efficient parallel resolution of the simplified transport equations in mixed dual formulation a reactivity computation consists of computing the highest eigenvalue of a generalized eigenvalue problem for which an inverse power algorithm is commonly used very fine modelizations are difficult to treat for our sequential solver based on the simplified transport equations in terms of memory consumption and computational time a first implementation of a lagrangian based domain decomposition method brings to a poor parallel efficiency because of an increase in the power iterations 1 in order to obtain a high parallel efficiency we improve the parallelization scheme by changing the location of the loop over the subdomains in the overall algorithm and by benefiting from the characteristics of the raviart thomas finite element the new parallel algorithm still allows us to locally adapt the numerical scheme mesh finite element order however it can be significantly optimized for the matching grid case the good behavior of the new parallelization scheme is demonstrated for the matching grid case on several hundreds of nodes for computations based on a pin by pin discretization

the hp finite element method for problems in mechanics with boundary layers abstract we consider the numerical approximation of boundary layer phenomena occurring in many singularly perturbed problems in mechanics such as plate and shell problems we present guidelines for the effective resolution of such layers in the context of existing commercial p and hp finite element fe version codes we show that if high order spectral elements are available then just two elements are sufficient to approximate these layers at a near exponential rate independently of the problem parameters thickness or reynolds number we present hp mesh design principles for situations where both corner singularities and boundary layers are present

on computing the feedback capacity of channels and the feed forward rate distortion function of sources the problem of computing the capacity cost function of channels with feedback and the rate distortion function of sources with feed forward is considered sufficient conditions are derived on a the structure of the cost function for a chosen joint distribution to achieve the optimal feedback capacity cost function b the structure of the distortion function for a chosen joint distribution to achieve the optimal feed forward rate distortion function these structural results are useful since it is infeasible in general to directly compute the optimizations examples are provided to show how the results can help compute the performance limits with feedback and feed forward

error estimates for the combinedh andp versions of the finite element method in theh version of the finite element method convergence is achieved by refining the mesh while keeping the degree of the elements fixed on the other hand thep version keeps the mesh fixed and increases the degree of the elements in this paper we prove estimates showing the simultaneous dependence of the order of approximation on both the element degrees and the mesh in addition it is shown that a proper design of the mesh and distribution of element degrees lead to a better than polynomial rate of convergence with respect to the number of degrees of freedom even in the presence of corner singularities numerical results comparing theh version p version and combinedh p version for a one dimensional problem are presented

upper and lower bounds for higher order eigenvalues of some semilinear elliptic equations abstract we prove upper and lower bounds on the eigenvalues as the h 0 1 norm of the eigenfunction tends to zero in bifurcation problems for a class of semilinear elliptic equations in bounded domains of r n it is shown that these bounds are computable in terms of the eigenvalues of the associated linear equation

on the finite element method for singularly perturbed reaction diffusion problems in two and one dimensions second order elliptic boundary value problems which are allowed to degenerate into zero order equations are considered the behavior of the ordinary galerkin finite element method without special arrangements to treat singularities is studied as the problem ranges from true second order to singularly perturbed

a computational multilevel approach for solving 2d navier stokes equations over non matching grids a multilevel approach with parallel implementation is developed for obtaining fast solutions of the navier stokes equations solved on domains with non matching grids the method relies on computing solutions over different subdomains with different multigrid levels by using multiple processors a local vanka type relaxation operator for the multigrid solution of the navier stokes system allows solutions to be computed at the element level the natural implementation on a multiprocessor architecture results in a straightforward and flexible algorithm numerical computations are presented using benchmark applications in order to support the method parallelization is discussed to achieve proper accuracy load balancing and computational efficiency between different processors 2005 elsevier b v all rights reserved

