a semi circulant preconditioner for the convection diffusion equation in this paper we define and analyze a semi circulant preconditioner for the convection diffusion equation we derive analytical formulas for the eigenvalues and the eigenvectors of the preconditioned system of equations we show that for mesh peclet numbers less than 2 the rate of convergence depends only on the mesh peclet number and the direction of the convective field and not on the spatial grid ratio or the number of unknowns

indefinite overlapping schwarz methods for time dependent stokes problems abstract a class of indefinite overlapping schwarz methods is introduced and studied for time dependent stokes problems discretized with either mixed finite elements or mixed spectral elements the methods proposed are based on the solution of local time dependent stokes problems on overlapping subdomains and on the solution of a coarse time dependent stokes problem defined on the coarse subdomain mesh the resulting preconditioner is accelerated by an appropriate krylov space method producing a very efficient scalable and parallel solver

general constraint preconditioning iteration method for singular saddle point problems for the singular saddle point problems with nonsymmetric positive definite 1 1 block we present a general constraint preconditioning gcp iteration method based on a singular constraint preconditioner using the properties of the moore penrose inverse the convergence properties of the gcp iteration method are studied in particular for each of the two different choices of the 1 1 block of the singular constraint preconditioner a detailed convergence condition is derived by analyzing the spectrum of the iteration matrix numerical experiments are used to illustrate the theoretical results and examine the effectiveness of the gcp iteration method moreover the preconditioning effects of the singular constraint preconditioner for restarted generalized minimum residual gmres and quasi minimal residual qmr methods are also tested

a preconditioner for the steady state navier stokes equations we present a new method for solving the sparse linear system of equations arising from the discretization of the linearized steady state navier stokes equations also known as the oseen equations the solver is an iterative method of krylov subspace type for which we devise a preconditioner through a heuristic argument based on the fundamental solution tensor for the oseen operator the preconditioner may also be conceived through a weaker heuristic argument involving differential operators computations indicate that convergence for the preconditioned discrete oseen problem is only mildly dependent on the viscosity inverse reynolds number and most importantly that the number of iterations does not grow as the mesh size is reduced indeed since the preconditioner is motivated through analysis of continuous operators the number of iterations decreases for smaller mesh size which accords with better approximation of these operators

optimal control over multiple erasure channels using a data dropout compensation scheme in this paper we focus on networked control systems subject to data dropouts we consider that the controller communicates with the plant through multiple independent erasure channels and that each actuator sends acknowledgement packages to the controller we restrict our attention to the optimal design of a class of controllers that embeds a data dropout compensator and where besides this compensation mechanism all processing is an affine function of the available data our first result shows that the optimal design consists in solving separately an estimation problem and a control problem the optimal controller is a linear function of the estimated state which is obtained exploiting the information given by the acknowledgement packages and the data dropout compensator since we focus on a specific class of controllers our proposal is suboptimal however unlike the optimal lqg controller found in the literature our proposal is easier to implement and its steady state behaviour can be simply determined we also show that under standard conditions the affine part of the optimal controller converges to an lti filter which can be used as a simple alternative to the time varying controller

block diagonal and constraint preconditioners for nonsymmetric indefinite linear systems part i theory we study block diagonal preconditioners and an efficient variant of constraint preconditioners for general two by two block linear systems with zero 2 2 block we derive block diagonal preconditioners from a splitting of the 1 1 block of the matrix from the resulting preconditioned system we derive a smaller so called related system that yields the solution of the original problem solving the related system corresponds to an efficient implementation of constraint preconditioning we analyze the properties of both classes of preconditioned matrices in particular their spectra using analytical results we show that the related system matrix has the more favorable spectrum which in many applications translates into faster convergence for krylov subspace methods we show that fast convergence depends mainly on the quality of the splitting a topic for which a substantial body of theory exists our analysis also provides a number of new relations between block diagonal preconditioners and constraint preconditioners for constrained problems solving the related system produces iterates that satisfy the constraints exactly just as for systems with a constraint preconditioner finally for the lagrange multiplier formulation of a constrained optimization problem we show how scaling nonlinear constraints can dramatically improve the convergence for linear systems in a newton iteration our theoretical results are confirmed by numerical experiments on a constrained optimization problem we consider the general nonsymmetric nonsingular case our only additional requirement is the nonsingularity of the schur complement type matrix derived from the splitting that defines the preconditioners in particular the 1 2 block need not equal the transposed 2 1 block and the 1 1 block might be indefinite or even singular this is the first paper in a two part sequence in the second paper we will study the use of our preconditioners in a variety of applications

a parameterized deteriorated pss preconditioner and its optimization for nonsymmetric saddle point problems abstract based on the deteriorated positive definite and skew hermitian splitting dpss preconditioner we develop a parameterized dpss pdpss preconditioner for the nonsymmetric saddle point linear systems the spectral properties of the pdpss preconditioned coefficient matrix are analyzed moreover an upper bound of the degree of the minimal polynomial of the pdpss preconditioned matrix is also obtained inasmuch as the efficiency of the pdpss preconditioner depends on the values of its parameters we further derive a fast and effective method to compute the optimal parameter values of the pdpss preconditioner finally numerical examples are employed to illustrate the feasibility and the efficiency of the pdpss preconditioner

