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 and enumeration of absorbing sets for non binary graph based codes in this work we first provide the definition of absorbing sets for linear channel codes over non binary alphabets in a graphical representation of a non binary channel code an absorbing set can be described by a collection of topological and edge labeling conditions in the non binary case the equations relating neighboring variable and check nodes are over a non binary field and the edge weights are given by the non zero elements of that non binary field as a consequence it becomes more difficult for a given structure to satisfy the absorbing set constraints compared to the binary case this observation in part explains the superior performance of non binary codes over their binary counterparts we show that the conditions in the non binary absorbing set definition can be simplified in the case of non binary elementary absorbing sets based on these simplified conditions we provide design guidelines for finite length non binary codes free of small non binary elementary absorbing sets these guidelines demonstrate that even under the preserved topology the performance of a non binary graph based code in the error floor region can be substantially improved by manipulating edge weights so as to avoid small absorbing sets our various simulation results suggest that the proposed non binary absorbing set definition is useful for a range of code constructions and decoders finally by using both insights from graph theory and combinatorial techniques we establish the asymptotic distribution of non binary elementary absorbing sets for regular code ensembles

stability and convergence of two level iterative methods for the stationary incompressible magnetohydrodynamics with different reynolds numbers abstract in the paper we develop some two level finite element iterative methods and use these methods to solve the stationary incompressible magnetohydrodynamics mhd with different reynolds numbers under some different uniqueness conditions firstly we use the stokes type iterative method newton type iterative method and oseen type iterative method of m times on a coarse mesh with mesh size h and then solve a linear problem with the stokes type newton type and oseen type correction of one time on a fine grid with mesh sizes h h furthermore we analyze the uniform stability and convergence of these methods with respect to reynolds numbers mesh sizes h and h and iterative times m finally the stationary incompressible mhd equations with large reynolds number are researched by the one level finite element method based on the oseen type iteration on a fine mesh under a weak uniqueness condition

two level method based on finite element and crank nicolson extrapolation for the time dependent navier stokes equations a fully discrete two level finite element method the two level method is presented for solving the two dimensional time dependent navier stokes problem the method requires a crank nicolson extrapolation solution u h tau 0 p h tau 0 on a spatial time coarse grid j h tau 0 and a backward euler solution u h tau p h tau on a space time fine grid j h tau the error estimates of optimal order of the discrete solution for the two level method are derived compared with the standard crank nicolson extrapolation method the one level method based on a space time fine grid j h tau the two level method is of the error estimates of the same order as the one level method in the h1 norm for velocity and the l2 norm for pressure however the two level method involves much less work than the one level method

weight distributions of non binary multi edge type ldpc code ensembles analysis and efficient evaluation non binary multi edge type ensembles of low density parity check codes are analyzed in terms of non binary codeword weight distribution and its growth rate in particular an exact expression of the growth rate for small weights is developed as a side result the stopping set distributions of these ensembles are developed examples of weight distributions are provided showing that the derived closed form expressions can be easily evaluated the obtained results can thus be exploited to analyze and design non binary low density parity check codes that fall within the multi edge type framework such as but not limited to protograph based codes

some uzawa methods for steady incompressible navier stokes equations discretized by mixed element methods some uzawa type iterative methods are devised to solve steady incompressible navier stokes equations discretized by mixed element methods compared with the most existing iterative methods these methods require no numerical solution of any saddle point system at each iteration step after some novel and technical derivation it is proved that the methods converge geometrically with a contraction number independent of the finite element mesh size h even for regular triangulations a series of numerical experiments are reported to show the computational performance and accuracy of our methods proposed

two level method and some a priori estimates in unsteady navier stokes calculations a two level method proposed for quasielliptic problems is adapted in this paper to the simulation of unsteady incompressible navier stokes flows the method requires a solution of a nonlinear problem on a coarse grid and a solution of linear symmetric problem on a fine grid the scaling between these two grids is superlinear approximation stability and convergence aspects of a fully discrete scheme are considered stability properties of the two level scheme are compared with those for a commonly used semi implicit scheme some new estimates are also proved for the latter

non binary decoder diversity for dense or locally dense parity check codes in this paper a new and promising framework called non binary decoder diversity is presented based on the observation that different non binary tanner graphs of the same binary code decoded with a non binary belief propagation decoder can have distinct convergence behaviors and fixed points the goal of this work is to propose a decoder with linear complexity in the blocklength and with performance close to maximum likelihood decoding this framework is especially interesting for binary codes which are dense or locally dense and for which the usual binary iterative decoders perform far from the optimum curves by using the diversity brought by decoding distinct tanner graphs of the same code the proposed technique has very good decoding performance for three very different test cases which are known to be complex decoding problems i near maximum likelihood decoding mld of bch codes on the bpsk awgn channel ii performance results which outperform bounded distance decoding of bch codes over a binary symmetric channel bsc and finally iii decoding performance better than the bcjr based turbo decoder for parallel duo binary turbo codes

