iterative galerkin discretizations for strongly monotone problems in this article we investigate the use of fixed point iterations to solve the galerkin approximation of strictly monotone problems as opposed to newton s method which requires information from the previous iteration in order to linearize the iteration matrix and thereby to recompute it in each step the alternative method used in this article exploits the monotonicity properties of the problem and only needs the iteration matrix calculated once for all iterations of the fixed point method we outline the abstract a priori and a posteriori analyses for the iteratively obtained solutions and apply this to a finite element approximation of a second order elliptic quasilinear boundary value problem we show both theoretically as well as numerically how the number of iterations of the fixed point method can be restricted in dependence of the mesh size or of the polynomial degree to obtain optimal convergence using the a posteriori error analysis we also devise an adaptive algorithm for the generation of a sequence of galerkin spaces adaptively refined finite element meshes in the concrete example to minimize the number of iterations on each space

on the coefficients of integrated expansions of bessel polynomials a new formula expressing explicitly the integrals of bessel polynomials of any degree and for any order in terms of the bessel polynomials themselves is proved another new explicit formula relating the bessel coefficients of an expansion for infinitely differentiable function that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion of the function is also established an application of these formulae for solving ordinary differential equations with varying coefficients is discussed

distributed lossy averaging in this paper an information theoretic formulation of the distributed averaging problem previously studied in computer science and control is presented we assume a network with m nodes each observing a white gaussian noise wgn source the nodes communicate and perform local processing with the goal of computing the average of the sources to within a prescribed mean squared error distortion the network rate distortion function r d for a two node network with correlated gaussian sources is established a general cutset lower bound on r d is established and shown to be achievable to within a factor of 2 via a centralized protocol over a star network a lower bound on the network rate distortion function for distributed weighted sum protocols which is larger in order than the cutset bound by a factor of log m is established an upper bound on the network rate distortion function for gossip base weighted sum protocols which is only log log m larger in order than the lower bound for a complete graph network is established the results suggest that using distributed protocols results in a factor of log m increase in order relative to centralized protocols

guaranteed and robust a posteriori error estimates and balancing discretization and linearization errors for monotone nonlinear problems abstract we derive a posteriori error estimates for a class of second order monotone quasi linear diffusion type problems approximated by piecewise affine continuous finite elements our estimates yield a guaranteed and fully computable upper bound on the error measured by the dual norm of the residual as well as a global error lower bound up to a generic constant independent of the nonlinear operator they are thus fully robust with respect to the nonlinearity thanks to the choice of the error measure they are also locally efficient albeit in a different norm and hence suitable for adaptive mesh refinement moreover they allow to distinguish estimate separately and compare the discretization and linearization errors hence the iterative newton raphson fixed point linearization can be stopped whenever the linearization error drops to the level at which it does not affect significantly the overall error this can lead to important computational savings as performing an excessive number of unnecessary linearization iterations can be avoided a strategy combining the linearization stopping criterion and adaptive mesh refinement is proposed and numerically tested for the p laplacian

an efficient spectral method for ordinary differential equations with rational function coefficients the authors present some relations that allow the efficient approximate inversion of linear differential operators with rational function coefficients they employ expansions in terms of a large class of orthogonal polynomial families including all the classical orthogonal polynomials these families obey a simple 3 term recurrence relation for differentiation which implies that on an appropriately restricted domain the differentiation operator has a unique banded inverse the inverse is an integration operator for the family and it is simply the tridiagonal coefficient matrix for the recurrence since in these families convolution operators i e matrix representations of multiplication by a function are banded for polynomials the authors are able to obtain a banded representation for linear differential operators with rational coefficients this leads to a method of solution of initial or boundary value problems that besides having an operation count that scales linearly with the order of truncation n is computationally well conditioned among the applications considered is the use of rational maps for the resolution of sharp interior layers 19 refs 2 figs 3 tabs