a block product preconditioner for saddle point problems abstract in this paper a block product bp preconditioner is established for saddle point problems spectral properties of the bp preconditioned matrix are investigated a strategy for practical choice of quasi optimal parameter is given numerical results on saddle point linear systems arising from stokes problems and weighted least square problems show that the proposed bp preconditioner is more economic to implement within krylov subspace acceleration than some extensively studied preconditioners

an alternative look at the constant gain kalman filter for state estimation over erasure channels this technical note studies state estimation problems subject to data loss we consider a class of switched estimators where missing data is replaced by optimal estimates the considered class of estimators encompasses a number of estimation schemes proposed in the literature we show that the estimator that minimizes the steady state estimation error covariance within that class is given by a constant gain kalman filter which was previously proposed as an alternative to the kalman filter with intermittent observations as a by product of our results we derive expressions that allow one to compare analytically popular suboptimal data dropout compensation mechanisms

overlapping schwarz methods for mixed linear elasticity and stokes problems overlapping schwarz preconditioners are introduced and studied for saddle point problems with a penalty term such as stokes equations and mixed formulations of linear elasticity these preconditioners are based on the solution of local saddle point problems on overlapping subdomains and the solution of a coarse saddle point problem numerical experiments show that these are parallel and scalable preconditioners since the rate of convergence of the preconditioned operator is independent of the mesh size h the number of subdomains n and the penalty parameter

a taxonomy and comparison of parallel block multi level preconditioners for the incompressible navier stokes equations in recent years considerable effort has been placed on developing efficient and robust solution algorithms for the incompressible navier stokes equations based on preconditioned krylov methods these include physics based methods such as simple and purely algebraic preconditioners based on the approximation of the schur complement all these techniques can be represented as approximate block factorization abf type preconditioners the goal is to decompose the application of the preconditioner into simplified sub systems in which scalable multi level type solvers can be applied in this paper we develop a taxonomy of these ideas based on an adaptation of a generalized approximate factorization of the navier stokes system first presented in a quarteroni f saleri a veneziani factorization methods for the numerical approximation of navier stokes equations computational methods in applied mechanical engineering 188 2000 505 526 this taxonomy illuminates the similarities and differences among these preconditioners and the central role played by efficient approximation of certain schur complement operators we then present a parallel computational study that examines the performance of these methods and compares them to an additive schwarz domain decomposition dd algorithm results are presented for two and three dimensional steady state problems for enclosed domains and inflow outflow systems on both structured and unstructured meshes the numerical experiments are performed using mpsalsa a stabilized finite element code

qmr a quasi minimal residual method for non hermitian linear systems the biconjugate gradient bcg method is the natural generalization of the classical conjugate gradient algorithm for hermitian positive definite matrices to general non hermitian linear systems unfortunately the original bcg algorithm is susceptible to possible breakdowns and numerical instabilities in this paper we present a novel bcg like approach the quasi minimal residual qmr method which overcomes the problems of bcg an implementation of qmr based on a look ahead version of the nonsymmetric lanczos algorithm is proposed it is shown how bcg iterates can be recovered stably from the qmr process some further properties of the qmr approach are given and an error bound is presented finally numerical experiments are reported

a relaxed deteriorated pss preconditioner for nonsymmetric saddle point problems from the steady navier stokes equation for nonsymmetric saddle point problems arising from the steady navier stokes equations pan ng and bai presented a deteriorated positive definite and skew hermitian splitting dpss preconditioner pan et al 2006 to accelerate the convergence rates of the krylov subspace iteration methods such as gmres in this paper the unconditional convergence property of the dpss iteration method is proved and a relaxed dpss rdpss preconditioner is proposed the rdpss preconditioner is much closer to the coefficient matrix than the dpss preconditioner in certain norm which straightforwardly results in an rdpss iteration method the convergence conditions of the rdpss iteration are analyzed and the optimal parameter which minimizes the spectral radius of the rdpss iteration matrix is derived using the rdpss preconditioner to accelerate some krylov subspace methods like gmres is also studied the eigenproperty of the preconditioned matrix is described and an upper bound of the degree of the minimal polynomial of the preconditioned matrix is obtained finally numerical experiments of a model navier stokes equation are presented to illustrate the efficiency of the rdpss preconditioner

tannaka krein duality for compact matrix pseudogroups twisted su n groups the notion of concrete monoidalw category is introduced and investigated a generalization of the tannaka krein duality theorem is proved it leads to new examples of compact matrix pseudogroups among them we have twistedsu n groups denoted bys u n it is shown that the representation theory fors u n is similar to that ofsu n irreducible representations are labeled by young diagrams and formulae for dimensions and multiplicity are the same as in the classical case