flow through porous media due to high pressure gradients abstract while darcy s equations are adequate for studying a large class of flows through porous media there are several situations wherein it would be inappropriate to use darcy s equations one such example is a flow wherein the range of pressures involved is very large and high pressures and pressure gradients are at play here after developing an approximation for the flow through a porous solid that is a generalization of an equation developed by brinkman we study a simple boundary value problem that clearly delineates the difference between the solution to these equations and those due to the equations that are referred to as darcy law we find that the solutions for the equations under consideration exhibit markedly different characteristics from the counterpart for the brinkman equations or darcy s equations or the navier stokes equation if one neglects the porosity in that the solutions for the velocity as well as the vorticity lack symmetry and one finds the maximum value of the vorticity occurs at the boundary near which the fluid is less viscous in virtue of the pressure being lower we also find that for a certain range of values for the non dimensional parameters describing the flow boundary layers develop in that the vorticity is confined next to the boundary adjacent to which the viscosity is lower such boundary layers being absent in the other classical cases

a simplified two level method for the steady navier stokes equations abstract a simplified two level method for solving the steady two dimensional incompressible navier stokes equations is investigated in this article the simplified two level method involves solving one small nonlinear navier stokes problem on the coarse mesh and one linear stokes problem hence linear with positive definite symmetric part on the fine mesh the uniform optimal error estimates with respect to large for the standard galerkin method and the two level method are proven finally some numerically tests to confirm the theoretical results of the simplified two level method and the standard galerkin method are provided

multi level spectral galerkin method for the navier stokes problem i spatial discretization a multi level spectral galerkin method for the two dimensional non stationary navier stokes equations is presented the method proposed here is a multiscale method in which the fully nonlinear navier stokes equations are solved only on a low dimensional space subsequent approximations are generated on a succession of higher dimensional spaces j 2 j by solving a linearized navier stokes problem around the solution on the previous level error estimates depending on the kinematic viscosity 0 1 are also presented for the j level spectral galerkin method the optimal accuracy is achieved when we demonstrate theoretically that the j level spectral galerkin method is much more efficient than the standard one level spectral galerkin method on the highest dimensional space

a recursive approach to low complexity codes a method is described for constructing long error correcting codes from one or more shorter error correcting codes referred to as subcodes and a bipartite graph a graph is shown which specifies carefully chosen subsets of the digits of the new codes that must be codewords in one of the shorter subcodes lower bounds to the rate and the minimum distance of the new code are derived in terms of the parameters of the graph and the subeodes both the encoders and decoders proposed are shown to take advantage of the code s explicit decomposition into subcodes to decompose and simplify the associated computational processes bounds on the performance of two specific decoding algorithms are established and the asymptotic growth of the complexity of decoding for two types of codes and decoders is analyzed the proposed decoders are able to make effective use of probabilistic information supplied by the channel receiver e g reliability information without greatly increasing the number of computations required it is shown that choosing a transmission order for the digits that is appropriate for the graph and the subcodes can give the code excellent burst error correction abilities the construction principles

on error estimates of the projection method for the time dependent natural convection problem in this article a projection method or fractional step method is proposed and analyzed for the time dependent natural convection problem in two dimensions based on this method the considered problem is decoupled into two subproblems one is for the velocity and temperature and the other is for the pressure firstly a linear elliptic equation is treated for the temperature then a linear elliptic equation is solved for the velocity finally a poisson equation is considered for the pressure stability is performed and the convergence results of the projection method are derived furthermore a modified projection scheme which leads to some improved error estimates is also proposed finally some numerical examples are presented to verify the established theoretical findings

numerical solution of the stationary navier stokes equations using a multilevel finite element method the numerical implementation of a multilevel finite element method for the steady state navier stokes equations is considered the multilevel method proposed here for the navier stokes equations is a multiscale method in which the full nonlinear navier stokes equations are only solved on a single coarse grid subsequent approximations are generated on a succession of refined grids by solving a linearized navier stokes problem two numerical examples are considered the first is an example for which an exact solution is known and the second is the driven cavity problem we demonstrate numerically that for an appropriate choice of grids a two or three level finite element method is significantly more efficient than the standard one level finite element method

solving steady incompressible navier stokes equations by the arrow hurwicz method this article is devoted to analyzing an arrow hurwicz type method for solving incompressible navier stokes equations discretized by mixed element methods under several reasonable conditions it is proved by a subtle argument that the method converges geometrically with a contraction number independent of the finite element mesh size h even for regular triangulations a series of numerical examples are provided to illustrate the computational performance of the method

interface evolution the hele shaw and muskat problems we study the dynamics of the interface between two incompressible 2 d ows where the evolution equation is obtained from darcy s law the free boundary is given by the discontinuity among the densities and viscosities of the uids this physical scenario is known as the two dimensional muskat problem or the two phase hele shaw ow we prove local existence in sobolev spaces when initially the dierence of the gradients of the pressure in the normal direction has the proper sign an assumption which is also known as the rayleigh taylor condition

a local and parallel uzawa finite element method for the generalized navier stokes equations abstract in this paper we propose and develop a local and parallel uzawa finite element method for the generalized navier stokes equations the uzawa finite element method is no need to deal with the saddle point problem and only solves one vector valued elliptic equation and one simple scalar valued equation it has the geometric convergence with a crispation number what has nothing to do with the mesh size h as for the local and parallel uzawa finite element method each subproblem is a global problem but most of degrees of freedom originate from the subdomain moreover the presented method is easy to be applied with less communication requirements and has good parallelism finally numerical results verify the performance of the proposed method

