experimental study of ilu preconditioners for indefinite matrices incomplete lu factorization preconditioners have been surprisingly successful for many cases of general nonsymmetric and indefinite matrices however their failure rate is still too high for them to be useful as black box library software for general matrices besides fatal breakdowns due to zero pivots the major causes of failure are inaccuracy and instability of the triangular solves when there are small pivots both these problems can occur but these problems can also occur without small pivots through examples from actual problems this paper shows how these problems evince themselves how these problems can be detected and how these problems can sometimes be circumvented through pivoting reordering scaling perturbing diagonal elements and preserving symmetric structure the goal of this paper is to gain a better practical understanding of ilu preconditioners and help improve their reliability

domain decomposition schemes with high order accuracy and unconditional stability parallel finite difference schemes with high order accuracy and unconditional stability for solving parabolic equations are presented the schemes are based on domain decomposition method i e interface values between subdomains are computed by the explicit scheme interior values are computed by the implicit scheme the numerical stability and error are derived in the h 1 norm in one dimensional case numerical results of both one and two dimensions examining the stability accuracy and parallelism of the procedure are also presented

compressed graphs and the minimum degree algorithm the minimum degree algorithm is well known and widely used to find a permutation matrix p such that a sparse symmetric positive definite matrix a can be factored as pap t ll t to yield a lower triangular matrix l with relatively few nonzero entries in this paper we show that by compressing the graph of a when possible the run time of the minimum degree algorithm can be significantly reduced some empirical results for a collection of test matrices are presented

high performance pcg solvers for fem structural analysis the preconditioned conjugate gradient algorithm is a well known and powerful method used to solve large sparse symmetric positive definite linear systems such system are generated by the finite element discretisation in structural analysis but users of finite elements in this contest generally still rely on direct methods it is our purpose in the present work to highlight the improvement brought forward by some new preconditioning techniques and show that the preconditioned conjugate gradient method performs better than efficient direct methods

joint data and channel estimation using blind trellis search techniques a novel method of maximum likelihood sequence estimation is proposed for data that are transmitted over unknown linear channels this procedure does not require a startup sequence for estimating the channel impulse response rather the data and the channel are simultaneously estimated it is implemented without any loss of optimality by a trellis search algorithm which searches for the best data sequence from among a number of hypothesized trellises which are constructed from the observed sequence the number of states in each trellis and the number of trellises grow exponentially with the channel memory a suboptimal trellis search algorithm is proposed whose complexity at best is slightly higher than that of the adaptive viterbi algorithm operating with a known channel response a simplified channel estimation algorithm when the number of data alphabets is greater than 2 is also proposed fast convergence of the algorithm in estimating the channel is demonstrated for binary pulse amplitude modulation over a variety of channels convergence over a wide range of snr occurs within 100 symbols the channel estimation algorithm for multi level signals converges within 500 1000 symbols we finally present an application of this algorithm to the problem of sequence estimation in the presence of rapidly time varying intersymbol interference the algorithm provides reliable zero delay decisions which results in a better channel tracking algorithm when compared to previously proposed schemes

incomplete cholesky factorizations with limited memory we propose an incomplete cholesky factorization for the solution of large scale trust region subproblems and positive definite systems of linear equations this factorization depends on a parameter p that specifies the amount of additional memory in multiples of n the dimension of the problem that is available there is no need to specify a drop tolerance our numerical results show that the number of conjugate gradient iterations and the computing time are reduced dramatically for small values of p we also show that in contrast with drop tolerance strategies the new approach is more stable in terms of number of iterations and memory requirements

alternating direction method for generalized sylvester matrix equation axb cyd e this paper presents alternating direction methods of multipliers for finding the solution the best approximate solution and the nonnegative solution of the generalized sylvester matrix equation axb cyd e where a b c d and e are given matrices of suitable sizes preliminary convergence properties of the proposed algorithms are given numerical experiments show that the proposed algorithms tend to deliver higher quality solutions with less iteration steps and less computing times than recent algorithms on the tested problems

