an information theoretic framework for deriving canonical decision feedback receivers in gaussian channels a framework is presented that allows a number of known results relating feedback equalization linear prediction and mutual information to be easily understood a lossless additive decomposition of mutual information in a general class of gaussian channels is introduced and shown to produce an information preserving canonical decision feedback receiver the approach is applied to intersymbol interference isi channels to derive the well known minimum mean square error mmse decision feedback equalizer dfe when applied to the synchronous code division multiple access cdma channel the result is the mmse or signal to interference ratio sir maximizing decision feedback detector which is shown to achieve the channel sum capacity at the vertices of the capacity region finally in the case of the asynchronous cdma channel we are able to give new connections between information theory decision feedback receivers and structured factorizations of multivariate spectra

analysis of dual consistency for discontinuous galerkin discretizations of source terms the effects of dual consistency on discontinuous galerkin discretizations of solution and solution gradient dependent source terms are examined two common discretizations are analyzed the standard weighting technique for source terms and the mixed formulation it is shown that if the source term depends on the first derivative of the solution the standard weighting technique leads to a dual inconsistent scheme a straightforward procedure for correcting this dual inconsistency and arriving at a dual consistent discretization is demonstrated the mixed formulation where the solution gradient in the source term is replaced by an additional variable that is solved for simultaneously with the state leads to an asymptotically dual consistent discretization numerical results for a one dimensional test problem confirm that the dual consistent and asymptotically dual consistent schemes achieve higher asymptotic convergence rates with grid refinement than a similar dual inconsistent scheme for both the primal and adjoint solutions as well as a simple functional output

error estimation and adaptation for functional outputs in time dependent flow problems the paper presents an adjoint based approach for determining global error in the time domain that is relevant to functional outputs from unsteady flow simulations the algorithm is derived for the unsteady euler equations that are discretized for second order accuracy in both space and time and takes into account the effect of dynamic meshes in addition to error due to temporal resolution the formulation also takes into account algebraic error arising from partial convergence of the governing equations at each implicit time step the resulting error distributions are then used to drive adaptation of the temporal resolution and the convergence tolerances for the governing equations at each time step the method is demonstrated in the context of both time integrated and instantaneous functionals and the results are compared against traditional adaptation methods

a monte carlo method for solving unsteady adjoint equations traditionally solving the adjoint equation for unsteady problems involves solving a large structured linear system this paper presents a variation on this technique and uses a monte carlo linear solver the monte carlo solver yields a forward time algorithm for solving unsteady adjoint equations when applied to computing the adjoint associated with burgers equation the monte carlo approach is faster for a large class of problems while preserving sufficient accuracy

duality based boundary conditions and dual consistent finite difference discretizations of the navier stokes and euler equations in this paper we derive new far field boundary conditions for the time dependent navier stokes and euler equations in two space dimensions the new boundary conditions are derived by simultaneously considering well posedness of both the primal and dual problems we moreover require that the boundary conditions for the primal and dual navier stokes equations converge to well posed boundary conditions for the primal and dual euler equations we perform computations with a high order finite difference scheme on summation by parts form with the new boundary conditions imposed weakly by the simultaneous approximation term we prove that the scheme is both energy stable and dual consistent and show numerically that both linear and non linear integral functionals become superconvergent

adjoint and defect error bounding and correction for functional estimates we present two error estimation approaches for bounding or correcting the error in functional estimates such as lift or drag adjoint methods quantify the error in a particular output functional that results from residual errors in approximating the solution to the partial differential equation defect methods can be used to bound or reduce the error in the entire solution with corresponding improvements to functional estimates both approaches rely on smooth solution reconstructions and may be used separately or in combination to obtain highly accurate solutions with asymptotically sharp error bounds the adjoint theory is presented for both smooth and shocked problems numerical experiments confirm fourth order error estimates for a pressure integral of shocked quasi id euler flow by employing defect and adjoint methods together and accounting for errors in approximating the geometry it is possible to obtain functional estimates that exceed the order of accuracy of the discretization process and the reconstruction approach superconvergent drag estimates are obtained for subsonic euler flow over a lifting airfoil