an exact solution to the one phase zero surface tension hele shaw free boundary problem abstract a family of explicit solutions are presented to the one phase zero surface tension hele shaw problem with a single source sink the free boundary is described as a function of three parameters whose dependence on time is evaluated through two different yet complementary methods possible mechanisms for nonexistence beyond a critical time are found in the case of both a contracting and expanding fluid region

performance comparison of ldpc block and spatially coupled codes over gf q in this paper we compare the finite length performance of protograph based spatially coupled low density paritycheck sc ldpc codes and ldpc block codes ldpc bcs over gf q to reduce computational complexity and latency a sliding window decoder with a stopping rule based on a soft belief propagation bp estimate is used for the q ary sc ldpc codes two regimes are considered one when the constraint length of q ary sc ldpc codes is equal to the block length of q ary ldpc bcs and the other when the two decoding latencies are equal simulation results confirm that in both regimes 3 6 3 9 and 3 12 regular non binary sc ldpc codes can significantly outperform both binary and non binary ldpc bcs and binary sc ldpc codes finally we present a computational complexity comparison of q ary sc ldpc codes and q ary ldpc bcs under equal decoding latency and equal decoding performance assumptions

iterative schemes for the non homogeneous navier stokes equations based on the finite element approximation in this paper we consider the stability and convergence of three iterative schemes for the non homogeneous steady navier stokes equations as a nonlinear problem we will get a nonlinear discrete system if approximating the non homogeneous navier stokes equations after proving the stability and error estimates of the finite element method for the non homogeneous navier stokes equations three iterative schemes are investigated for solving the resulted nonlinear discrete system the stability and convergence conditions for these iterative methods are also analyzed respectively furthermore new results for the stop criterion are proved finally we show some numerical experiments to illustrate the theoretical prediction

non binary protograph based ldpc codes enumerators analysis and designs this paper provides a comprehensive analysis of nonbinary low density parity check ldpc codes built out of protographs we consider both random and constrained edge weight labeling and refer to the former as the unconstrained nonbinary protograph based ldpc codes u nbpb codes and to the latter as the constrained nonbinary protograph based ldpc codes c nbpb codes equipped with combinatorial definitions extended to the nonbinary domain ensemble enumerators of codewords trapping sets stopping sets and pseudocodewords are calculated the exact enumerators are presented in the finite length regime and the corresponding growth rates are calculated in the asymptotic regime an exit chart tool for computing the iterative decoding thresholds of protograph based ldpc codes is presented followed by several examples of finite length u nbpb and c nbpb codes with high performance throughout this paper we provide accompanying examples which demonstrate the advantage of nonbinary protograph based ldpc codes over their binary counterparts and over random constructions the results presented in this paper advance the analytical toolbox of nonbinary graph based codes

stability and convergence of iterative methods related to viscosities for the 2d 3d steady navier stokes equations abstract some finite element iterative methods related to viscosities are designed to solve numerically the steady 2d 3d navier stokes equations the two level finite element iterative methods are designed to solve numerically the steady 2d 3d navier stokes equations for a large viscosity such that a strong uniqueness condition holds the two level finite element iterative methods consist of using the stokes newton and oseen iterations of m times on a coarse mesh with mesh size h and computing the stokes newton and oseen correction of one time on a fine grid with mesh size h h moreover the one level oseen finite element iterative method based on a fine mesh with a small mesh size is designed to solve numerically the steady 2d 3d navier stokes equations for small viscosity such that a weak uniqueness condition holds the uniform stability and convergence of these methods with respect to and mesh sizes h and h and iterative times m are provided

convergence analysis of three finite element iterative methods for the 2d 3d stationary incompressible magnetohydrodynamics abstract three finite element iterative methods are designed and analyzed for solving 2d 3d stationary incompressible magnetohydrodynamics mhd by a new technique strong uniqueness conditions for both stokes type iterative method iterative method i and newton iterative method iterative method ii are obtained which are weaker than the ones reported in open literature stability and optimal convergence rates for the above two methods are derived where the iterative method ii has an exponential convergent part with respect to iterative step m moreover oseen type iterative method iterative method iii is unconditionally stable and convergent under a uniqueness condition finally performance of the three proposed methods is investigated by numerical experiments

high re solutions for incompressible flow using the navier stokes equations and a multigrid method the vorticity stream function formulation of the two dimensional incompressible navierstokes equations is used to study the effectiveness of the coupled strongly implicit multigrid csi mg method in the determination of high re fine mesh flow solutions the driven flow in a square cavity is used as the model problem solutions are obtained for configurations with reynolds number as high as 10 000 and meshes consisting of as many as 257 x 257 points for re 1000 the 129 x 129 grid solution required 1 5 minutes of cpu time on the amdahl 470 v 6 computer because of the appearance of one or more secondary vortices in the flow field uniform mesh refinement was preferred to the use of one dimensional gridclustering coordinate transformations the past decade has witnessed a great deal of progress in the area of computational fluid dynamics developments in computer technology hardware as well as in advanced numerical algorithms have enabled attempts to be made towards analysis and numerical solution of highly complex flow problems for some of these applications the use of simple iterative techniques to solve the navier stokes equations leads to a rather slow convergence rate for the solutions the solution convergence rate can be seriously affected if the coupling among the various governing differential equations is not properly honored either in the interior of the solution domain or at its boundaries the rate of convergence is also generally strongly dependent on such problem parameters as the reynolds number the mesh size and the total number of computational points this has led several researchers to examine carefully the recently emerging multigrid mg technique as a useful means for enhancing the convergence rate of iterative numerical methods for solving discretized equations at a number of computational grid points so large as to be considered impractical previously