maximum likelihood priority first search decodable codes for combined channel estimation and error correction the coding technique that combines channel estimation and error correction has received attention recently and has been regarded as a promising approach to counter the effects of multipath fading it has been shown by simulation that a proper code design that jointly considers channel estimation can improve the system performance subject to a fixed code rate as compared to a conventional system which performs channel estimation and error correction separately nevertheless the major obstacle that prevents the practice of such coding technique is that the existing codes are mostly searched by computers and subsequently exhibit no apparent structure for efficient decoding hence the operation intensive exhaustive search becomes the only decoding option and the decoding complexity increases dramatically with codeword length in this paper a systematic construction is derived for a class of structured codes that support joint channel estimation and error correction it is confirmed by simulation that these codes have comparable performance to the best simulated annealing based computer searched codes moreover the systematically constructed codes can now be maximum likelihoodly decoded with respect to the unknown channel criterion in terms of a newly derived recursive metric for use by the priority first search decoding algorithm thus the decoding complexity is significantly reduced as compared with that of an exhaustive decoder

regular incomplete factorizations of real positive definite matrices abstract we study the classes of matrices which admit a regular incomplete factorization with respect to any graph set we extend the construction to the class of real positive definite matrices the convergence of the basic iteration associated with the splittings we have is discussed

on a variable smoothing procedure for krylov subspace methods abstract it is known that the convergence behavior of galerkin krylov subspace methods for solving linear systems can be very erratic a smoothing technique or a minimal residual seminorm variant of these galerkin methods can be proposed to eliminate this problem in this paper we examine a class of minimal residual seminorm methods and show that this class of methods can be obtained from a variable smoothing technique applied to galerkin methods

iterative solution of linear systems in the 20th century this paper sketches the main research developments in the area of iterative methods for solving linear systems during the 20th century although iterative methods for solving linear systems nd their origin in the early 19th century work by gauss the eld has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences the past ve decades have been particularly rich in new developments ending with the availability of large toolbox of specialized algorithms for solving the very large problems which arise in scientic and industrial computational models as in any other scientic area research in iterative methods has been a journey characterized by a chain of contributions building on each other it is the aim of this paper not only to sketch the most signicant of these contributions during the past century but also to relate them to one another c 2000 elsevier science b v all rights reserved

preconditioning highly indefinite and nonsymmetric matrices standard preconditioners like incomplete factorizations perform well when the coefficient matrix is diagonally dominant but often fail on general sparse matrices we experiment with nonsymmetric permutations and scalings aimed at placing large entries on the diagonal in the context of preconditioning for general sparse matrices the permutations and scalings are those developed by olschowka and neumaier linear algebra appl 240 1996 pp 131 151 and by duff and koster siam j matrix anal appl 20 1999 pp 889 901 tech report ral tr 99 030 rutherford appleton laboratory chilton uk 1999 we target highly indefinite nonsymmetric problems that cause difficulties for preconditioned iterative solvers our numerical experiments indicate that the reliability and performance of preconditioned iterative solvers are greatly enhanced by such preprocessing

robust approximate inverse preconditioning for the conjugate gradient method we present a variant of the ainv factorized sparse approximate inverse algorithm which is applicable to any symmetric positive definite matrix the new preconditioner is breakdown free and when used in conjunction with the conjugate gradient method results in a reliable solver for highly ill conditioned linear systems we also investigate an alternative approach to a stable approximate inverse algorithm based on the idea of diagonally compensated reduction of matrix entries the results of numerical tests on challenging linear systems arising from finite element modeling of elasticity and diffusion problems are presented

an iterative algorithm for the least frobenius norm least squares solution of a class of generalized coupled sylvester transpose linear matrix equations abstract the iterative algorithm of a class of generalized coupled sylvester transpose matrix equations is presented we prove that if the system is consistent a solution can be obtained within finite iterative steps in the absence of round off errors for any initial matrices if the system is inconsistent the least squares solution can be obtained within finite iterative steps in the absence of round off errors furthermore we provide a method for choosing the initial matrices to obtain the least frobenius norm least squares solution of the problem finally numerical examples are presented to demonstrate that the algorithm is efficient

