design of capacity approaching irregular low density parity check codes we design low density parity check ldpc codes that perform at rates extremely close to the shannon capacity the codes are built from highly irregular bipartite graphs with carefully chosen degree patterns on both sides our theoretical analysis of the codes is based on the work of richardson and urbanke see ibid vol 47 no 2 p 599 618 2000 assuming that the underlying communication channel is symmetric we prove that the probability densities at the message nodes of the graph possess a certain symmetry using this symmetry property we then show that under the assumption of no cycles the message densities always converge as the number of iterations tends to infinity furthermore we prove a stability condition which implies an upper bound on the fraction of errors that a belief propagation decoder can correct when applied to a code induced from a bipartite graph with a given degree distribution our codes are found by optimizing the degree structure of the underlying graphs we develop several strategies to perform this optimization we also present some simulation results for the codes found which show that the performance of the codes is very close to the asymptotic theoretical bounds

analysis of dynamic algorithms in knuth s model abstract this paper analyzes the average behaviour of algorithms that operate on dynamically varying data structures subject to insertions i deletions d positive resp negative queries q resp q under the following assumptions if the size of the data structure is k k n then the number of possibilities for the operations d and q is a linear function of k whereas the number of possibilities for the ith insertion or negative query is equal to i this statistical model was introduced by francon 6 7 and knuth 12 and differs from the model used in previous analyses 2 7 integrated costs for these dynamic structures are defined as averages of costs taken over the set of all their possible histories i e evolutions considered up to order isomorphism of length n we show that the costs can be calculated for the data structures serving as implementations of linear lists priority queues and dictionaries the problem of finding the limiting distributions is also considered and the linear list case is treated in detail the method uses continued fractions and orthogonal polynomials but in a paper in preparation we show that the same results can be recovered with the help of a probabilistic model

how to get around a simple quadratic fold we analyse from a theoretical point of view a computational technique previously introduced 2 5 for tracing branches of solutions of nonlinear equations near simple quadratic folds

boundary conditions for the numerical solution of elliptic equations in exterior regions elliptic equations in exterior regions frequently require a boundary condition at infinity to ensure the weill posedness of the problem examples of practical applications include the helmholtz equation and laplace s equation computational procedures based on a direct discretization of the elliptic problem require the replacement of the condition at infinity by a boundary condition on a finite artificial surface direct imposition of the condition at infinity along the finite boundary results in large errors a sequence of boundary conditions is developed which provides increasingly accurate approximations to the problem in the infinite domain estimates of the error due to the finite boundary are obtained for several cases computations are presented which demonstrate the increased accuracy that can be obtained by the use of the higher order boundary conditions the examples are based on a finite element formulation but finite difference methods can also be used

an analysis of upwind schemes for the navier stokes equations we state a general result on approximation of branches of nonsingular solutions of nonlinear problems the abstract theory is then applied to the study of two mixed finite element schemes for the navier stokes equations using an upwind discretization of the convective terms

extrinsic information transfer functions model and erasure channel properties extrinsic information transfer exit charts are a tool for predicting the convergence behavior of iterative processors for a variety of communication problems a model is introduced that applies to decoding problems including the iterative decoding of parallel concatenated turbo codes serially concatenated codes low density parity check ldpc codes and repeat accumulate ra codes exit functions are defined using the model and several properties of such functions are proved for erasure channels one property expresses the area under an exit function in terms of a conditional entropy a useful consequence of this result is that the design of capacity approaching codes reduces to a curve fitting problem for all the aforementioned codes a second property relates the exit function of a code to its helleseth klove levenshtein information functions and thereby to the support weights of its subcodes the relation is via a refinement of information functions called split information functions and via a refinement of support weights called split support weights split information functions are used to prove a third property that relates the exit function of a linear code to the exit function of its dual

nonlinear analysis of the iterative decoding of parallel concatenated convolutional codes in this correspondence we introduce a simple one dimensional 1 d nonlinear map to describe the iterates of the bit error rate ber of parallel concatenated convolutional codes pccc on the binary input gaussian channel a lower bound on this map is derived based upon the weight enumerator of the constituent codes thus enabling the characterization of the dynamics of the decoder in terms of fixed points along with the associated stability analysis

numerical solution of the three dimensional ginzburg landau models using artificial boundary for three dimensional simulations of the vortex phenomena in superconductors based on the ginzburg landau gl theory the physical variables must be solved in the whole space in general exact boundary conditions on an artificial boundary are discussed to reformulate the problem in a finite domain approximations of these boundary conditions with high accuracy are given and their convergence properties are examined error estimates are derived for the finite element approximations with the approximate boundary conditions