delayed decision feedback sequence estimation the authors delayed decision feedback sequence estimation ddfse for uncoded pam signals is considered estimates on the performance of the algorithm are given and simulation results are provided for several examples a more general form of ddfse applicable to coded modulation systems is also presented as an example detection of trellis coded qpsk quadrature phase shift keyed signals over intersymbol interference channels is discussed

capacity penalty due to ideal zero forcing decision feedback equalization we consider the capacity c of a continuous time channel with frequency response h f and additive white gaussian noise if h f sup 2 behaves like a polynomial of order spl rho at high frequencies we show that the per symbol capacity approaches spl rho 2 nats per channel use at high signal powers if the receiver uses an ideal zero forcing decision feedback equalizer dfe consisting of a sampled whitened matched filter followed by a zero forcing tail canceler that is free of error propagation the overall system is free of intersymbol interference and has a well defined capacity c sub zf by comparing this capacity with the capacity c of the underlying channel we quantify the loss of information inherent in the tail canceling operation that typifies zero forcing dfe and zero forcing precoding systems for strictly bandlimited channels we find that the capacity penalty approaches zero in the limit of large signal power on the other hand for nonstrictly bandlimited channels the asymptotic penalty is nonzero however with bandwidth optimization the asymptotic penalty is at most 0 59 db and the asymptotic ratio c sub zf c is at least 93 6 depending on the asymptotic order spl rho of the channel response

summation by parts in time we develop a new high order accurate time integration technique for initial value problems we focus on problems that originate from a space approximation using high order finite difference methods on summation by parts form with weak boundary conditions and extend that technique to the time domain the new time integration method is global high order accurate unconditionally stable and together with the approximation in space it generates optimally sharp fully discrete energy estimates in particular it is shown how stable fully discrete high order accurate approximations of the maxwells equations the elastic wave equations and the linearized euler and navier stokes equations can obtained even though we focus on finite difference approximations we stress that the methodology is completely general and suitable for all semi discrete energy stable approximations numerical experiments show that the new technique is very accurate and has limited order reduction for stiff problems

superconvergence of the velocity in mimetic finite difference methods on quadrilaterals superconvergence of the velocity is established for mimetic finite difference approximations of second order elliptic problems over h 2 uniform quadrilateral meshes the superconvergence result holds for a full tensor coefficient the analysis exploits the relation between mimetic finite differences and mixed finite element methods via a special quadrature rule for computing the scalar product in the velocity space the theoretical results are confirmed by numerical experiments

adjoint recovery of superconvergent functionals from pde approximations motivated by applications in computational fluid dynamics a method is presented for obtaining estimates of integral functionals such as lift or drag that have twice the order of accuracy of the computed flow solution on which they are based this is achieved through error analysis that uses an adjoint pde to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy numerical results are presented for the poisson equation in one and two dimensions and for the nonlinear quasi one dimensional euler equations the theory is equally applicable to nonlinear equations in complex multi dimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations

mmse decision feedback equalizers and coding ii coding results for pt i see ibid vol 43 no 10 p 2582 1995 the minimum mean squared error decision feedback equalizer mmse dfe has properties that suggest that it is a canonical equalization structure in systems that combine equalization with coded modulation with a given symbol rate 1 t and transmit spectrum the output signal to noise ratio snr sub mmse dfe u of a mmse dfe with an unbiased decision rule is a single parameter that characterizes the channel for coding purposes indeed the transmit spectrum that maximizes snr sub mmse dfe u is the capacity achieving water pouring spectrum and the capacity c t in bits per two dimensions is given by c t log sub 2 1 snr sub mmse dfe u regardless of the channel characteristics the performance of a coded system with a mmse dfe equalization structure may be accurately estimated using the gain of the coding scheme at a given pr e this performance is shown to be approximately the same as that of a multicarrier system using the same transmit spectrum and similar coding such systems are known to be able to approach capacity arbitrarily closely the mmse dfe can perform significantly better than a zero forcing decision feedback equalizer particularly at moderate to low snr s and on severe isi channels simulation results indicate that performance of the mmse dfe is surprisingly insensitive to transmit spectral shaping as long as the transmit spectrum exceeds the capacity achieving band but that there is an optimal symbol rate that should approximately be used