computational methods for linear matrix equations given the square matrices a b d e and the matrix c of conforming dimensions we consider the linear matrix equation a mathbf x e d mathbf x b c in the unknown matrix mathbf x our aim is to provide an overview of the major algorithmic developments that have taken place over the past few decades in the numerical solution of this and related problems which are producing reliable numerical tools in the formulation and solution of advanced mathematical models in engineering and scientific computing

on global hessenberg based methods for solving sylvester matrix equations abstract in the first part of this paper we investigate the use of hessenberg based methods for solving the sylvester matrix equation a x x b c to achieve this goal the sylvester form of the global generalized hessenberg process is presented using this process different methods based on a petrov galerkin or on a minimal norm condition are derived in the second part we focus on the sgl cmrh method which is based on the sylvester form of the hessenberg process with pivoting strategy combined with a minimal norm condition in order to accelerate the sgl cmrh method a preconditioned framework of this method is also considered it includes both fixed and flexible variants of the sgl cmrh method moreover the connection between the flexible preconditioned sgl cmrh method and the fixed one is studied and some upper bounds for the residual norm are obtained in particular application of the obtained theoretical results is investigated for the special case of solving linear systems of equations with several right hand sides finally some numerical experiments are given in order to evaluate the effectiveness of the proposed methods

orderings for factorized sparse approximate inverse preconditioners the influence of reorderings on the performance of factorized sparse approximate inverse preconditioners is considered some theoretical results on the effect of orderings on the fill in and decay behavior of the inverse factors of a sparse matrix are presented it is shown experimentally that certain reorderings like minimum degree and nested dissection can be very beneficial the benefit consists of a reduction in the storage and time required for constructing the preconditioner and of faster convergence of the preconditioned iteration in many cases of practical interest

a four order alternating segment crank nicolson scheme for the dispersive equation in this paper a new alternating segment crank nicolson scheme for the dispersive equation with a periodic boundary condition is derived the scheme has a four order truncation error in space and unconditional stability its theoretical results are conformed to the numerical simulation a comparison of the accuracy of this method with the prior asei and age methods is also included

a matrix crs iterative method for solving a class of coupled sylvester transpose matrix equations abstract in this paper we apply kronecker product and vectorization operator to extend the conjugate residual squared crs method for solving a class of coupled sylvester transpose matrix equations some numerical examples are given to compare the accuracy and efficiency of the new matrix iterative method with other methods presented in the literature numerical results validate that the proposed method can be much more efficient than some existing methods

a new implementation of the cmrh method for solving dense linear systems the cmrh method h sadok methodes de projections pour les systemes lineaires et non lineaires habilitation thesis university of lille1 lille france 1994 h sadok cmrh a new method for solving nonsymmetric linear systems based on the hessenberg reduction algorithm numer algorithms 20 1999 303 321 is an algorithm for solving nonsymmetric linear systems in which the arnoldi component of gmres is replaced by the hessenberg process which generates krylov basis vectors which are orthogonal to standard unit basis vectors rather than mutually orthogonal the iterate is formed from these vectors by solving a small least squares problem involving a hessenberg matrix like gmres this method requires one matrix vector product per iteration however it can be implemented to require half as much arithmetic work and less storage moreover numerical experiments show that this method performs accurately and reduces the residual about as fast as gmres with this new implementation we show that the cmrh method is the only method with long term recurrence which requires not storing at the same time the entire krylov vectors basis and the original matrix as in the gmres algorithm a comparison with gaussian elimination is provided

some history of the conjugate gradient and lanczos methods this paper gives some of the history of the conjugate gradient and lanczos algorithms and an annotated bibliography for the period 1948 1976