a multilevel mesh independence principle for the navier stokes equations multilevel finite element discretization methods for the navier stokes equations are considered in contrast to usual multilevel methods a superlinear scaling of the consecutive meshwidths h j 1 mathcal o h j alpha j is used on the coarsest mesh the discretized system is solved which is small and nonlinear on each subsequent mesh only one or two newton correction steps are performed so that only one or two larger linear systems are solved the scalings of the meshwidths that lead to optimal accuracy of the approximate solution in both the h 1 and l 2 norm are investigated an error analysis of independent interest is also presented for the basic finite element method using the conservation form of the nonlinear term this formulation leads to a nonlinear system whose nonlinearity is easier to resolve than the systems arising when either the convective or explicitly skew symmetric corms are used for the nonlinear term

finite element approximation of the nonstationary navier stokes problem i regularity of solutions and second order error estimates for spatial discretization this is the first part of a work dealing with the rigorous error analysis of finite element solutions of the nonstationary navier stokes equations second order error estimates are proven for spatial discretization using conforming or nonconforming elements the results indicate a fluid like behavior of the approximations even in the case of large data so long as the solution remains regular the analysis is based on sharp a priori estimates for the solution particularly reflecting its behavior as t to 0 and as t to infty it is shown that the regularity customarily assumed in the error analysis for corresponding parabolic problems cannot be realistically assumed in the case of the navier stokes equations as it depends on nonlocal compatibility conditions for the data the results which are presented here are independent of such compatibility conditions which cannot be verified in practice

nonlinear galerkin methods and mixed finite elements two grid algorithms for the navier stokes equations a nonlinear galerkin method using mixed finite elements is presented for the two dimensional incompressible navier stokes equations the scheme is based on two finite element spaces x h and x h for the approximation of the velocity defined respectively on one coarse grid with grid size h and one fine grid with grid size h ll h and one finite element space m h for the approximation of the pressure nonlinearity and time dependence are both treated on the coarse space we prove that the difference between the new nonlinear galerkin method and the standard galerkin solution is of the order of h 2 both in velocity h 1 omega and pressure l 2 omega norm we also discuss a penalized version of our algorithm which enjoys similar properties

trellis based extended min sum algorithm for non binary ldpc codes and its hardware structure in this paper we present an improvement and a new implementation of a simplified decoding algorithm for non binary low density parity check codes nb ldpc in galois fields gf q the base algorithm that we use is the extended min sum ems algorithm which has been widely studied in the recent literature and has been shown to approach the performance of the belief propagation bp algorithm with limited complexity in our work we propose a new way to compute modified configuration sets using a trellis representation of incoming messages to check nodes we call our modification of the ems algorithm trellis ems t ems in the t ems the algorithm operates directly on the deviation space by considering a trellis built from differential messages which serves as a new reliability measure to sort the configurations we show that this new trellis representation reduces the computational complexity without any performance degradation in addition we show that our modifications of the algorithm allows to greatly reduce the decoding latency by using a larger degree of hardware parallelization

convergence of three iterative methods based on the finite element discretization for the stationary navier stokes equations abstract this paper considers three iterative methods for solving the stationary navier stokes equations iterative method i consists in solving the stationary stokes equations iterative method ii consists in solving the stationary linearized navier stokes equations and iterative method iii consists in solving the stationary oseen equations under the finite element discretization respectively at each iterative step also we discuss the stability and convergence of three iterative methods the iterative methods i and ii are stability and convergence under the strong uniqueness conditions where the iterative method ii is the second order convergence furthermore the iterative method iii is uncondition stability and convergence under the uniqueness condition finally some numerical tests show that the efficiency of the theoretical analysis

some iterative finite element methods for steady navier stokes equations with different viscosities two level iterative finite element methods are designed to solve numerically the steady 2d 3d navier stokes equations for a large viscosity n such that a strong uniqueness condition holds moreover the one level oseen iterative finite element method based on a fine mesh with small mesh size h is designed to solve numerically the steady 2d 3d navier stokes equations for small viscosity n such that a weak uniqueness condition holds meanwhile the numerical investigations are provided to show that the proposed methods are efficient for solving the 2d 3d steady navier stokes equations for different viscosities