the relation between primal and dual boundary conditions for hyperbolic systems of equations abstract in this paper we study boundary conditions for linear hyperbolic systems of equations and the corresponding dual problem in particular we show that the primal and dual boundary conditions are related by a simple scaling relation it is also shown that the weak dual problem can be derived directly from the weak primal problem based on the continuous analysis we discretize and perform computations with a high order finite difference scheme on summation by parts form with weak boundary conditions it is shown that the results obtained in the continuous analysis lead directly to stability results for the primal and dual discrete problems numerical experiments corroborate the theoretical results

on pseudo spectral time discretizations in summation by parts form abstract fully implicit discrete formulations in summation by parts form for initial boundary value problems must be invertible in order to provide well functioning procedures we prove that under mild assumptions pseudo spectral collocation methods for the time derivative lead to invertible discrete systems when energy stable spatial discretizations are used

superconvergent functional output for time dependent problems using finite differences on summation by parts form finite difference operators satisfying the summation by parts sbp rules can be used to obtain high order accurate energy stable schemes for time dependent partial differential equations when the boundary conditions are imposed weakly by the simultaneous approximation term sat in general an sbp sat discretization is accurate of order p 1 with an internal accuracy of 2p and a boundary accuracy of p despite this it is shown in this paper that any linear functional computed from the time dependent solution will be accurate of order 2p when the boundary terms are imposed in a stable and dual consistent way the method does not involve the solution of the dual equations and superconvergent functionals are obtained at no extra computational cost four representative model problems are analyzed in terms of convergence and errors and it is shown in a systematic way how to derive schemes which gives superconvergent functional outputs

a dual consistent summation by parts formulation for the linearized incompressible navier stokes equations posed on deforming domains in this article well posedness and dual consistency of the linearized incompressible navier stokes equations posed on time dependent spatial domains are studied to simplify the derivation of the

superconvergence of discontinuous galerkin and local discontinuous galerkin schemes for linear hyperbolic and convection diffusion equations in one space dimension in this paper we study the superconvergence property for the discontinuous galerkin dg and the local discontinuous galerkin ldg methods for solving one dimensional time dependent linear conservation laws and convection diffusion equations we prove superconvergence towards a particular projection of the exact solution when the upwind flux is used for conservation laws and when the alternating flux is used for convection diffusion equations the order of superconvergence for both cases is proved to be k frac 3 2 when piecewise p k polynomials with k geq1 are used the proof is valid for arbitrary nonuniform regular meshes and for piecewise p k polynomials with arbitrary k geq1 improving upon the results in y cheng and c w shu j comput phys 227 2008 pp 9612 9627 y cheng and c w shu computers and structures 87 2009 pp 630 641 in which the proof based on fourier analysis was given only for uniform meshes with periodic boundary condition and piecewise p 1 polynomials

adjoint sensitivity analysis for time dependent partial differential equations with adaptive mesh refinement a new adjoint sensitivity analysis approach is presented for time dependent partial differential equations with adaptive mesh refinement the new approach called adda combines the best features of both the adjoint of the discretization ad and discretization of the adjoint da approaches it removes the obstacles of applying ad to adaptive methods and in contrast to da requires for its use only a minimal amount of knowledge about the formulation of adjoint pdes and their boundary conditions the effectiveness and efficiency of adda are demonstrated for several numerical examples

superconvergent functional estimates from summation by parts finite difference discretizations diagonal norm summation by parts sbp operators can be used to construct time stable high order accurate finite difference schemes however to achieve both stability and accuracy these operators must use s order accurate boundary closures when the interior scheme is 2s order accurate the boundary closure limits the solution to s 1 order global accuracy despite this bound on solution accuracy we show that functional estimates can be constructed that are 2s order accurate this superconvergence requires dual consistency which depends on the sbp operators the boundary condition implementation and the discretized functional the theory is developed for scalar hyperbolic and elliptic partial differential equations in one dimension in higher dimensions we show that superconvergent functional estimates remain viable in the presence of curvilinear multiblock grids with interfaces the generality of the theoretical results is demonstrated using a two dimensional poisson problem and a nonlinear hyperbolic system the euler equations of fluid mechanics