the fast multipole method we study integral methods applied to the resolution of the maxwell equations where the linear system is solved using an iterative method which requires only matrix vector products the fast multipole method fmm is one of the most efficient methods used to perform matrix vector products and accelerate the resolution of the linear system a problem involving n degrees of freedom may be solved in cniternlogn floating operations where c is a constant depending on the implementation of the method in this article several techniques allowing one to reduce the constant c are analyzed this reduction implies a lower total cpu time and a larger range of application of the fmm in particular new interpolation and anterpolation schemes are proposed which greatly improve on previous algorithms several numerical tests are also described these confirm the efficiency and the theoretical complexity of the fmm

explicit implicit conservative domain decomposition procedures for parabolic problems based on block centered finite differences domain decomposition procedures for solving parabolic equations are considered the underlying discretization consists of block centered finite differences in these procedures fluxes at subdomain interfaces are calculated from the solution at the previous time level these fluxes serve as neumann boundary data for implicit block centered discretizations in the subdomains a priori error estimates are presented and numerical results examining the stability accuracy and parallelism of the scheme are presented

flexible global generalized hessenberg methods for linear systems with multiple right hand sides a variant of the global generalized hessenberg method is presented which allows varying preconditioning at each restart theoretical results that relate the residual norm of this new method with its original version are developed as two special variants the flexible global gmres method and the flexible global cmrh method are investigated both theoretically and experimentally numerical examples are conducted to illustrate the performance of these two flexible global methods in comparison with both the original global methods and weighted global methods

preconditioning techniques for large linear systems a survey this article surveys preconditioning techniques for the iterative solution of large linear systems with a focus on algebraic methods suitable for general sparse matrices covered topics include progress in incomplete factorization methods sparse approximate inverses reorderings parallelization issues and block and multilevel extensions some of the challenges ahead are also discussed an extensive bibliography completes the paper

low rank approximate solutions to large sylvester matrix equations in the present paper we propose an arnoldi based method for solving large and sparse sylvester matrix equations with low rank right hand sides we will show how to extract low rank approximations via a matrix krylov subspace method we give some theoretical results such an expression of the exact solution and upper bounds for the norm of the error and for the residual numerical experiments will also be given to show the effectiveness of the proposed method

a note on the numerical approximate solutions for generalized sylvester matrix equations with applications abstract in the present paper we propose a krylov subspace method for solving large and sparse generalized sylvester matrix equations the proposed method is an iterative projection method onto matrix krylov subspaces as a particular case we show how to adapt the ilu and the ssor preconditioners for solving large sylvester matrix equations numerical examples and applications to some pde s will be given

identification of nonminimum phase systems using higher order statistics a method is presented for identification of linear time variant nonminimum phase systems when only output data are available the input sequence need not be independent but it must be non gaussian with some special properties described in the test the authors model a finite dimensional system as an arma autoregressive moving average rational function of known orders but the special cases of ar ma and all pass models are also considered to estimate the parameters of their model the authors utilize both second and higher order statistics of the output which may be contaminated by additive zero mean gaussian white noise of unknown variance the parameter estimators obtained are proved under mild conditions to be consistent simulations verify the performance of the proposed method in the case of relatively low signal to noise ratios and when there is a model order mismatch

adaptive deconvolution and identification of nonminimum phase fir systems based on cumulants a novel adaptive deconvolution and system identification scheme is introduced for a linear non minimum phase finite impulse response fir system driven by non gaussian white noise the adaptive scheme is based on approximating the fir system by noncausal autoregressive ar models and using higher order cumulants of the system output as such it is a blind equalization deconvolution scheme the set of updated ar parameters is obtained by using a gradient type algorithm and by using higher order cumulants instead of time samples of the output signal it is demonstrated by means of extensive simulations that the adaptive scheme works well for both stationary and nonstationary cases as expected it outperforms the autocorrelation based gradient method for nonminimum phase system identification and deconvolution performance comparisons to existing methods are given using as figures of merit the probability of errors in the restored input sequence computational complexity and convergence rate