computable bounds for rate distortion with feed forward for stationary and ergodic sources in this paper we consider the rate distortion problem of discrete time ergodic and stationary sources with feed forward at the receiver we derive a sequence of achievable and computable rates that converge to the feed forward rate distortion we show that for ergodic and stationary sources the rate rn d 1 n min ixn xn is achievable for any n where the minimization is performed over the transition conditioning probability p xn xn such that e d xn xn d we also show that the limit of rn d exists and is the feed forward rate distortion we follow gallager s proof where there is no feed forward and with appropriate modification obtain our result we provide an algorithm for calculating rn d using the alternating minimization procedure and present several numerical examples we also present a dual form for the optimization of rn d and transform it into a geometric programming problem

duality between source coding and channel coding and its extension to the side information case we explore the information theoretic duality between source coding with side information at the decoder and channel coding with side information at the encoder we begin with a mathematical characterization of the functional duality between classical source and channel coding formulating the precise conditions under which the optimal encoder for one problem is functionally identical to the optimal decoder for the other problem we then extend this functional duality to the case of coding with side information by invoking this duality we are able to generalize the result of wyner and ziv 1976 relating to no rate loss for source coding with side information from gaussian to more arbitrary distributions we consider several examples corresponding to both discrete and continuous valued cases to illustrate our formulation for the gaussian cases of coding with side information we invoke geometric arguments to provide further insights into their duality our geometric treatment inspires the construction and dual use of practical coset codes for a large class of emerging applications for coding with side information such as distributed sensor networks watermarking and information hiding communication systems

error estimates using the cell discretization method for steady state convection diffusion equations the cell discretization algorithm is applied to generate approximate solutions for some second order non self adjoint elliptic equations general convergence for homogeneous problems is shown by obtaining suitable error estimates the method is applied using polynomial bases this provides a nonconforming extension of the finite element method that can also produce the continuous approximations of an h p finite element method numerical tests on convection diffusion problems are made that confirm the theoretical estimates and methods for dealing with boundary layer problems are illustrated

source coding when the side information may be delayed for memoryless sources delayed side information at the decoder does not improve the rate distortion function however this is not the case for sources with memory as demonstrated by a number of works focusing on the special case of delayed feedforward in this paper a setting is studied in which the encoder is potentially uncertain about the delay with which measurements of the side information which is available at the encoder are acquired at the decoder assuming a hidden markov model for the source sequences at first a single letter characterization is given for the setup where the side information delay is arbitrary and known at the encoder and the reconstruction at the destination is required to be asymptotically lossless then with delay equal to zero or one source symbol a single letter characterization of the rate distortion region is given for the case where unbeknownst to the encoder the side information may be delayed or not finally examples for binary and gaussian sources are provided

nodal properties and variational characterizations of solutions to nonlinear sturm liouville problems on considere des problemes aux valeurs propres non lineaires pour des equations differentielles de la forme p t u q t u f t u r t u avec des conditions aux limites de dirichlet

on the global structure of the set of positive solutions of some semilinear elliptic boundary value problems abstract in this work we analyze the structure of the set of positive solutions of a class of semilinear boundary value problems it is shown that the global continuum of positive solutions emanating from the trivial equilibrium at the principal eigenvalue of the linearization is constituted by a regular curve if the slope of the kinetic at the trivial solution is large enough and is convex the same result holds if the support region of the species is a bounded simply connected domain of r 2 close to a convex domain in a sense to be precised later to prove these results we have to find out the exact width of the boundary layer of a singular perturbation problem the results about the singular perturbation problem are new and of great interest by themselves

le nombre de solutions de certains problemes semi lineaires elliptiques abstract in this paper we compute the exact number of solutions for certain semi linear elliptic problems in the first part we derive some complementary results to a theorem of ambrosetti and prodi the global behaviour of bifurcating branches is then studied for some classes of problems in the context of a concave or a convex nonlinearity in the second part we consider the branch of positive solutions lastly in the third part in dimension one for an autonomous equation we obtain a precise description of all the solutions to these problems the proofs rely on simple eigenvalue comparison arguments

