the positivity of the solutions and the asymptotic analysis we also give the rate of the asymptotic decay in order to solve the new coupled problem we propose to use the transmission time marching algorithm introduced and studied in refs 10 13 15 we then study the convergence of the resulting algorithm these studies are based in an essential way on the methods we introduced in refs 14 15

does ell _ p minimization outperform ell _ 1 minimization in many application areas ranging from bioinformatics to imaging we are faced with the following question can we recover a sparse vector x o in mathbb r n from its undersampled set of noisy observations y in mathbb r n y a x o w the last decade has witnessed a surge of algorithms and theoretical results to address this question one of the most popular schemes is the ell p regularized least squares given by the following formulation hat x gamma p in arg min x 1 2 y ax 2 2 gamma x p p where p in 0 1 among these optimization problems the case p 1 also known as lasso is the best accepted in practice for the following two reasons first thanks to the extensive studies performed in the fields of high dimensional statistics and compressed sensing we have a clear picture of lasso s performance second it is convex and efficient algorithms exist for finding its global minima unfortunately neither of the above two properties hold for 0 leq p however they are still appealing because of the following folklores in the high dimensional statistics first hat x gamma p is closer to x o than hat x gamma 1 second if we employ iterative methods that aim to converge to a local minima of arg min x 1 2 y ax 2 2 gamma x p p then under good initialization these algorithms converge to a solution that is still closer to x o than hat x gamma 1 in spite of the existence of plenty of empirical results that support these folklore theorems the theoretical progress to establish them has been very limited this paper aims to study the above mentioned folklore theorems and establish their scope of validity starting with approximate message passing amp algorithm as a heuristic method for solving ell p regularized least squares we study the following questions first what is the impact of initialization on the performance of the algorithm second when does the algorithm recover the sparse signal x o under a good initialization third when does the algorithm converge to the sparse signal regardless of the initialization studying these questions will not only shed light on the second folklore theorem but also lead us to the answer the first one i e the performance of the global optima hat x gamma p for that purpose we employ the replica analysis 1 to show the connection between the solution of amp and hat x gamma p in the asymptotic settings this enables us to compare the accuracy of hat x gamma p and hat x gamma 1 in particular we will present an accurate characterization of the phase transition and noise sensitivity of ell p regularized least squares for every 0 leq p leq 1 our results in the noiseless setting confirm that ell p regularized least squares if gamma is tuned optimally exhibits the same phase transition for every 0 leq p and this phase transition is much better than that of lasso furthermore we show that in the noisy setting there is a major difference between the performance of ell p regularized least squares with different values of p for instance we will show that for very small and very large measurement noises p 0 and p 1 outperform the other values of p respectively 1 replica method is a widely accepted heuristic in statistical physics for analyzing large disordered systems

diagnosis of random effect model misspecification in generalized linear mixed models for binary response summary generalized linear mixed models glmms are widely used in the analysis of clustered data however the validity of likelihood based inference in such analyses can be greatly affected by the assumed model for the random effects we propose a diagnostic method for random effect model misspecification in glmms for clustered binary response we provide a theoretical justification of the proposed method and investigate its finite sample performance via simulation the proposed method is applied to data from a longitudinal respiratory infection study

diagnosing misspecification of the random effects distribution in mixed models it is traditionally assumed that the random effects in mixed models follow a multivariate normal distribution making likelihood based inferences more feasible theoretically and computationally however this assumption does not necessarily hold in practice which may lead to biased and unreliable results we introduce a novel diagnostic test based on the so called gradient function proposed by verbeke and molenberghs 2013 to assess the random effects distribution we establish asymptotic properties of our test and show that under a correctly specified model the proposed test statistic converges to a weighted sum of independent chi squared random variables each with one degree of freedom the weights which are eigenvalues of a square matrix can be easily calculated we also develop a parametric bootstrap algorithm for small samples our strategy can be used to check the adequacy of any distribution for random effects in a wide class of mixed models including linear mixed models generalized linear mixed models and non linear mixed models with univariate as well as multivariate random effects both asymptotic and bootstrap proposals are evaluated via simulations and a real data analysis of a randomized multicenter study on toenail dermatophyte onychomycosis