stability and convergence of the explicit implicit conservative domain decomposition procedure for parabolic problems abstract in this note we present an improved stability condition of a finite difference domain decomposition procedure for the parabolic equation this procedure is proposed in 1 in which interface fluxes are calculated from the solution at the previous time level and then these fluxes serve as neumann boundary conditions for implicit subdomain problems

blind equalization by direct examination of the input sequences this paper presents a novel approach to blind equalization deconvolution which is based on direct examination of possible input sequences in contrast to many other approaches it does not rely on a model of the approximative inverse of the channel dynamics to start with the blind equalization identifiability problem for a noise free finite impulse response channel model is investigated a necessary condition for the input which is algorithm independent for blind deconvolution is derived this condition is expressed in an information measure of the input sequence a sufficient condition for identifiability is also inferred which imposes a constraint on the true channel dynamics the analysis motivates a recursive algorithm where all permissible input sequences are examined the exact solution is guaranteed to be found as soon as it is possible an upper bound on the computational complexity of the algorithm is given this algorithm is then generalized to cope with time varying infinite impulse response channel models with additive noise the estimated sequence is an arbitrary good approximation of the maximum a posteriori estimate the proposed method is evaluated on a rayleigh fading communication channel the simulation results indicate fast convergence properties and good tracking abilities

conservative domain decomposition procedure with unconditional stability and second order accuracy a new derivation of the conservative domain decomposition procedure for solving the parabolic equation is presented in this procedure fluxes at subdomain interfaces are calculated from the solution at the previous time level then these fluxes serve as neumann boundary data for implicit block centered discretization in the subdomain the unconditional stability and the second order accuracy of solution values as well as fluxes are proved numerical results examining the stability accuracy and parallelism of the procedure are also presented

combined coding and training for unknown isi channels the traditional method of sending a training signal to identify a channel followed by data may be viewed as a simple code for the unknown channel results in blind sequence detection suggest that performance similar to this traditional approach can be obtained without training however for short packets and or time recursive algorithms significant error floors exist due to the presence of sequences that are indistinguishable without knowledge of the channel in this paper we reconsider training signal design in light of recent results in blind sequence detection specifically we consider the tradeoff between the complexity of receiver processing and the amount of training overhead required more generally we design training codes which combine modulation and training in order to design these codes we find an expression for the pairwise error probability of the joint maximum likelihood jml channel and sequence estimator this expression motivates a pairwise distance for the jml receiver based on principal angles between the range spaces of data matrices the general code design problem generalized sphere packing is formulated as the clique problem associated with an unweighted undirected graph we provide optimal and heuristic algorithms for this clique problem for both long and short packets we demonstrate that significant improvements are possible by jointly considering the design of the training modulation and receiver processing

bayesian equalization for ldpc channel decoding we describe the channel equalization problem and its prior estimate of the channel state information csi as a joint bayesian estimation problem to improve each symbol posterior estimates at the input of the channel decoder our approach takes into consideration not only the uncertainty due to the noise in the channel but also the uncertainty in the csi estimate however this solution cannot be computed in linear time because it depends on all the transmitted symbols hence we also put forward an approximation for each symbol s posterior using the expectation propagation algorithm which is optimal from the kullback leibler divergence viewpoint and yields an equalization with a complexity identical to the bcjr algorithm we also use a graphical model representation of the full posterior in which the proposed approximation can be readily understood the proposed posterior estimates are more accurate than those computed using the ml estimate for the csi in order to illustrate this point we measure the error rate at the output of a low density parity check decoder which needs the exact posterior for each symbol to detect the incoming word and it is sensitive to a mismatch in those posterior estimates for example for qpsk modulation and a channel with three taps we can expect gains over 0 5 db with same computational complexity as the ml receiver