robust exponential convergence of hp fem in balanced norms for singularly perturbed reaction diffusion problems corner domains abstract the h p version of the finite element method is applied to singularly perturbed reaction diffusion type equations on polygonal domains the solution exhibits boundary layers as well as corner layers on a class of meshes that are suitably refined near the boundary and the corners robust exponential convergence in the polynomial degree is shown in both a balanced norm and the maximum norm

on spectral asymptotics and bifurcation for elliptic operators with odd superlinear term on considere le probleme aux valeurs propes semi lineaire lu f x u u dans u 0 sur ou est un sous ensemble ouvert borne de r n n 1 a frontiere lisse ou l est un operateur uniformement elliptique dans formellement auto adjoint et f une application continu de r r

on the influence of the geometry on skin effect in electromagnetism we consider the equations of electromagnetism set on a domain made of a dielectric and a conductor subdomain in a regime where the conductivity is large assuming smoothness for the dielectric conductor interface relying on recent works we prove that the solution of the maxwell equations admits a multiscale asymptotic expansion with profile terms rapidly decaying inside the conductor this skin effect is measured by introducing a skin depth function that turns out to depend on the mean curvature of the boundary of the conductor we then confirm these asymptotic results by numerical experiments in various axisymmetric configurations we also investigate numerically the case of a nonsmooth interface namely a cylindrical conductor

the optimal convergence rate of the p version of the finite element method optimal error estimates for the p version of the finite element method in two dimensions are proven for the case when u in h k omega or u has singularities induced by the corners of the domain the case of nonhomogenous essential boundary conditions is also analyzed

the p and hp versions of the finite element method for problems with boundary layers we study the uniform approximation of boundary layer functions exp x d for x 0 1 d 0 1 by the p and hp versions of the finite element method for the p version with fixed mesh we prove super exponential convergence in the range p 1 2 e 2d we also establish for this version an overall convergence rate of o p 1 ln p in the energy norm error which is uniform in d and show that this rate is sharp up to the ln p term when robust estimates uniform in d 0 1 are considered for the p version with variable mesh i e the hp version we show that exponential convergence uniform in d 0 1 is achieved by taking the first element at the boundary layer to be of size o pd numerical experiments for a model elliptic singular perturbation problem show good agreement with our convergence estimates even when few degrees of freedom are used and when d is as small as e g 10 8 they also illustrate the superiority of the hp approach over other methods including a low order h version with optimal exponential mesh refinement the estimates established in this paper are also applicable in the context of corresponding spectral element methods

on solenoidal high degree polynomial approximations to solutions of the stationary navier stokes equations the cell discretization algorithm a nonconforming extension of the finite element method is used to obtain approximations to the navier stokes equations for viscous steady state incompressible flow an implementation using polynomial bases is described that permits the use of approximations that exactly satisfy the solenoidal requirement a galerkin basis is constructed and starting with the solution of the associated stokes problem newton s method can be used to generate an approximate solution a unique solution exists if the viscosity is sufficiently large here an a posteriori computation is described that tests if uniqueness is likely to hold and gives an error estimate showing closeness of the approximation tests of the algorithm applied to problems in polygonal domains are described where approximations are 10th degree polynomials

existence and bifurcation theorems for nonlinear elliptic eigenvalue problems on unbounded domains abstract we consider nonlinear elliptic eigenvalue problems on unbounded domains g r n using an extended ljusternik schnirelman theory we prove the existence of infinitely many eigenfunctions on every sphere in l 2 g moreover we establish that the infimum of the spectrum of the linearized problem l is always a bifurcation point in addition there is an infinity of branches emanating at from the trivial line of solutions if belongs to the essential spectrum of l