the coefficients of differentiated expansions and derivatives of ultraspherical polynomials abstract a formula expressing the ultraspherical coefficients of the general order derivative of an infinitely differentiable function in terms of its original ultraspherical coefficients is stated in a more compact form and proved in a simpler way than the formula suggested by karageorghis and phillips in their recent report 5 formulas expressing explicitly the derivatives of ultraspherical polynomials of any degree and for any order in terms of the ultraspherical polynomials are given the special cases of chebyshev polynomials of the first and second kinds and of legendre polynomials are considered an application of how to use ultraspherical polynomials for solving ordinary and partial differential equations is described

computable error estimators for the approximation of nonlinear problems by linearized models in the modeling of nonlinear phenomena a nonlinear model may often be replaced by a linear one giving rise to a modeling or linearization error this is in addition to the discretization error introduced when this linear model is solved using e g the finite element method we investigate the a posteriori estimation of these errors for a general class of problems characterized by strongly monotone operators our results lead to the construction of computable upper estimators for the total error with identifiable components from each of the these error sources several numerical tests evaluating the efficiency of our estimators are provided

on the connection coefficients and recurrence relations arising from expansions in series of laguerre polynomials a formula expressing the laguerre coefficients of a general order derivative of an infinitely differentiable function in terms of its original coefficients is proved and a formula expressing explicitly the derivatives of laguerre polynomials of any degree and for any order as a linear combination of suitable laguerre polynomials is deduced a formula for the laguerre coefficients of the moments of one single laguerre polynomial of certain degree is given formulae for the laguerre coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its laguerre coefficients are also obtained a simple approach in order to build and solve recursively for the connection coefficients between jacobi laguerre and hermite laguerre polynomials is described an explicit formula for these coefficients between jacobi and laguerre polynomials is given of which the ultra spherical polynomials of the first and second kinds and legendre polynomials are important special cases an analytical formula for the connection coefficients between hermite and laguerre polynomials is also obtained

construction of recurrences for the coefficients of expansions in q classical orthogonal polynomials an algorithmic approach is given to construct recurrence relations for the coefficients of the series expansion of a solution of a linear q difference equation in q classical orthogonal polynomials in the hahn s sense

a posteriori estimation of the linearization error for strongly monotone nonlinear operators we investigate the a posteriori estimation of the modeling or linearization error which arises when a nonlinear problem is replaced by a linear model using the context of strongly monotone operators we construct a computable upper estimator for this error and also provide an estimator that gives a lower bound several numerical results illustrating our theory are provided

second order recurrence relation for the linearization coefficients of the classical orthogonal polynomials abstract let pk be any sequence of the classical orthogonal polynomials we give explicitly a second order recurrence relation in k for the coefficients in p i p j k i j i j c i j k p k this result is obtained by a method which is alternative to the one proposed recently by a ronveaux et al applications of the result to some named systems of classical orthogonal polynomials are given

two way source coding with a fidelity criterion let x i y i i 1 2 cdots be a sequence of independent identically distributed bivariate random variables and consider the following communication situation the x component of the process is observed at some location say a while the y component is observed at a different location say b the x y component of the process has to be reproduced at location b a it is desired to find the amount of information that should be exchanged between the two locations so that the distortions incurred will not exceed some predetermined tolerance a single letter characterization of a certain region cal k subset cal r 4 of rates and distortions is given this region contains all and only the achievable rates and distortions for the special case where block coding is used and the information is conveyed through a one way link that can switch direction only k times per block

on the coefficients of differentiated expansions and derivatives of jacobi polynomials a formula expressing explicitly the derivatives of jacobi polynomials of any degree and for any order in terms of the jacobi polynomials themselves is proved another explicit formula which expresses the jacobi expansion coefficients of a general order derivative of an infinitely differentiable function in terms of its original jacobi coefficients is also given the results for the special case of ultraspherical polynomials are considered the results for chebyshev polynomials of the first and second kinds and for legendre polynomials are also noted an application of how to use jacobi polynomials for solving ordinary and partial differential equations is described