matrix krylov subspace methods for large scale model reduction problems the present paper considers matrix krylov subspace methods for solving large coupled lyapunov matrix equations of the form ap pa t bb t 0 and a tq qa c tc 0 where a is a real nxn matrix b and c t are real nxs and nxr matrices respectively with email protected n and email protected n such problems appear in many areas of control theory such as the computation of the controllability and observability gramians of a stable linear time invariant lti system the proposed methods are based on the global arnoldi and lanczos processes in the second part we show how to use matrix krylov subspace techniques to obtain a reduced order model for lti systems this will be done by approximating the corresponding transfer functions finally numerical experiments are reported to illustrate the behavior and the effectiveness of the global arnoldi and lanczos processes when applied to solve some large coupled lyapunov equations and to approximate some transfer functions

a linear algebraic analysis of diffusion synthetic acceleration for the boltzmann transport equation a linear algebraic formulation of discretized mono energetic steady state linear boltzmann transport equations in slab geometry is presented the discretization consists of a discrete ordinates collocation in angle and a diamond difference method in space by expressing diffusion synthetic acceleration in this formalism asymptotic results are obtained that prove the effectiveness of the associated preconditioner in various asymptotic regimes including the asymptotic diffusion limit these results hold for problems with nonconstant coefficients and nonuniform spatial zoning posed on finite domains with an incident flux and or reflection prescribed at the boundaries numerical results are also presented which demonstrate these results

mlse for an unknown channel i optimality considerations the problem of performing joint maximum likelihood ml estimation of a digital sequence and unknown dispersive channel impulse response is considered starting from a continuous time ct model previous investigations of this problem have not considered the front end fe processing in detail rather a discrete time signal model has been assumed we show that a fractionally spaced whitened matched filter matched to the known data pulse provides a set of sufficient statistics when a tapped delay line channel model is assumed and that the problem is ill posed when the channel impulse response is generalized to a ct finite length model practical approximations are considered that circumvent this ill posed condition recursive computation of the joint ml metric is developed together the fe processing and metric recursion provide a receiver structure which may be interpreted as the theoretical foundation for the previously introduced technique of per survivor processing and they lead directly to generalizations several fe processors representative of those suggested in the literature are developed and related to the practically optimal fe

sparse approximate inverse preconditioners using norm minimization techniques we investigate the use of sparse approximate inverse preconditioners for the iterative solution of unsymmetric linear systems of equations we consider the approximations proposed by cosgrove diaz and griewank internat j comput math 44 1992 pp 91 110 and huckle and grote a new approach to parallel preconditioning with sparse approximate inverses tech report sccm 94 03 stanford university 1994 which are based on norm minimization techniques such methods are of particular interest because of the considerable scope for parallelization we propose a number of enhancements which may improve their performance when run in a sequential environment these methods can perform unfavorably when compared with other techniques however they can be successful when other methods fail and simulations indicate that they can be competitive when considered in a parallel environment

a sparse approximate inverse preconditioner for nonsymmetric linear systems this paper is concerned with a new approach to preconditioning for large sparse linear systems a procedure for computing an incomplete factorization of the inverse of a nonsymmetric matrix is developed and the resulting factorized sparse approximate inverse is used as an explicit preconditioner for conjugate gradient type methods some theoretical properties of the preconditioner are discussed and numerical experiments on test matrices from the harwell boeing collection and from tim davis s collection are presented our results indicate that the new preconditioner is cheaper to construct than other approximate inverse preconditioners furthermore the new technique insures convergence rates of the preconditioned iteration which are comparable with those obtained with standard implicit preconditioners

explicit implicit and crank nicolson domain decomposition methods for parabolic partial differential equations abstract two parallel non overlapping domain decomposition algorithms for solving parabolic partial differential equations are proposed the algorithms combine crank nicolson scheme with implicit galerkin finite element methods in sub domains and explicit flux approximation along inner boundaries at each time step thus parallelism can be easily achieved l 2 norm error estimates for these explicit implicit procedures are presented in which time step constraints are proved to be less severe than that of fully explicit schemes numerical experiments are also performed to verify the theoretical analysis