the approximation of solutions of elliptic boundary value problems via the p version of the finite element method the approximation theory developed in 11 is used to determine the piecewise polynomial approximability of solutions of elliptic problems on polygonal domains in mathbb r 2 and polyhedra in mathbb r 3 from these estimates convergence orders for the p version of the finite element method applied to such problems are readily obtained the critical issue is the approximation of the singularities which occur at the nonsmooth parts of the domain boundaries it is seen that the estimates of 11 involving the weighted sobolev spaces z l s are well suited for treating such singular functions yielding directly the optimal approximation degree up to an arbitrarily small varepsilon numerical results for two problems from two dimensional linear elasticity are also presented the computations show that the predicted order of convergence is achieved even for low values of p moreover in contrast to the usual h version of the finite element method the point at which the p version enters the asym

achievable rates for multiple descriptions with feed forward the two channel multiple descriptions problem for an independent and identically distributed i i d source with feed forward to one or both side decoders is considered a single letter achievable rate region is derived it enlarges the best known rate region for multiple descriptions without feed forward the proof of the result uses a block markov superposition source coding strategy in point to point source coding feed forward does not decrease the rate distortion function of an i i d source in contrast an example is provided to show that the derived region can be strictly larger than the optimal multiple description rate distortion region without feed forward

stability and convergence of nonconforming hp finite element methods the stability and convergence of nonconforming hp finite element methods in par ticular the mortar finite element method and its variants are established based on a new stability measure for these methods using a generalized eigenvalue analysis estimates for this measure are computed numerically our numerical results demonstrate that these nonconforming methods prove to be good candidates for hp implementation and also behave ss well as conforming finite element methods the discussion here ls primarily in two dimensions but some extensions to three dimensions

uniqueness of global positive solution branches of nonlinear elliptic problems has been studied in many publications the first global description was given in 19 nonetheless there were open questions some of which are answered in this paper we motivate some of these questions now for a suitable class of smooth fimctions f ir ir problem 1 1 has infinitely many solutions for 2 1 this was shown in 15 via the method of suband supersolutions which also gives an order of these solutions 0 0 which is not excluded in 15 there is also a global branch continuum c of solutions 2 u bifurcating from the trivial solution line 2 0 at 2 21 where 1 21 f 0 is the principal eigenvalue of the linearization

an explicit finite element method for convection dominated steady state convection diffusion equations a finite element method for hyperbolic equations is extended to steady state convection diffusion equations dominated by convection the resulting finite element method is of arbitrarily high order and can be applied in an explicit fashion the method is stable for sufficiently small sigma h where sigma is the diffusion coefficient and h is the meshsize for problems where this stability condition holds only over part of the domain the method can be used in combination with galerkin s method error estimates are derived and corroborative numerical results given

approximation and convergence rate of nonlinear eigenvalues lipschitz perturbations of a bounded self adjoint operator abstract we consider nonlinear eigenvalue problems of the form t x b x x where t is a self adjoint bounded linear operator acting in a real hilbert space h and b h h is a possibly nonlinear continuous perturbation term assuming that 0 is an isolated eigenvalue of finite multiplicity of t we ask if for 0 and small there are eigenvalues of near 0 that is numbers for which is satisfied by some normalized eigenvector x of t b in this paper we recall some recent results giving an affirmative answer to this question and for these cases we prove assuming in addition lipschitz continuity on b upper and lower bounds for the perturbed eigenvalues which are determined by those for the nonlinear rayleigh quotient b v v v v with v in the eigenspace ker t 0 i this yields in particular information on the rate of convergence of to 0 as 0 applications are given in the sequence space l 2 and in the sobolev space h 0 1 to deal with some nonlinearly perturbed ordinary or partial differential equations

capacity results for the discrete memoryless network a discrete memoryless network dmn is a memoryless multiterminal channel with discrete inputs and outputs a sequence of inner bounds to the dmn capacity region is derived by using code trees capacity expressions are given for three classes of dmns 1 a single letter expression for a class with a common output 2 a two letter expression for a binary symmetric broadcast channel bc with partial feedback and 3 a finite letter expression for push to talk dmns the first result is a consequence of a new capacity outer bound for common output dmns the third result demonstrates that the common practice of using a time sharing random variable does not include all time sharing possibilities namely time sharing of channels several techniques for improving the bounds are developed 1 causally conditioned entropy and directed information simplify the inner bounds 2 code trellises serve as simple code trees 3 superposition coding and binning with code trees improves rates numerical computations show that the last technique enlarges the best known rate regions for a multiple access channel mac and a bc both with feedback in addition to the rate bounds a sequence of inner bounds to the dmn reliability function is derived a numerical example for a two way channel illustrates the behavior of the error exponents