on the error of estimating the sparsest solution of underdetermined linear systems let a be an n m matrix with m n and suppose that the underdetermined linear system as x admits a sparse solution s0 for which s0 0 1 2 spark a such a sparse solution is unique due to a well known uniqueness theorem suppose now that we have somehow a solution s as an estimation of s0 and suppose that s is only approximately sparse that is many of its components are very small and nearly zero but not mathematically equal to zero is such a solution necessarily close to the true sparsest solution more generally is it possible to construct an upper bound on the estimation error s s0 2 without knowing s0 the answer is positive and in this paper we construct such a bound based on minimal singular values of submatrices of a we will also state a tight bound which is more complicated but besides being tight enables us to study the case of random dictionaries and obtain probabilistic upper bounds we will also study the noisy case that is where x as n moreover we will see that where s0 0 grows to obtain a predetermined guaranty on the maximum of s s0 2 s is needed to be sparse with a better approximation this can be seen as an explanation to the fact that the estimation quality of sparse recovery algorithms degrades where s0 0 grows

fractional diffusion limit for collisional kinetic equations this paper is devoted to diffusion limits of linear boltzmann equations when the equilibrium distribution function is a maxwellian distribution it is well known that for an appropriate time scale the small mean free path limit gives rise to a diffusion equation in this paper we consider situations in which the equilibrium distribution function is a heavy tailed distribution with infinite variance we then show that for an appropriate time scale the small mean free path limit gives rise to a fractional diffusion equation

computation of interface reflection and regular or diffuse transmission of the planar symmetric radiative transfer equation with isotropic scattering and its diffusion limit in this paper we develop a numerical scheme for the interface problem in the planar symmetric radiative transfer equation with isotropic scattering such problems arise in the modeling of the propagation of energy density for waves in heterogeneous media with weak random fluctuation in the high frequency regime the idea following the earlier work of jin and wen for regular transmission and reflection is to build the interface condition which characterizes the reflection and transmission into the numerical flux the new contribution of this article is to deal with the diffuse transmission at the interface the positivity of the scheme is proven moreover we show via numerical examples that this new scheme is able to capture the correct regular or diffuse transmission and reflection through the interface and as the mean free path goes to zero capture the diffusion limit with the correct interface conditions

multiple description l_ 1 compression multiple descriptions mds is a method to obtain reliable signal transmissions on erasure channels an md encoder forms several descriptions of the signal and each description is independently transmitted across an erasure channel the reconstruction quality then depends on the set of received descriptions in this paper we consider the design of redundant descriptions in an md setup using l1 minimization with euclidean distortion constraints in this way we are able to obtain sparse descriptions using convex optimization the proposed method allows for an arbitrary number of descriptions and supports both symmetric and asymmetric distortion design we show that mds with partial overlapping information corresponds to enforcing coupled constraints in the proposed convex optimization problem to handle the coupled constraints we apply dual decompositions which makes first order methods applicable and thereby admit solutions for large scale problems e g coding entire images or image sequences we show by examples that the proposed framework generates non trivial sparse descriptions and non trivial refinements we finally show that the sparse descriptions can be quantized and encoded using off the shell encoders such as the set partitioning in hierarchical trees spiht encoder however the proposed method shows a rate distortion loss compared to state of the art image md encoders

a note on the bias of estimators with missing data it is well known that many standard analyses including maximum likelihood estimation and the generalized estimating equation approach liang and zeger 1986 biometrika 73 13 22 can result in biased estimation when there are missing observations in such cases it is of interest to calculate the magnitude of the bias incurred under specific assumptions about the process generating the full data and the nonresponse mechanism in this paper we give a condition that identifies the limit in probability of estimators that are solutions of estimating equations computed from the incomplete data with discrete data this condition suggests a simple algorithm to compute the asymptotic bias of these estimators that can be easily implemented with existing statistical software we illustrate our approach with asthma prevalence data in children

majorization minimization algorithms for wavelet based image restoration standard formulations of image signal deconvolution under wavelet based priors regularizers lead to very high dimensional optimization problems involving the following difficulties the non gaussian heavy tailed wavelet priors lead to objective functions which are nonquadratic usually nondifferentiable and sometimes even nonconvex the presence of the convolution operator destroys the separability which underlies the simplicity of wavelet based denoising this paper presents a unified view of several recently proposed algorithms for handling this class of optimization problems placing them in a common majorization minimization mm framework one of the classes of algorithms considered when using quadratic bounds on nondifferentiable log priors shares the infamous singularity issue si of iteratively re weighted least squares irls algorithms the possibility of having to handle infinite weights which may cause both numerical and convergence issues in this paper we prove several new results which strongly support the claim that the si does not compromise the usefulness of this class of algorithms exploiting the unified mm perspective we introduce a new algorithm resulting from using bounds for nonconvex regularizers the experiments confirm the superior performance of this method when compared to the one based on quadratic majorization finally an experimental comparison of the several algorithms reveals their relative merits for different standard types of scenarios