recurrence relations for connection coefficients between two families of orthogonal polynomials we describe a simple approach in order to build recursively the connection coefficients between two families of orthogonal polynomial solutions of second and fourth order differential equations

minimal recurrence relations for connection coefficients between classical orthogonal polynomials discrete case abstract we present a simple approach in order to compute recursively the connection coefficients between two families of classical discrete orthogonal polynomials charlier meixner kravchuk hahn i e the coefficients cm n in the expression p n x n m 0 c m n q m x where pn x and qm x belong to the aforementioned class of polynomials this is scv2 done by adapting a general and systematic algorithm recently developed by the authors to the discrete classical situation moreover extensions of this method allow to give new addition formulae and to estimate cm n asymptotics in limit relations between some families

expansions in series of orthogonal hypergeometric polynomials abstract let us consider an arbitrary hypergeometric polynomial qj x and a set of orthogonal hypergeometric polynomials p n x in the domain of orthogonality here the expansion coefficients of xm and xmqj x m n0 in series of the set p n x are found in terms of the polynomials x and x characterizing the second order differential equations satisfied by the involved hypergeometric polynomials the resulting general expressions which are given in an explicit and compact form are used to produce known for checking and unknown expansions for various concrete classical orthogonal polynomials

a note on the chebyshev coefficients of the general order derivative of an infinitely differentiable function abstract a formula expressing the chebyshev coefficients of the general order derivative of an infinitely differentiable function in terms of its chebyshev coefficients is suggested

finite element solution of nonlinear diffusion problems abstract in this paper we describe the rothe finite element numerical scheme to find an approximate solution of a nonlinear diffusion problem modeled as a parabolic partial differential equation of even order this scheme is based on the rothe s approximation in time and on the finite element method fem approximation in the spatial discretization a proof of convergence of the approximate solution is given and error estimates are shown

results for some inversion problems for classical continuous and discrete orthogonal polynomials explicit expressions for the coefficients in the expansion of classical discrete orthogonal polynomials charlier meixner krawtchouck hahn hahn eberlein into the falling factorial basis are given the corresponding inversion problems are solved explicitly this is done by using a general algorithm recently developed by the authors which is also applied to this kind of inversion problem but relating the basis and the classical continuous orthogonal polynomials of jacobi laguerre hermite and bessel

rate distortion theory for gaussian multiterminal source coding systems with several side informations at the decoder in this paper we consider the separate coding problem for l 1 correlated gaussian memoryless sources we deal with the case where l sources work as partial side information at the decoder for the reconstruction of the remaining source the determination problem of the rate distortion region for this system is the so called many help one problem and it has been known as a highly challenging problem for almost 20 years in this paper we give a partial solution to this problem we determine the rate distortion region in the case where the l sources working as partial side information are conditionally independent if the remaining source we wish to reconstruct is given the additive white gaussian noise ceo problem is a special case of this we also discuss the relation of the result previous results of ours

representations of orthogonal polynomials abstract zeilberger s algorithm provides a method to compute recurrence and differential equations from given hypergeometric series representations and an adaption of almquist and zeilberger computers recurrence and differential equations for hyperexponential integrals further versions of this algorithm allow the computation of recurrence and differential equations from rodrigues type formulas and from generating functions in particular these algorithms can be used to compute the differential difference and recurrence equations for the classical continuous and discrete orthogonal polynomials from their hypergeometric representations and from their rodrigues representations and generating functions in recent work we used an explicit formula for the recurrence equation of families of classical continuous and discrete orthogonal polynomials in terms of the coefficients of their differential difference equations to give an algorithm to identify the polynomial system from a given recurrence equation in this article we extend these results by presenting a collection of algorithms with which any of the conversions between the differential difference equation the hypergeometric representation and the recurrence equation is possible the main technique is again to use explicit formulas for structural identities of the given polynomial systems