high degree efficient symmetrical gaussian quadrature rules for the triangle gaussian quadrature is required for the computation of matrices based on the isoparametric formulztion of the finite element method a brief review of existing quadrature rules for the triangle is given and the method for the determination of high degree efficient symmetrical rules for the triangle is discussed new quadrature rules of degree 12 20 are presented and a short fortran program is included

discrete dispersion relation for hp version finite element approximation at high wave number the dispersive properties of high order finite element schemes are analyzed in the setting of the helmholtz equation and an explicit form of the discrete dispersion relation is obtained for elements of arbitrary order it is shown that the numerical dispersion displays three different types of behavior depending on the size of the order of the method relative to the mesh size and the wave number quantitative estimates are obtained for the behavior and rates of decay of the dispersion error in the differing regimes all estimates are fully explicit and are shown to be sharp limits are obtained where transitions between the different regimes occur and used to provide guidelines for the selection of the mesh size and the polynomial order in terms of the wave number so that the dispersion error is controlled

the treatment of nonhomogeneous dirichlet boundary conditions by thep version of the finite element method the paper addresses the problem of the implementation of nonhomogeneous essential dirichlet type boundary conditions in thep version of the finite element method

spectral methods and a maximum principle etude de diverses approximations spectrales de tchebychev d un probleme de couche limite pour un operateur d helmholtz et d advection diffusion comportement des solutions spectrales dans l espace des frequences et dans l espace physique estimations d erreur

uniform high order difference schemes for a singularly perturbed two point boundary value problem on construit une famille de schemas aux differences finies uniformement precis pour le probleme du type eu a x u b x u f x on donne une analyse complete de l erreur de discretisation basee sur les resultats de stabilite on presente des essais numeriques

asymptotic expansion of the variational eigencurve for two parameter simple pendulum type equations abstract we consider the nonlinear two parameter simple pendulum type equation u t u p sin u t u t 0 t i t t u t 0 where p 1 is a constant and 0 are parameters for a given 0 there exists a solution triple u r 2 c 2 i which is obtained by a variational method such that u 0 as we establish the precise asymptotic formula for the variational eigencurve as whose growth order is sublinear

a priori bounds and asymptotics on the eigenvalues in bifurcation problems for perturbed self adjoint operators abstract we prove upper and lower bounds on the eigenvalues and discuss their asymptotic behaviour as the norm of the eigenvector tends to zero in bifurcation problems from the line of trivial solutions considering perturbations of linear self adjoint operators in a hilbert space the proofs are based on the lyapounov schmidt reduction the results are applied to a class of semilinear elliptic operators in bounded domains of r n and in particular to sturm liouville operators

local structure of bifurcation curves for nonlinear sturm liouville problems abstract we consider the nonlinear bifurcation problem arising in population dynamics and nonlinear schrodinger equation u t f u t u t 0 t i 0 1 u 0 u 1 0 where 0 is a parameter we mainly treat the case where f u u u p p 1 and establish the precise asymptotic expansion formulas for the bifurcation curve near the bifurcation point 2 in l q framework together with the result of the global behavior of the bifurcation curve we understand completely the structure of the bifurcation curve we also consider the nodal solution u n of the equation u t u t u t p 1 u t with u 0 u 0 and establish an asymptotic expansion formula for with respect to the gradient norm of the solution associated with as n 2