newton iterative method based on finite element discretization for the stationary darcy brinkman equations abstract the stationary darcy brinkman equations in the double diffusive convection which model the heat and mass transfer phenomena are considered in this paper based on a suitable contractive operator the existence and uniqueness of the problem are firstly proved by using the fixed point theorem the regularities of the weak solution are also derived then the newton iterative method is studied for solving the nonlinear discrete system generated from the finite element approximation including the stability and the optimal error estimates regarding the spatial mesh size and the iterative factor the analysis indicates that the viscosity coefficient has more impact on the numerical algorithm than the thermal conductivity and the mass diffusivity coefficients finally many numerical examples are shown to confirm the correctness of the theoretical prediction

uniform error estimates of finite element method for a 3d semilinear elliptic problem with small viscosity abstract a finite element method based on p 1 element for the 3d semilinear elliptic boundary value problem with small viscosity is considered the existence uniqueness and uniform optimal h 1 and suboptimal l 2 error estimates of the finite element solution are provided where 0 1 also the numerical tests are made to show the efficiency of the error estimates of the finite element solution for the 3d semilinear elliptic boundary value problem with small viscosity

multiple steady bubbles in a hele shaw cell this paper presents new solutions in analytical form for the shapes of an assembly of steady co travelling bubbles in a hele shaw cell the associated velocity field is also derived the assembly can consist of any finite number of bubbles the solutions are expressed in terms of schottky klein prime functions and are derived by exploiting results on multiply connected conformal slit mappings

design of spatially coupled ldpc codes over gf q for windowed decoding in this paper we study spatially coupled low density parity check sc ldpc codes over finite fields gf q q geq 2 and develop design rules for q ary sc ldpc code ensembles based on their iterative belief propagation decoding thresholds with particular emphasis on low latency windowed decoding wd we consider transmission over both the binary erasure channel bec and the binary input additive white gaussian noise channel biawgnc and present results for a variety of j k regular sc ldpc code ensembles constructed over gf q using protographs thresholds are calculated using the protograph versions of q ary density evolution for the bec and the q ary extrinsic information transfer analysis for the biawgnc we show that the wd of q ary sc ldpc codes provides significant threshold gains compared with corresponding uncoupled q ary ldpc block code ldpc bc ensembles when the window size w is large enough and that these gains increase as the finite field size q 2 m increases moreover we demonstrate that the new design rules provide wd thresholds that are close to capacity even when both m and w are relatively small thereby reducing decoding complexity and latency the analysis further shows that compared with standard flooding schedule decoding the wd of q ary sc ldpc code ensembles results in significant reductions in both the decoding complexity and the decoding latency and that these reductions increase as m increases for the applications with a near threshold performance requirement and a constraint on decoding latency we show that using q ary sc ldpc code ensembles with moderate q 2 instead of their binary counterparts results in reduced decoding complexity

reliability based joint detection decoding algorithm for nonbinary ldpc coded modulation systems this paper studies an extension and improvement of the joint detection decoding algorithm for nonbinary ldpc coded modulation systems the iterative joint detection decoding ijdd algorithm in 1 combines nonbinary ldpc decoding with signal detection based on the hard message passing strategy resulting in significantly reduced decoding complexity however it applies only to majority logic decodable nonbinary ldpc codes with high column weight for nonbinary ldpc codes with low column weight a noticeable performance loss will be incurred to handle this problem we propose a reliability based iterative joint detection decoding also termed improved ijdd algorithm which combines the accumulated reliability of symbols based on the one step majority logic decoding mlgd algorithm and a chase like local list decoding algorithm simulation results show that the improved ijdd algorithm outperforms the ijdd algorithm by about 0 3 db using nonbinary ldpc codes with high column weight and by about 3 db using nonbinary ldpc codes with low column weight d v 4 while maintaining the low complexity of decoding compared to the fft qspa the proposed algorithm has a performance degradation of 0 5 db in the high column weight regime and about 1 db in the low column weight regime

strongly φ like functions of order α in two dimensional free boundary problems abstract in this paper we apply certain results in the theory of univalent functions to investigate the time evolution of the free boundary of a viscous fluid for a planar flow problem in the hele shaw cell model under injection to this end we prove that the property of strongly likeness of order 0 1 is preserved in time for both inner and outer problems under the assumption of nonzero small surface tension

design and analysis of nonbinary ldpc codes for arbitrary discrete memoryless channels we present an analysis under the iterative decoding of coset low density parity check ldpc codes over gf q designed for use over arbitrary discrete memoryless channels particularly nonbinary and asymmetric channels we use a random coset analysis to produce an effect that is similar to output symmetry with binary channels we show that the random selection of the nonzero elements of the gf q parity check matrix induces a permutation invariance property on the densities of the decoder messages which simplifies their analysis and approximation we generalize several properties including symmetry and stability from the analysis of binary ldpc codes we show that under a gaussian approximation the entire q 1 dimensional distribution of the vector messages is described by a single scalar parameter like the distributions of binary ldpc messages we apply this property to develop extrinsic information transfer exit charts for our codes we use appropriately designed signal constellations to obtain substantial shaping gains simulation results indicate that our codes outperform multilevel codes at short block lengths we also present simulation results for the additive white gaussian noise awgn channel including results within 0 56 db of the unrestricted shannon limit i e not restricted to any signal constellation at a spectral efficiency of 6 bits s hz