the convergence of numerical transfer schemes in diffusive regimes i discrete ordinate method in highly diffusive regimes the transfer equation with anisotropic boundary conditions has an asymptotic behavior as the mean free path epsilon tends to zero that is governed by a diffusion equation and boundary conditions obtained through a matched asymptotic boundary layer analysis a numerical scheme for solving this problem has an epsilon 1 contribution to the truncation error that generally gives rise to a nonuniform consistency with the transfer equation for small epsilon thus degrading its performance in diffusive regimes in this paper we show that whenever the discrete ordinate method has the correct diffusion limit both in the interior and at the boundaries its solutions converge to the solution of the transport equation uniformly in epsilon our proof of the convergence is based on an asymptotic diffusion expansion and requires error estimates on a matched boundary layer approximation to the solution of the discrete ordinate method

existence and uniqueness of weak solutions for nonlocal parabolic problems via the galerkin method abstract fractional differential equations are becoming increasingly popular as a modeling tool to describe a wide range of non classical phenomena with spatial heterogeneities throughout the applied science and engineering a recently developed nonlocal vector calculus is exploited to provide a variational analysis for a general class of nonlocal operators which include fractional laplacians on bounded domains in r n we develop the galerkin method to prove existence and uniqueness of weak solutions to nonlocal parabolic problems moreover we study the existence of orthonormal basis of eigenvectors associated to these nonlocal operators

type i and type ii error under random effects misspecification in generalized linear mixed models generalized linear mixed models glmms have become a frequently used tool for the analysis of non gaussian longitudinal data estimation is based on maximum likelihood theory which assumes that the underlying probability model is correctly specified recent research is showing that the results obtained from these models are not always robust against departures from the assumptions on which these models are based in the present work we have used simulations with a logistic random intercept model to study the impact of misspecifying the random effects distribution on the type i and ii errors of the tests for the mean structure in glmms we found that the misspecification can either increase or decrease the power of the tests depending on the shape of the underlying random effects distribution and it can considerably inflate the type i error rate additionally we have found a theoretical result which states that whenever a subset of fixed effects parameters not included in the random effects structure equals zero the corresponding maximum likelihood estimator will consistently estimate zero this implies that under certain conditions a significant effect could be considered as a reliable result even if the random effects distribution is misspecified

a smooth nonparametric estimate of a mixing distribution using mixtures of gaussians abstract we propose a method of estimating mixing distributions using maximum likelihood over the class of arbitrary mixtures of gaussians subject to the constraint that the component variances be greater than or equal to some minimum value h this approach can lead to estimates of many shapes with smoothness controlled by parameter h we show that the resulting estimate will always be a finite mixture of gaussians each having variance h the nonparametric maximum likelihood estimate can be viewed as a special case with h 0 the method can be extended to estimate multivariate mixing distributions examples and the results of a simulation study are presented

generalized solutions of the radiative transfer equations in a singular case this paper is devoted to the study of the radiative transfer equations first we prove a global existence theorem which allows a blow up of the opacity v when 0 thus it extends mercier s previous result 13 this proof relies mainly on a nonlinear version of hille yosida theorem see crandall ligett 9 then we prove the uniqueness of the semigroup solving tr and some regularity results in the class of functions with bounded variation finally we prove the convergence of some splitting algorithms associated to tr

diffusion approximations and domain decomposition method of linear transport equations in this paper we construct numerical schemes to approximate linear transport equations with slab geometry by diffusion equations we treat both the case of pure diffusive scaling and the case where kinetic and diffusive scalings coexist the diffusion equations and their data are derived from asymptotic and layer analysis which allows general scattering kernels and general data we apply the half space solver in 20 to resolve the boundary layer equation and obtain the boundary data for the diffusion equation the algorithms are validated by numerical experiments and also by error analysis for the pure diffusive scaling case

a proof of conjecture on restricted isometry property constants delta _ tk left 0 t frac 4 3 right in this paper we give a complete answer to the conjecture on restricted isometry property rip constants delta tk 0 which was proposed by t cai and a zhang we have shown that when 0 the condition delta tk is sufficient to guarantee the exact recovery for all k sparse signals in the noiseless case via the constrained ell 1 norm minimization these bounds are sharp in the sense that for any epsilon 0 delta tk cannot guarantee the exact recovery of some k sparse signals furthermore it will be shown that similar characterizations also hold for low rank matrix recovery thus combined with t cai and a zhang s work a complete characterization for sharp rip constants delta tk for all t 0 is obtained to guarantee the exact recovery of all k sparse signals and matrices with rank at most k by ell 1 norm minimization and nuclear norm minimization respectively noisy cases and approximately sparse cases are also considered to solve the conjecture we construct a few identities so that rip of order tk which is the target of our main results can be perfectly applied to them