analysis of the iterative decoding of ldpc and product codes using the gaussian approximation we propose a novel density evolution approach to analyze the iterative decoding algorithms of low density parity check ldpc codes and product codes based on gaussian densities namely for these classes of codes we derive a one dimensional 1d map whose iterates directly represent the error probability both for the additive white gaussian noise awgn and the rayleigh fading channel these simple models allow a qualitative analysis of the nonlinear dynamics of the decoding algorithm as an application we compute the decoding thresholds and show that they are consistent with the simulation results available in the literature

analysis and approximation of the ginzburg landau model of superconductivity the authors consider the ginzburg landau model for superconductivity first some well known features of superconducting materials are reviewed and then various results concerning the model the resultant differential equations and their solution on bounded domains are derived then finite element approximations of the solutions of the ginzburg landau equations are considered and error estimates of optimal order are derived

analysis of sum product decoding of low density parity check codes using a gaussian approximation density evolution is an algorithm for computing the capacity of low density parity check ldpc codes under message passing decoding for memoryless binary input continuous output additive white gaussian noise awgn channels and sum product decoders we use a gaussian approximation for message densities under density evolution to simplify the analysis of the decoding algorithm we convert the infinite dimensional problem of iteratively calculating message densities which is needed to find the exact threshold to a one dimensional problem of updating the means of the gaussian densities this simplification not only allows us to calculate the threshold quickly and to understand the behavior of the decoder better but also makes it easier to design good irregular ldpc codes for awgn channels for various regular ldpc codes we have examined thresholds can be estimated within 0 1 db of the exact value for rates between 0 5 and 0 9 codes designed using the gaussian approximation perform within 0 02 db of the best performing codes found so far by using density evolution when the maximum variable degree is 10 we show that by using the gaussian approximation we can visualize the sum product decoding algorithm we also show that the optimization of degree distributions can be understood and done graphically using the visualization

the analysis of simple list structures abstract we present an analysis of simple lists either sorted or unsorted under the set of all their possible histories i e evolutions considered up to order isomorphism of length n using the theory of continued fractions and orthogonal polynomials flajolet francon and vuillemin have determined average costs of sequences of operations for many data structures of the dictionary or priority queue type we show here that for the simplest structures variance estimates can also be obtained the method uses continued fractions and properties of nonclassical q generalizations of hermite and laguerre polynomials

une methode numerique pour le calcul des points de retournement application a un probleme aux limites non lineaire an iterative method is presented for determining a turning point in a branch of solutions of an algebraic system of equations depending on a parameter one shows that the convergence is superlinear and that this method may constitute a complement for the newton method with a finite element method numerical results are given testing the process on the discrete version of a mildly nonlinear boundary value problem to determine the turning point in the positive solution branch

a finite element method for solving helmholtz type equations in waveguides and other unbounded domains a finite element method is described for solving helmholtz type boundary value problems in unbounded regions including those with infinite boundaries typical examples include the propagation of acoustic or electromagnetic waves in waveguides the radiation condition at infinity is based on separation of variables and differs from the classical sommerfeld radiation condition it is shown that the problem may be replaced by a boundary value problem on a fixed bounded domain the behavior of the solution near infinity is incorporated in a nonlocal boundary condition this problem is given a weak or variational formulation and the finite element method is then applied it is proved that optimal error estimates hold

finite dimensional approximation of nonlinear problems part ii limit points note univ paris 6 f 75230 paris 05 france ecole polytech ctr math appl cnrs equipe rech 747 f 91128 palaiseau france univ pavia cnr ist anal numer i 27100 pavia italy brezzi f univ pavia cnr ist math appl i 27100 pavia italy isi document delivery no lr761times cited 54cited reference count 13 reference asn article 1981 002doi 10 1007 bf01396184 record created on 2006 08 24 modified on 2017 05 12

analysis of low density parity check codes based on exit functions we exploit extrinsic information transfer functions of single parity check and repetition codes over the binary input additive white gaussian noise biawgn channel derived by the authors for asymptotic performance analysis of belief propagation decoding of low density parity check codes the approach is based on a gaussian approximation ga of the density evolution algorithm using the mutual information measure we show that this method allows more accurate prediction of the decoding threshold in the biawgn channel than the earlier known ga methods