the intersymbol interference channel lower bounds on capacity and channel precoding loss the discrete time additive gaussian intersymbol interference isi channel with i i d not necessarily gaussian input signals is considered several new and old lower bounds on the capacity are derived in a unified manner by assuming different front end receiver filters in particular the sampled whitened matched filter swmf and the minimum mean squared error decision feedback equalizer mmse dfe filter the features of the bounds are demonstrated and compared in several examples with binary and quaternary input signals it is also shown that the effect of an ideal post cursor or tail cancellation in an information preserving context depends primarily on the front end filter while as is well known ideal post cursor cancellation at the output of the swmf decreases the information the opposite trend is seen when an mmse dfe front filter is considered this observation reflects the basic theoretical obstacles in precoding i e ideal post cursor cancellation in the presence of a pre cursor it is used to assess the inherent loss in terms of information rates as compared to the rates achievable with the hypothetical ideal post cursor cancellation associated with any post cursor cancellation technique such as precoding dfe or other variants when operating in synergy with the mmse dfe front end filter the effect of the front end filter on an ideally interleaved precoded coded modulation system is also addressed

well posed boundary conditions for the navier stokes equations in this article we propose a general procedure that allows us to determine both the number and type of boundary conditions for time dependent partial differential equations with those well posedness can be proven for a general initial boundary value problem the procedure is exemplified on the linearized navier stokes equations in two and three space dimensions on a general domain

a stable and high order accurate conjugate heat transfer problem this paper analyzes well posedness and stability of a conjugate heat transfer problem in one space dimension we study a model problem for heat transfer between a fluid and a solid the energy method is used to derive boundary and interface conditions that make the continuous problem well posed and the semi discrete problem stable the numerical scheme is implemented using 2nd 3rd and 4th order finite difference operators on summation by parts sbp form the boundary and interface conditions are implemented weakly we investigate the spectrum of the spatial discretization to determine which type of coupling that gives attractive convergence properties the rate of convergence is verified using the method of manufactured solutions

summation by parts projections and stability ii we have derived stability results for high order finite difference approximations of mixed hyperbolic parabolic initial boundary value problems ibvp the results are obtained using summation by parts and a new way of representing general linear boundary conditions as an orthogonal projection by slightly rearranging the analytic equations we can prove strict stability for hyperbolic parabolic ibvp furthermore we generalize our technique so as to yield strict stability on curvilinear non smooth domains in two space dimensions finally we show how to incorporate inhomogeneous boundary data while retaining strict stability using the same procedure one can prove strict stability in higher dimensions as well

mmse decision feedback equalizers and coding i equalization results the minimum mean squared error decision feedback equalizer mmse dfe has properties that suggest that it is a canonical equalization structure for systems that combine equalization with coded modulation the structure and performance of the mmse dfe are succinctly derived using linear estimation theoretic principles in this first part of this two part paper the front end of the mmse dfe called the mean square whitened matched filter ms wmf is preferable in some ways to a matched filter or a whitened matched filter as a canonical receiver front end in a coded system the feedback filter of the mmse dfe may be implemented in the transmitter using precoding the mmse dfe can perform significantly better than a zero forcing decision feedback equalizer particularly at moderate to low snr s and on severe isi channels the mmse dfe is biased the optimum unbiased mmse dfe is the mmse dfe with the bias removed removing bias improves error probability but reduces the snr to snr sub mmse dfe u snr sub mmse dfe 1 it is shown that this snr relationship is a particular case of a very general result and that snr sub mmse dfe u gives a more realistic estimate of snr the results are extended to partial response equalization and to equalization with correlated inputs in an appendix

vector coding for partial response channels a linear technique for combining equalization and coset codes on partial response channels with additive white gaussian noise is developed the technique vector coding uses a set of transmit filters or vectors to partition the channel into an independent set of parallel intersymbol interference isi free channels for any given finite or infinite block length the optimal transmit vectors for such channel partitioning are shown to be the eigenvectors of the channel covariance matrix for the specified block length and the gains of the individual channels are the eigenvalues an optimal bit allocation and energy distribution are derived for the set of parallel channels under an accurate extension of the continuous approximation for power in optimal multidimensional signal sets for constellations with unequal signal spacing in different dimensions examples are presented that demonstrate performance advantages with respect to zero forcing decision feedback methods that use the same coset code on the same partial response channel only resampling the channel at an optimal rate and assuming no errors in the feedback path will bring the performance of the decision feedback methods up to the level of the vector coded system