an asymptotic preserving scheme for linear kinetic equation with fractional diffusion limit we present a new asymptotic preserving scheme for the linear boltzmann equation which under appropriate scaling leads to a fractional diffusion limit our scheme rests on novel micro macro decomposition to the distribution function which splits the original kinetic equation following a reshuffled hilbert expansion as opposed to classical diffusion limit a major difficulty comes from the fat tail in the equilibrium which makes the truncation in velocity space depending on the small parameter our idea is while solving the macro micro part in a truncated velocity domain truncation only depends on numerical accuracy to incorporate an integrated tail over the velocity space that is beyond the truncation and its major component can be precomputed once with any accuracy such an addition is essential to drive the solution to the correct asymptotic limit numerical experiments validate its efficiency in both kinetic and fractional diffusive regimes

restricted isometry constants where ell p sparse recovery can fail for 0 p leq 1 this paper investigates conditions under which the solution of an underdetermined linear system with minimal lscrp norm 0 p les 1 is guaranteed to be also the sparsest one matrices are constructed with restricted isometry constants ric delta2m arbitrarily close to 1 radic2 ap 0 707 where sparse recovery with p 1 fails for at least one m sparse vector as well as matrices with delta2m arbitrarily close to one where lscr1 minimization succeeds for any m sparse vector this highlights the pessimism of sparse recovery prediction based on the ric and indicates that there is limited room for improving over the best known positive results of foucart and lai which guarantee that lscr1 minimization recovers all m sparse vectors for any matrix with delta2m 2 3 radic2 7 ap 0 4531 these constructions are a by product of tight conditions for lscrp recovery 0 les p les 1 with matrices of unit spectral norm which are expressed in terms of the minimal singular values of 2m column submatrices compared to lscr1 minimization lscrp minimization recovery failure is shown to be only slightly delayed in terms of the ric values furthermore in this case the minimization is nonconvex and it is important to consider the specific minimization algorithm being used it is shown that when lscrp optimization is attempted using an iterative reweighted lscr1 scheme failure can still occur for delta2m arbitrarily close to 1 radic2

a two surface problem of the electron flow in a semiconductor on the basis of kinetic theory a steady flow of electrons in a semiconductor between two parallel plane ohmic contacts is studied on the basis of the semiconductor boltzmann equation assuming a relaxation time collision term and the poisson equation for the electrostatic potential a systematic asymptotic analysis of the boltzmann poisson system for small knudsen numbers scaled mean free paths is carried out in the case where the debye length is of the same order as the distance between the contacts and where the applied potential is of the same order as the thermal potential a system of drift diffusion type equations and their boundary conditions is obtained up to second order in the knudsen number a numerical comparison is made between the obtained system and the original boltzmann poisson system

micro macro schemes for kinetic equations including boundary layers we introduce a new micro macro decomposition of collisional kinetic equations in the specific case of the diffusion limit which naturally incorporates the incoming boundary conditions the idea is to write the distribution function f in all its domain as the sum of an equilibrium adapted to the boundary which is not the usual equilibrium associated with f and a remaining kinetic part this equilibrium is defined such that its incoming velocity moments coincide with the incoming velocity moments of the distribution function a consequence of this strategy is that no artificial boundary condition is needed in the micro macro models and the exact boundary condition on f is naturally transposed to the macro part of the model this method provides an asymptotic preserving numerical scheme that generates a very good approximation of the space boundary values at the diffusive limit without any mesh refinement in the boundary layers our numerical results are in very good agreement with the exact so c

the milne problem for high field kinetic equations half space problems of the linear boltzmann equation with a constant driving force are considered such problems model boundary layers between kinetic zones and fluid zones described by a high field limit of the boltzmann equation existence uniqueness and asymptotic behavior of solutions are studied for positive and negative driving forces in the positive case the force field accelerates the particles and we show that the solution of the half space problem is determined only by the inflow data in contrast for negative forces the behavior at infinity has to be prescribed in order to insure uniqueness due to the nonvanishing forces the problem does not possess any entropy the existence and uniqueness issues are dealt with by supersolution techniques while the asymptotic behavior is analyzed by semiexplicit integration of the equations along the characteristics in the case of relaxation time approximation a fast numerical method for computing the asymptotic state method is presented and tested