bounds on the belief propagation threshold of non binary ldpc codes we consider low density parity check ldpc code ensembles over non binary galois fields when used for transmission over arbitrary discrete memoryless channels belief propagation decoding for these codes has been shown to achieve excellent results however computing the decoding threshold using density evolution is usually impractical since one needs to propagate multi dimensional probability distributions and monte carlo simulations are required instead by considering the evolution of the message bhattacharyya parameter and the message expected value parameter we derive a simple lower bound on the performance of the algorithm this bound applies for both regular and irregular non binary ldpc ensembles

error correction method for navier stokes equations at high reynolds numbers in this paper we present a new iterative method for solving the stationary navier stokes equations nses at high reynolds numbers the method consists of first solving the nses by the oseen iterative scheme and then an error correction strategy is implemented to control the error arising from the linearization of the nonlinear nses the new method retains the advantage of the classical oseen scheme but it leads to a rapid rate of convergence and also enhances the capability for solving problems with higher reynolds numbers it will be shown that under the uniqueness condition the proposed method accelerates up to a factor of three in the convergence rate the stability analysis and error estimate are presented furthermore numerical simulations using the new method and other classical schemes are reported to verify the superior performance of the proposed method

the stabilized extrapolated trapezoidal finite element method for the navier stokes equations abstract we consider a fully discrete stabilized finite element method for the navier stokes equations which is unconditionally stable and has second order temporal accuracy of o k 2 hk spatial error the method involves a simple artificial viscosity stabilization of the linear system for the approximation of the new time level connected to anti diffusion of its effects at the old time level lowering the cell reynolds number of the oseen problem to o 1 the method requires only the solution of one linear system arising from an oseen problem per time step the effectiveness of the method is illustrated in several numerical experiments

two enhanced reliability based decoding algorithms for nonbinary ldpc codes the weighted bit reliability based wbrb algorithm for nonbinary ldpc codes suffers certain loss of symbol reliability thus this paper enhances its soft decision version by passing multiple symbol reliability instead of bit reliability furthermore it demonstrates that plurality robustly indicates symbol reliability of extrinsic information sums thus this paper enhances the hard decision version by introducing symbol reliability from plurality analysis results show that these two enhanced decoding algorithms significantly outperform the wbrb algorithm with reasonable overhead

a rational high order compact difference method for the steady state stream functionvorticity formulation of the navierstokes equations a rational high order compact rhoc finite difference fd method on the nine point stencil is proposed for solving the steady state two dimensional navierstokes equations in the stream functionvorticity form the resulting system of algebra equations can be solved by using the point successive over or under relaxation sor iteration numerical experiments involving two linear and two nonlinear problems with their analytical solutions and two flow problems including the lid driven cavity and backward facing step flows are carried out to validate the performance of the newly proposed method numerical solutions of the driven cavity problem with different grid mesh sizes maximum being 513513 for reynolds numbers ranging from 0 to 17500 are obtained and compared with some of the accurate results available in the literature

decoding algorithms for nonbinary ldpc codes over gf q in this letter we address the problem of decoding nonbinary low density parity check ldpc codes over finite fields gf q with reasonable complexity and good performance in the first part of the letter we recall the original belief propagation bp decoding algorithm and its fourier domain implementation we show that the use of tensor notations for the messages is very convenient for the algorithm description and understanding in the second part of the letter we introduce a simplified decoder which is inspired by the min sum decoder for binary ldpc codes we called this decoder extended min sum ems we show that it is possible to greatly reduce the computational complexity of the check node processing by computing approximate reliability measures with a limited number of values in a message by choosing appropriate correction factors or offsets we show that the ems decoder performance is quite good and in some cases better than the regular bp decoder the optimal values of the factor and offset correction are obtained asymptotically with simulated density evolution our simulations on ultra sparse codes over very high order fields show that nonbinary ldpc codes are promising for applications which require low frame error rates for small or moderate codeword lengths the ems decoder is a good candidate for practical hardware implementations of such codes

galerkin method and optimal error algorithms in hilbert spaces we consider operator equations of the formlu f wherel belongs to the class of linear bounded by a constantm and coercive with a constantm operators from a hilbert spacev onto its dualv andf belongs to a hilbert spacew v we study optimality of the galerkin methodp n lu n p n f whereu n v n v n is subspace ofv p n is the orthogonal projector ontov n andp n is dual top n we show that the galerkin method is quasi optimal independently of the choice of the subspacev n and the spacew ifm m in the casem m optimality of the method depends strongly on the choice ofv n andw therefore we define a new algorithm which is always optimal independently of the choice ofv n andw and relations betweenm andm

parallel vs sequential belief propagation decoding of ldpc codes over gf q and markov sources abstract a sequential updating scheme sus for belief propagation bp decoding of ldpc codes over galois fields gf q and correlated markov sources is proposed and compared with the standard parallel updating scheme pus a thorough experimental study of various transmission settings indicates that the convergence rate in iterations of the bp algorithm for the sus is about one half of that for the pus independent of the finite field size q moreover this 1 2 factor appears regardless of the correlations of the source and the channel s noise model while the error correction performance remains unchanged these results may imply on the universality of the one half convergence speed up of sus decoding a comparison to the dynamics of physical spin systems is also addressed