numerical approximations of the ginzburg landau models for superconductivity in this paper we review various methods for the numerical approximations of the ginzburg landau models of superconductivity particular attention is given to the different treatment of gauge invariance in both the finite element finite difference and finite volume settings representative theoretical results typical numerical simulations and computational challenges are presented generalizations to other relevant models are also discussed

random walks gaussian processes and list structures abstract an asymptotic analysis of list structure properties leads to limiting gaussian markovian processes several costs functions are shown to have asymptotic normal distributions

a new density evolution approximation for ldpc and multi edge type ldpc codes this paper considers density evolution for low density parity check ldpc and multi edge type ldpc met ldpc codes over the binary input additive white gaussian noise channel we first analyze three single parameter gaussian approximations for density evolution and discuss their accuracy under several conditions namely at low rates with punctured and degree one variable nodes we observe that the assumption of symmetric gaussian distribution for the density evolution messages is not accurate in the early decoding iterations particularly at low rates and with punctured variable nodes thus single parameter gaussian approximation methods produce very poor results in these cases based on these observations we then introduce a new density evolution approximation algorithm for ldpc and met ldpc codes our method is a combination of full density evolution and a single parameter gaussian approximation where we assume a symmetric gaussian distribution only after density evolution messages closely follow a symmetric gaussian distribution our method significantly improves the accuracy of the code threshold estimation additionally the proposed method significantly reduces the computational time of evaluating the code threshold compared with full density evolution thereby making it more suitable for code design

finite dimensional approximation of nonlinear problems in the first two papers of this series 4 5 we have studied a general method of approximation of nonsingular solutions and simple limit points of nonlinear equations in a banach space we derive here general approximation results of the branches of solutions in the neighborhood of a simple bifurcation point the abstract theory is applied to the galerkin approximation of nonlinear variational problems and to a mixed finite element approximation of the von karman equations

finite element approximation of a periodic ginzburg landau model for type ii superconductors we consider efficient finite element algorithms for the computational simulation of type ii superconductors the algorithms are based on discretizations of a periodic ginzburg landau model periodicity is defined with respect to a non orthogonal lattice that is not necessarily aligned with the coordinate axes also the primary dependent variables employed in the model satisfy non standard quasi periodic boundary conditions after introducing the model we define finite element schemes derive error estimates of optimal order and present the results of some numerical calculations for a similar quality of simulation the resulting algorithms seem to be significantly less costly than are previously used numerical approximation methods

numerical solution to a mixed navier stokes darcy model by the two grid approach we study numerical methods for a coupled navier stokes darcy model which describes a fluid flow filtrating through porous media a decoupled and linearized two grid algorithm is proposed numerical analysis and experiments are presented to show the efficiency and effectiveness of the decoupled and linearized algorithm

an optimization based domain decomposition method for the navier stokes equations an optimization based nonoverlapping domain decomposition method for the solution of the navier stokes equations is presented the crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains the constraints are the navier stokes equations in the subdomains with suitably chosen boundary conditions along the interfaces we show that solutions of the minimization problem exist and derive an optimality system from which these solutions may be determined finite element approximations of the solutions of the optimality system are examined the domain decomposition method is also reformulated as a nonlinear least squares problem and the results of some numerical experiments are given

approximation of hopf bifurcation we make several assumptions on a nonlinear evolution problem ensuring the existence of a hopf bifurcation under a fairly general approximation condition we define a discrete problem which retains the bifurcation property and we prove an error estimate between the branches of exact and approximate periodic solutions

analysis and convergence of a covolume approximation of the ginzburg landau model of superconductivity in this paper we present the mathematical analysis of a covolume method for the approximations of the ginzburg landau gl model for superconductivity a nice feature of this approach is that the gauge invariance properties are retained in discrete approximations based on triangular grids we also use properties of discrete vector fields to study issues such as the gauge choices and their enforcement

an analysis of belief propagation on the turbo decoding graph with gaussian densities motivated by its success in decoding turbo codes we provide an analysis of the belief propagation algorithm on the turbo decoding graph with gaussian densities in this context we are able to show that under certain conditions the algorithm converges and that somewhat surprisingly though the density generated by belief propagation may differ significantly from the desired posterior density the means of these two densities coincide since computation of posterior distributions is tractable when densities are gaussian use of belief propagation in such a setting may appear unwarranted indeed our primary motivation for studying belief propagation in this context stems from a desire to enhance our understanding of the algorithm s dynamics in a non gaussian setting and to gain insights into its excellent performance in turbo codes nevertheless even when the densities are gaussian belief propagation may sometimes provide a more efficient alternative to traditional inference methods