inverse bifurcation problems for diffusive logistic equation of population dynamics abstract we consider the nonlinear eigenvalue problem arising in population dynamics u t f u t u t u t 0 t i 0 1 u 0 u 1 0 where 0 is a parameter f u u p g u p 1 and g u is assumed to be an unknown nonlinear term the purpose of this paper is to study the inverse bifurcation problem in l 1 framework more precisely we suppose that g u has compact support in 0 and the precise global behavior of the l 1 bifurcation curve of the equation is given then we show that g u 0

on the use of lagrange multipliers in domain decomposition for solving elliptic problems the primal hybrid method for solving second order elliptic equa tions is extended from finite element approximations to general bases varia tional techniques are used to show convergence of approximations to the so lution of the homogeneous dirichlet problem for selfadjoint equations error estimates are obtained and examples are given

precise spectral asymptotics for nonlinear sturm liouville problems we consider the nonlinear sturm liouville problem u t vert u t vert p 1 u t f u t lambda u t enskip t in i 0 1 enskip u 0 u 1 0 where p 1 is a constant and lambda 0 is an eigenvalue parameter we establish the three term asymptotics of n th eigencurve lambda n alpha associated with eigenfunction u n alpha with n 1 simple interior zeros and vert u n alpha vert 2 alpha as alpha to infty we also obtain the corresponding asymptotics of the width of the boundary layer of u n alpha as alpha to infty

source coding with feed forward rate distortion theorems and error exponents for a general source in this work we consider a source coding model with feed forward we analyze a system with a noiseless feed forward link where the decoder has knowledge of all previous source samples while reconstructing the present sample the rate distortion function for an arbitrary source with feed forward is derived in terms of directed information a variant of mutual information we further investigate the nature of the rate distortion function with feed forward for two common types of sources discrete memory less sources and gaussian sources we then characterize the error exponent for a general source with feed forward the results are then extended to feed forward with an arbitrary delay larger than the block length

hp submeshing via non conforming finite element methods abstract non conformity in the hp version can involve incompatibility in both the degrees and the meshes between adjoining subdomains in this paper we show how the mortar finite element method m0 and two new variants m1 m2 can be used to join together such incompatible hp sub discretizations our results show optimality of the resulting non conforming method for various h p and hp discretizations including the case of exponential hp convergence over geometric meshes we also present numerical results for the lagrange multiplier when the method is implemented via a mixed method three dimensional considerations suggest that our methods m1 m2 are easier to generalize to arbitrary meshes than m0

solution of wave propagation problems by the cell discretization method abstract the cell discretization algorithm cd is first applied to a fairly complex steady state wave scattering problem and the results compared with the same problem solved by the finite element method the p d e solved is the elliptic helmholtz equation for the components of the complex scattered amplitude the method is then directly applied to some hyperbolic problems in space time using a total discretization instead of semidiscretization the problem then becomes one of solving a discretized elliptic equation in successive time slabs as the process progresses through time the main difficulty is in choosing the right combination of collocation moments peculiar to the cd method intracell polynomials and the cfl ratio i e the ratio of the cell length in the time direction to its length s in the space direction s apart from this serious difficulty there is little problem in solving several problems involving first and second order wave equations in one and two space dimensions and one solution of a second order wave equation in two space dimensions copious figures are supplied showing surface plots of the solutions and plots of the spectrum in the complex plane of the propagation matrix associated with the problem are supplied

on using the cell discretization algorithm for mixed boundary value problems and domain decomposition the cell discretization algorithm is used to approximate solutions to self adjoint elliptic equations with general nonhomogeneous dirichlet or neumann or mixed boundary values error estimates are obtained showing general convergence this provides the framework for a nonoverlapping iterative algorithm for domain decomposition the domain of a dirichlet problem is partitioned into two subdomains solutions on the two subdomains can be patched together to form a solution to the original problem provided the solutions agree across the common interface of the subdomains and the normal derivatives there have equal absolute values and are opposite in sign we generate such a solution by alternating between imposing neumann and dirichlet conditions on the interface with boundary data adapted from the results of the previous approximation a posteriori error estimates show that we have global convergence provided computed interface errors are sufficiently small