a two level method with backtracking for the navier stokes equations we consider a two level method for resolving the nonlinearity in finite element approximations of the equilibrium navier stokes equations the method yields l2 and h1 optimal velocity approximations and an l2 optimal pressure approximation the two level method involves solving one small nonlinear coarse mesh system one oseen problem hence linear with positive definite symmetric part on the fine mesh and one linear correction problem on the coarse mesh the algorithm we study produces an approximate solution with the optimal asymptotic in h accuracy for any fixed reynolds number we do not consider the behavior of the error for fixed h as re rightarrow infty i e for flows in transition to turbulence

iterative methods in penalty finite element discretization for the steady mhd equations this paper characterizes one penalty finite element method for the incompressible mhd equations the method is an interesting combination of the classic iterative schemes stokes newton and oseen iterations with two different finite element pairs p1b p1 p1b and p1 p0 p1 moreover the rigorous analysis of stability and error estimate for the proposed methods are given finally the applicability and effectiveness of the presented schemes are illustrated in several numerical experiments

starlike approximations of analytic functions abstract when an analytic function is not univalent it is often of interest to approximate it by univalent functions in this paper we introduce a measure of the non univalency of a function and we derive a method for constructing the best starlike univalent approximations of analytic functions with respect to it suitable for both practical problems and numerical implementation

symbol flipping decoding algorithms based on prediction for non binary ldpc codes this paper constructs an objective function for symbol flipping decoding algorithms considering not only soft reliability but also hard reliability the maximization of this objective function indicates that the flipping metric should involve both the information before and after symbol flipping while the existing algorithms consider the information before symbol flipping theoretical analysis shows that such prediction mechanism together with hard reliability can significantly improve the error performance of symbol flipping algorithms simulation results show that the proposed algorithms provide effective tradeoff between error performance and complexity for decoding non binary ldpc codes

stability and convergence of a finite element method for a semilinear elliptical problem with small viscosity abstract in this paper we propose and analyze a new finite element method for a semilinear elliptical problem with small viscosity firstly we prove the existence and uniqueness of solution for its variational problem then we obtain the stability and convergence of the corresponding finite element galerkin approximation furthermore we derive the optimal error estimates in the l 2 and h 1 norms for the finite element approximations respectively finally several numerical experiments are provided to confirm the above theoretical results

shuffled iterative decoding shuffled versions of iterative decoding of low density parity check codes and turbo codes are presented the proposed schemes have about the same computational complexity as the standard versions and converge faster simulations show that the new schedules offer better performance complexity tradeoffs especially when the maximum number of iterations has to remain small

iterative finite element variational multiscale method for the incompressible navier stokes equations abstract three iterative finite element variational multiscale methods are proposed and applied to the numerical simulation of the navier stokes equations the main idea of these methods is to combine the finite element variational multiscale method based on two local gauss integrations with three different iterative schemes the existence and uniqueness of approximate solutions of these iterative finite element variational multiscale methods are proved firstly and then the convergence and error estimates of them are deduced finally some numerical examples are given to support the theoretical analysis the numerical results show that the iterative finite element variational multiscale method has a wider range of reynolds numbers than standard galerkin iterative finite element method and the oseen iterative scheme is much more efficient than the other two under high reynolds numbers

a novel two grid method for semilinear elliptic equations a new finite element discretization technique based on two coarse and fine subspaces is presented for a semilinear elliptic boundary value problem the solution of a nonlinear system on the fine space is reduced to the solution of two small one linear and one nonlinear systems on the coarse space and a linear system on the fine space it is shown both theoretically and numerically that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation as a result the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation

low complexity decoding for non binary ldpc codes in high order fields in this paper we propose a new implementation of the extended min sum ems decoder for non binary ldpc codes a particularity of the new algorithm is that it takes into accounts the memory problem of the non binary ldpc decoders together with a significant complexity reduction per decoding iteration the key feature of our decoder is to truncate the vector messages of the decoder to a limited number nm of values in order to reduce the memory requirements using the truncated messages we propose an efficient implementation of the ems decoder which reduces the order of complexity to nm log2 nm this complexity starts to be reasonable enough to compete with binary decoders the performance of the low complexity algorithm with proper compensation is quite good with respect to the important complexity reduction which is shown both with a simulated density evolution approach and actual simulations

global classical solution of quasi stationary stefan free boundary problem we consider an one phase quasi stationary stefan problem hele shaw problem in multidimensional case under some reasonable conditions we prove that the problem has a classical solution globally in time the method can be used in two phase problem as well we also discuss asymptotic behavior of solution as t the method developed here can be extended to a general class of free boundary problems

bit reliability based low complexity decoding algorithms for non binary ldpc codes this paper presents bit reliability based majority logic decoding mlgd algorithms for non binary ldpc codes the proposed algorithms pass only one galois field element and its reliability along each edge of the tanner graph of a non binary ldpc code since their reliability updates are in terms of bits rather than symbols they are more efficient than traditional mlgd based decoding algorithms by weighting the soft reliability of the extrinsic information sums based on their hard reliability the proposed algorithms can achieve good error performance for non binary ldpc codes with various column weights moreover their computational complexity and memory consumption are remarkably reduced compared with existing mlgd based decoding algorithms as a result they provide effective tradeoffs between error performance and complexity for decoding of non binary ldpc codes