recurrences and explicit formulae for the expansion and connection coefficients in series of bessel polynomials a formula expressing explicitly the derivatives of bessel polynomials of any degree and for any order in terms of the bessel polynomials themselves is proved another explicit formula which expresses the bessel expansion coefficients of a general order derivative of an infinitely differentiable function in terms of its original bessel coefficients is also given a formula for the bessel coefficients of the moments of one single bessel polynomial of certain degree is proved a formula for the bessel coefficients of the moments of a general order derivative of an infinitely differentiable function in terms of its bessel coefficients is also obtained application of these formulae for solving ordinary differential equations with varying coefficients by reducing them to recurrence relations in the expansion coefficients of the solution is explained an algebraic symbolic approach using mathematica in order to build and solve recursively for the connection coefficients between bessel bessel polynomials is described an explicit formula for these coefficients between jacobi and bessel polynomials is given of which the ultraspherical polynomial and its consequences are important special cases two analytical formulae for the connection coefficients between laguerre bessel and hermite bessel are also developed

linearization and connection coefficients for hypergeometric type polynomials we consider the problem of finding closed analytical formulas for both the linearization and connection coefficients for hypergeometric type polynomials directly in terms of the corresponding differential equations we illustrate the method by producing explicit formulas for hermite polynomials

coding with side information for rate constrained consensus average consensus algorithms are protocols to compute the average value of all sensor measurements via near neighbors communications they offer a natural tradeoff between the number of messages exchanged among terminals and the accuracy in the computation most of the models adopted for the message exchange in the literature however neither include explicit rate constraints nor explore the rate distortion tradeoff associated with the algorithm the contribution of our work is in examining the impact of such constraints and in finding strategies to minimize the communication cost in terms of rate the main motivation behind the proposed coding strategies is the observation that consensus algorithms offer the perfect example of a network communication problem where there is an increasing correlation between the data exchanged as the algorithm iterates henceforth it is possible to utilize previously exchanged data and current side information to significantly reduce the demands of quantization bit rate for a certain precision we analyze the case of a network where the links are assumed to be reliable at a constant bit rate we explore the conditions on the quantization noise which lead to a consensus value whose mean squared distance from the initial average is bounded in the case of infinite length vector coding with gaussian states we show that our proposed schemes achieve bounded convergence with vanishing rates as the iteration index tends to infinity

efficient spectral galerkin algorithms for direct solution of second order equations using ultraspherical polynomials it is well known that spectral methods tau galerkin collocation have a condition number of o n4 n is the number of retained modes of polynomial approximations this paper presents some efficient spectral algorithms which have a condition number of o n2 based on the ultraspherical galerkin methods for second order elliptic equations in one and two space variables the key to the efficiency of these algorithms is to construct appropriate base functions which lead to systems with specially structured matrices that can be efficiently inverted the complexities of the algorithms are a small multiple of nd 1 operations for a d dimensional domain with n 1 d unknowns while the convergence rates of the algorithms are exponential for problems with smooth solutions

an accurate solution of the poisson equation by the chebyshev tau method abstract a new tau method is presented for the two dimensional poisson equation comparison of the results for the test problem u x y sin 4 x sin 4 y with those computed by haidvogel and zang using the matrix diagonalization method and dang vu and delcarte using the chebyshev collocation method indicates that our method would be more accurate

a class of nonlinear elliptic problems we obtain a strict coercivity estimate generalizing that of t i seidman j differential equations 19 1975 242 257 in considering spatial variation for second order elliptic operators a u u with radial in the gradient a for rm the estimate is then applied to obtain existence of solutions of boundary value problems a u u u f u u with dirichlet conditions

on the coefficients of integrated expansions of ultraspherical polynomials the problem of expressing the coefficients of an expansion of orthogonal polynomials that has been integrated an arbitrary number of times in terms of the coefficients of the original expansion is considered a formula is proved for the ultraspherical polynomials of which the chebyshev and legendre polynomials are important special cases