error estimates using the cell discretization method for some parabolic problems the cell discretization algorithm is used to approximate solutions to parabolic problems with coefficients independent of time error estimates are obtained showing general convergence for homogeneous problems a polynomial implementation of the algorithm is described that can be viewed as a nonconforming extension of the finite element method that can also produce the continuous approximations of an h p finite element method numerical tests that confirm the theoretical estimates are discussed

a general formula for channel capacity a formula for the capacity of arbitrary single user channels without feedback not necessarily information stable stationary etc is proved capacity is shown to equal the supremum over all input processes of the input output inf information rate defined as the liminf in probability of the normalized information density the key to this result is a new converse approach based on a simple new lower bound on the error probability of m ary hypothesis tests among equiprobable hypotheses a necessary and sufficient condition for the validity of the strong converse is given as well as general expressions for spl epsiv capacity

nodal properties and bifurcation from the essential spectrum for a class of nonlinear sturm liouville problems is taken as the boundary condition at infinity our interest in this problem derives from the fact that it provides an example of bifurcation from the essential spectrum the terms bifurcation from the essential spectrum or singular bifurcation have recently been used to describe the following situation suppose a is a self adjoint hilbert space operator arising from a singular elliptic boundary value problem so that the essential spectrum of a may be non empty when a is perturbed by a nonlinearity f such that f u vanishes of higher order for u 0 the points of the essential spectrum of a are candidates for bifurcation points of the nonlinear eigenvalue problem

on approximating the solution of the non stationary stokes equations using the cell discretization algorithm the cell discretization algorithm a nonconforming extension of the finite element method is used to obtain approximations to the velocity and pressure satisfying the nonstationary stokes equations error estimates show convergence of the approximations an implementation using polynomial bases is described that permits the use of the continuous approximations of the h p finite element method and exactly satisfies the solenoidal requirement we express the error estimates in terms of the diameter h of a cell and the degree p of the approximation on each cell results of an experiment with p 10 are presented that confirm the theoretical estimates

on the convergence rate of the cell discretization algorithm for solving elliptic problems error estimates for the cell discretization algorithm are obtained for polynomial bases used to approximate both h k and analytic solutions to selfadjoint elliptic problems the polynomial implementation of this algorithm can be viewed as a nonconforming version of the h p finite element method that also can produce the continuous approximations of the h p method the examples provided by our experiments provide discontinuous approximations that have errors similar to the finite element results

the capacity of channels with feedback in this paper we introduce a general framework for treating channels with memory and feedback first we prove a general feedback channel coding theorem based on massey s concept of directed information second we present coding results for markov channels this requires determining appropriate sufficient statistics at the encoder and decoder we give a recursive characterization of these sufficient statistics third a dynamic programming framework for computing the capacity of markov channels is presented fourth it is shown that the average cost optimality equation acoe can be viewed as an implicit single letter characterization of the capacity fifth scenarios with simple sufficient statistics are described sixth error exponents for channels with feedback are presented

a non conforming finite element method for sub meshing in engineering applications there is a constant need for solving problems using finite elements over large and complicated domains such analysis is often accomplished by decomposing the global domain into several sub domains and performing the analysis individually on each local sub domain the global analysis can then be realized by piecing together the information that is available from each sub domain non conforming finite elements are particularly useful in this regard in this paper we discuss such a technique for a model problem we derive necessary and sufficient conditions in order for the mixed method formulation of such a non conforming method to have a unique solution error estimates for the finite element solution are obtained finally we present computational experiments to demonstrate that this non conforming technique is a good candidate for finite element implementation