two level picard and modified picard methods for the navier stokes equations abstract iterative methods of picard type for the navier stokes equations are known to converge only for quite small reynolds numbers however we study methods involving just one such iteration at general reynolds numbers for the initial approximation a coarse mesh of width h0 is used the corrected approximation is computed by just one picard or modified picard step on a fine mesh of width h1 for example h1 may be of order o h 0 2 when linear velocity elements are used the resulting method requires the solution of a small system of nonlinear equations on the coarse mesh and only one larger linear system on the fine mesh this two level picard method is proven to converge for fixed reynolds number as h 0 further the fine mesh solution satisfies a quasi optimal error bound the error constants grow as re as for the usual finite element method one very heuristic explanation why one step of the divergent picard method might work when beginning with a coarse mesh approximation is that the terms neglected involve lower order derivatives thus they are approximated with higher accuracy on the coarse mesh this is linked with a smoothing property of the fine mesh step

additive structural and multiplicative transformations for the construction of quasi cyclic ldpc matrices the construction of a quasi cyclic low density parity check qc ldpc matrix is usually carried out in two steps in the first step a prototype matrix is defined according to certain criteria size girth check and variable node degrees and so on the second step involves the expansion of the prototype matrix during this last phase an integer value is assigned to each non null position in the prototype matrix corresponding to the right rotation of the identity matrix the problem of determining these integer values is complex the state of the art solutions use either some mathematical constructions to guarantee a given girth of the final qc ldpc code or a random search of values until the target girth is satisfied in this paper we propose an alternative complementary method that reduces the search space by defining large equivalence classes of topologically identical matrices through row and column permutations using additive structural and multiplicative transformations selecting only a single element per equivalence class can reduce the search space by a few orders of magnitude then we use the formalism of constraint programming to list the exhaustive sets of solutions for a given girth and a given expansion factor an example is presented in all sections of the paper to illustrate the methodology

adaptive multiset stochastic decoding of non binary ldpc codes we propose a non binary stochastic decoding algorithm for low density parity check ldpc codes over gf q with degree two variable nodes called adaptive multiset stochastic algorithm amsa the algorithm uses multisets an extension of sets that allows multiple occurrences of an element to represent probability mass functions that simplifies the structure of the variable nodes the run time complexity of one decoding cycle using amsa is o q for conventional memory architectures and o 1 if a custom memory architecture is used two fully parallel amsa decoders are implemented on fpga for two 192 96 2 4 regular codes over gf 64 and gf 256 both achieving a maximum clock frequency of 108 mhz the gf 64 decoder has a coded throughput of 65 mb s at eb n0 2 4 db when using conventional memory while a decoder using the custom memory version can achieve 698 mb s at the same eb n0 at a frame error rate fer of 2 10 6 the gf 64 version of the algorithm is only 0 04 db away from the floating point spa performance and for the gf 256 code the difference is 0 2 db to the best of our knowledge this is the first fully parallel non binary ldpc decoder over gf 256 reported in the literature

a high throughput trellis based layered decoding architecture for non binary ldpc codes using max log qspa this paper presents a high throughput decoder architecture for non binary low density parity check ldpc codes where the q ary sum product algorithm qspa in the log domain is considered we reformulate the check node processing such that an efficient trellis based implementation can be used where forward and backward recursions are involved in order to increase the decoding throughput bidirectional forward backward recursion is used in addition layered decoding is adopted to reduce the number of iterations based on a given performance finally a message compression technique is used to reduce the storage requirements and hence the area using a 90 nm cmos process a 32 ary 837 726 ldpc decoder was implemented to demonstrate the proposed techniques and architecture this decoder can achieve a throughput of 233 53 mb s at a clock frequency of 250 mhz based on the post layout results compared to the decoders presented in previous literature the proposed decoder can achieve the highest throughput based on a similar better error rate performance

reduced complexity decoding of ldpc codes various log likelihood ratio based belief propagation llr bp decoding algorithms and their reduced complexity derivatives for low density parity check ldpc codes are presented numerically accurate representations of the check node update computation used in llr bp decoding are described furthermore approximate representations of the decoding computations are shown to achieve a reduction in complexity by simplifying the check node update or symbol node update or both in particular two main approaches for simplified check node updates are presented that are based on the so called min sum approximation coupled with either a normalization term or an additive offset term density evolution is used to analyze the performance of these decoding algorithms to determine the optimum values of the key parameters and to evaluate finite quantization effects simulation results show that these reduced complexity decoding algorithms for ldpc codes achieve a performance very close to that of the bp algorithm the unified treatment of decoding techniques for ldpc codes presented here provides flexibility in selecting the appropriate scheme from performance latency computational complexity and memory requirement perspectives

nonlinear galerkin methods this article presents a new method of integrating evolution differential equations the non linear galerkin method that is well adapted to the long term integration of such equations while the usual galerkin method can be interpreted as a projection of the considered equation on a linear space the methods considered here are related to the projection of the equation on a nonlinear manifold from the practical point of view some terms have been identified as small and sometimes but not always disregarded