polynomial preconditioners for conjugate gradient calculations dubois greenbaum and rodrigue proposed using a truncated neumann series as an approximation to the inverse of a matrix a for the purpose of preconditioning conjugate gradient iterative approximations to ax b if we assume that a has been symmetrically scaled to have unit diagonal and is thus of the form i g then the neumann series is a power series in g with unit coefficients the incomplete inverse was thought of as a replacement of the incomplete cholesky decomposition suggested by meijerink and van der vorst in the family of methods iccg n the motivation for the replacement was the desire to have a preconditioned conjugate gradient method which only involved vector operations and which utilized long vectors we here suggest parameterizing the incomplete inverse to form a preconditioning matrix whose inverse is a polynomial in g we then show how to select the parameters to minimize the condition number of the product of the polynomial and i g theoretically the resulting algorithm

a new preconditioning technique for solving large sparse linear systems abstract a new sparse approximate triangular factorization technique for solving large sparse linear system by iterative methods is proposed the method is based on the description of the triangular factorization of a matrix as a product of elementary matrices and provides a general scheme for constructing incomplete preconditioners for the given matrix in particular the familiar incomplete choleski decomposition can be incorporated into this scheme the algorithm based on choice by value compares favorably with the incomplete choleski preconditioner and equipped with a user controlled parameter is able to tackle extremely ill conditioned problems arising in structural analysis semiconductor simulation oil reservoir modelling and other applications when applied to a positive definite symmetric matrix the algorithm produces a preconditioning matrix preserving that property

fast approximate solution of large scale sparse linear systems abstract in this paper we apply the fast approximate inversion algorithm of frederickson to solve a large scale sparse linear system arising from the triangular finite element solution to the second order elliptic problems the main advantage of this algorithm is that the rate of convergence is independent of n for equations in the class considered several approximate inversion techniques for the algorithm are proposed and numerical results for these techniques are also provided

a finite difference domain decomposition algorithm for numerical solution of the heat equation a domain decomposition algorithm for numerically solving the heat equation in one and two space dimensions is presented in this procedure interface values between subdomains are found by an explicit finite difference formula once these values are calculated interior values are determined by backward differencing in time a natural extension of this method allows for the use of different time steps in different subdomains maximum norm error estimates for these procedures are derived which demonstrate that the error incurred at the interfaces is higher order in the discretization parameters

a tensor format for the generalized hessenberg method for solving sylvester tensor equations abstract in this paper a general framework using tensor krylov projection techniques is proposed for solving high order sylvester tensor equations after describing the tensor format of the generalized hessenberg process we combine the obtained different processes with a galerkin orthogonality condition or with a minimal norm condition in order to derive the hess btf and cmrh btf methods which are based on the tensor format of the hessenberg process in addition we also recover the fom btf and gmres btf which are known methods based on the tensor format of the arnoldi process to accelerate the convergence or prevent a possible stagnation of the different obtained methods we incorporate a weighting strategy based on the use of a weighted inner product instead of the classical one when building a basis for the tensor krylov subspace numerical experiments are described in order to compare the new proposed methods that are hess btf and cmrh btf with the known methods fom btf and gmres btf and to show the efficiency of the weighting strategy moreover we utilize a flexible preconditioning framework for the unweighted and weighted forms of the proposed methods and the flexible version is validated by satisfactory numerical results

on the least squares generalized hamiltonian solution of generalized coupled sylvester conjugate matrix equations abstract in this paper we discuss the finite iterative algorithm to solve a class of generalized coupled sylvester conjugate matrix equations we prove that if the system is consistent an exact generalized hamiltonian solution can be obtained within finite iterative steps in the absence of round off errors for any initial matrices if the system is inconsistent the least squares generalized hamiltonian solution can be obtained within finite iterative steps in the absence of round off errors furthermore we provide a method for choosing the initial matrices to obtain the minimum norm least squares generalized hamiltonian solution of the system finally numerical examples are presented to demonstrate the algorithm is efficient