minimal repetition dynamic checkpointing algorithm for unsteady adjoint calculation adjoint equations of differential equations have seen widespread applications in optimization inverse problems and uncertainty quantification a major challenge in solving adjoint equations for time dependent systems has been the need to use the solution of the original system in the adjoint calculation and the associated memory requirement in applications where storing the entire solution history is impractical checkpointing methods have frequently been used however traditional checkpointing algorithms such as revolve require a priori knowledge of the number of time steps making these methods incompatible with adaptive time stepping we propose a dynamic checkpointing algorithm applicable when the number of time steps is a priori unknown our algorithm maintains a specified number of checkpoints on the fly as time integration proceeds for an arbitrary number of time steps the resulting checkpoints at any snapshot during the time integration have the optimal repetition number the efficiency of our algorithm is demonstrated both analytically and experimentally in solving adjoint equations this algorithm also has significant advantage in automatic differentiation when the length of execution is variable

summation by parts operators and high order quadrature summation by parts sbp operators are finite difference operators that mimic integration by parts the sbp operator definition includes a weight matrix that is used formally for discrete integration however the accuracy of the weight matrix as a quadrature rule is not explicitly part of the sbp definition we show that sbp weight matrices are related to trapezoid rules with end corrections whose accuracy matches the corresponding difference operator at internal nodes for diagonal weight matrices the accuracy of sbp quadrature extends to curvilinear domains provided the jacobian is approximated with the same sbp operator used for the quadrature this quadrature has significant implications for sbp based discretizations in particular the diagonal norm accurately approximates the l 2 norm for functions and multi dimensional sbp discretizations accurately approximate the divergence theorem

output error estimation for summation by parts finite difference schemes the paper develops a posteriori error estimates of integral output functionals for summation by parts finite difference methods the error estimates are based on the adjoint weighted residual method and take advantage of a variational interpretation of summation by parts discretizations the estimates are computed on a fixed grid and do not require an embedded grid or explicit interpolation operators for smooth boundary value problems containing first and second derivatives the error estimates converge to the exact error as the mesh is refined the theory is verified using linear boundary value problems and the euler equations

the sbp sat technique for initial value problems a detailed account of the stability and accuracy properties of the sbp sattechnique for numerical time integration is presented we show how thetechnique can be used to formulate both global and mu

on the impact of boundary conditions on dual consistent finite difference discretizations in this paper we derive well posed boundary conditions for a linear incompletely parabolic system of equations which can be viewed as a model problem for the compressible navier stokes equations we show a general procedure for the construction of the boundary conditions such that both the primal and dual equations are well posed the form of the boundary conditions is chosen such that reduction to first order form with its complications can be avoided the primal equation is discretized using finite difference operators on summation by parts form with weak boundary conditions it is shown that the discretization can be made energy stable and that energy stability is sufficient for dual consistency since reduction to first order form can be avoided the discretization is significantly simpler compared to a discretization using dirichlet boundary conditions we compare the new boundary conditions with standard dirichlet boundary conditions in terms of rate of convergence errors and discrete spectra it is shown that the scheme with the new boundary conditions is not only far simpler but also has smaller errors error bounded properties and highly optimizable eigenvalues while maintaining all desirable properties of a dual consistent discretization

stable and accurate schemes for the compressible navier stokes equations minimal stencil width discretizations of combined mixed and non mixed second order derivatives are analyzed with respect to accuracy and stability we show that these discretizations lead to stability for cauchy problems with a careful boundary treatment we also show that the stability holds for initial boundary value problems the analysis is verified by numerical simulations of burgers and navier stokes equations in two and three space dimensions

optimal time splitting for two and three dimensional navier stokes equations with mixed derivatives a new explicit time splitting algorithm has been developed for finite difference modelling of the full two and three dimensional time dependent compressible viscous navier stokes equations of fluid mechanics the scheme is optimal in the sense that the split operators achieve their maximum allowable time step i e the corresponding courant number the algorithm allows a conservation form formulation stability is proven analytically and verified numerically in proving stability it was shown that all nine matrix coefficients of the navier stokes equations are simultaneously symmetrizable by a similarity transformation two such transformations and their resulting symmetric matrix coefficients are presented explicitly