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

uniform hp convergence results for the mortar finite element method the mortar finite element is an example of a non conforming method which can be used to decompose and re compose a domain into sub domains without requiring compatibility between the meshes on the separate components we obtain stability and convergence results for this method that are uniform in terms of both the degree and the mesh used without assuming quasiuniformity for the meshes our results establish that the method is optimal when non quasiuniform h or hp methods are used such methods are essential in practice for good rates of convergence when the interface passes through a corner of the domain we also give an error estimate for when the p version is used numerical results for hp and hp mortar fems show that these methods behave as well as conforming fems an hp extension theorem is also proved

on locking and robustness in the finite element method a numerical scheme for the approximation of a parameter dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value a robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter precise mathematical definitions for these terms are developed their quantitative characterization is given and some general theorems involving locking and robustness are proven a model problem involving heat transfer is analyzed in detail using this mathematical framework and various related computational results are described applications to some different problems involving locking are presented

automatically stable discontinuous petrov galerkin methods for stationary transport problems quasi optimal test space norm we investigate the application of the discontinuous petrov galerkin dpg finite element framework to stationary convection diffusion problems in particular we demonstrate how the quasi optimal test space norm improves the robustness of the dpg method with respect to vanishing diffusion we numerically compare coarse mesh accuracy of the approximation when using the quasi optimal norm the standard norm and the weighted norm our results show that the quasi optimal norm leads to more accurate results on three benchmark problems in two spatial dimensions we address the problems associated to the resolution of the optimal test functions with respect to the quasi optimal norm by studying their convergence numerically in order to facilitate understanding of the method we also include a detailed explanation of the methodology from the algorithmic point of view

a coding theorem for a class of stationary channels with feedback a coding theorem is proved for a class of stationary channels with feedback in which the output yn f xn m n zn m n is the function of the current and past m symbols from the channel input xn and the stationary ergodic channel noise zn in particular it is shown that the feedback capacity is equal to limnrarr infin supp x n y n 1 1 n i xn rarr yn where i xn rarr yn sigmai 1 n i xi yi yi 1 denotes the massey directed information from the channel input to the output and the supremum is taken over all causally conditioned distributions p xn yn 1 pii 1 n p xi xi 1 yi 1 the main ideas of the proof are a classical application of the shannon strategy for coding with side information and a new elementary coding technique for the given channel model without feedback which is in a sense dual to gallager s lossy coding of stationary ergodic sources a similar approach gives a simple alternative proof of coding theorems for finite state channels by yang kavcic tatikonda chen berger and permuter weissman goldsmith

on competitive prediction and its relation to rate distortion theory consider the normalized cumulative loss of a predictor f on the sequence x sup n x sub 1 x sub n denoted l sub f x sup n for a set of predictors g let l g x sup n min sub f spl isin g l sub f x sup n denote the loss of the best predictor in the class on x sup n given the stochastic process x x sub 1 x sub 2 we look at el g x sup n termed the competitive predictability of g on x sup n our interest is in the optimal predictor set of size m i e the predictor set achieving min sub g spl les m el g x sup n when m is subexponential in n simple arguments show that min sub g spl les m el g x sup n coincides for large n with the bayesian envelope min sub f el sub f x sup n we investigate the behavior for large n of min sub g spl les e sup nr el g x sup n which we term the competitive predictability of x at rate r we show that whenever x has an autoregressive representation via a predictor with an associated independent and identically distributed i i d innovation process its competitive predictability is given by the distortion rate function of that innovation process indeed it will be argued that by viewing g as a rate distortion codebook and the predictors in it as codewords allowed to base the reconstruction of each symbol on the past unquantized symbols the result can be considered as the source coding analog of shannon s classical result that feedback does not increase the capacity of a memoryless channel for a general process x we show that the competitive predictability is lower bounded by the shannon lower bound slb on the distortion rate function of x and upper bounded by the distortion rate function of any not necessarily memoryless innovation process through which the process x has an autoregressive representation thus the competitive predictability is also precisely characterized whenever x can be autoregressively represented via an innovation process for which the slb is tight the error exponent i e the exponential behavior of min sub g spl les exp nr pr l g x sup n d is also characterized for processes that can be autoregressively represented with an i i d innovation process