grothendieck rings of universal quantum groups abstract we determine the grothendieck ring of finite dimensional comodules for the free hopf algebra on a matrix coalgebra and similarly for the free hopf algebra with bijective antipode and other related universal quantum groups the results turn out to parallel those for wang and van daele s universal compact quantum groups and bichon s generalization of those results to universal cosovereign hopf algebras in all cases the rings are isomorphic to those of non commutative polynomials over certain sets these sets varying from case to case in most cases we are able to give more precise information about the multiplication table of the grothendieck ring

a class of accelerated uzawa algorithms for saddle point problems in this paper we establish a class of accelerated uzawa au algorithms for solving the large sparse nonsingular saddle point problems by making use of the extrapolation technique this extrapolation technique is based on the eigenvalues of the iterative matrix these au algorithms involve two iteration parameters whose special choices can cover the known classical uzawa method as well as yield new ones firstly the accelerated model for the uzawa algorithm is established and the detail algorithm description of au method is presented then the convergence analyse of the au method is given moreover theoretical analyses show that the au algorithm converges faster than some uzawa type methods the uzawa method is also included in when the eigenvalues of the iterative matrix and the parameter satisfy some conditions numerical experiments on a few model problems are presented to illustrate the theoretical results and examine the numerical effectiveness of the au method

pressure correction algebraic splitting methods for the incompressible navier stokes equations in this paper we present a new family of methods for the effective numerical solution of the incompressible unsteady navier stokes equations these methods resort to an algebraic splitting of the discretized problem based on inexact lu block factorizations of the corresponding matrix following a quarteroni f saleri and a veneziani comput methods appl mech engrg 188 2000 pp 505 526 in particular we will start from inexact algebraic factorizations of algebraic chorin temam and yosida type and introduce a pressure correction step aimed at improving the time accuracy one of the schemes obtained in this way the algebraic chorin temam pressure correction method resembles a method previously introduced in the framework of differential projection schemes see l timmermans p minev and f v de vosse internat j numer methods fluids 22 1996 pp 673 688 a prohl projection and quasi compressibility methods for solving the incompressible navier stokes equations teubner stuttgart 1997 the stability and the dependence of splitting error on the time step of the new methods is investigated and tested on several numerical cases

development of a stokes flow solver robust to large viscosity jumps using a schur complement approach with mixed precision arithmetic we develop an iterative solution technique for solving stokes flow problems with smooth and discontinuous viscosity structures using a three dimensional staggered grid finite difference discretization two preconditioned iterative methodologies are applied to the saddle point arising from the discrete stokes problem they consist of a velocity pressure coupled approach fc and a decoupled schur complement approach sc within both of these methods we utilize either the scaled bfbt or an identity matrix scaled by the local cell viscosity lv to define a preconditioner for the schur complement additionally we propose to use a mixed precision krylov kernel to improve the convergence by reducing round off error in this approach standard double precision is used during the application of the preconditioner whilst higher precision arithmetic is used to define the matrix vector product dot products and norms required by the krylov method in our krylov kernel we utilize quad precision arithmetic which is emulated via the double double precision method we consider several simplified geodynamic problems with a viscosity contrast to demonstrate the robustness and scalability of our solution methods through a careful choice of stopping conditions we are able to quantitatively compare the residuals between the sc and fc approaches we examine the trade off relationship between the number of outer iterations required for convergence and the computational cost per iteration for the each solution methods we find that it is advantageous to use the fc approach utilizing relaxed tolerances for solution of the sub problems combined with the lv preconditioner we also observed that in general the sc approach is more robust than fc and that bfbt is more robust than lv when used in our numerical experimental in addition our mixed precision method produces improved convergence rates of arnoldi type krylov subspace methods without a drastic increasing the computational time the usage of a high precision krylov kernel is found to be useful for the solver associated with the velocity sub problem

on pss based constraint preconditioners for singular nonsymmetric saddle point problems for singular nonsymmetric saddle point problems a class of singular constraint preconditioners and the corresponding preconditioning iteration methods are proposed based on the positive definite and skew symmetric splitting pss of 1 1 block of the coefficient matrix which can be categorized into a recently proposed generalized constraint preconditioner gcp for singular nonsymmetric saddle point problems the convergence analysis of the preconditioned iteration method is presented and the convergence conditions are derived some numerical experiments are implemented to demonstrate the feasibility and effectiveness of the pss based constraint preconditioning both as a preconditioned iteration method and a preconditioner of the krylov subspace method

a generalization of the local hermitian and skew hermitian splitting iteration methods for the non hermitian saddle point problems abstract for large sparse saddle point problems jiang and cao studied a class of local hermitian and skew hermitian splitting lhss iteration methods see m q jiang y cao on local hermitian and skew hermitian splitting iteration methods for generalized saddle point problems j comput appl math 231 2009 973 982 in this paper we generalized these methods and propose a class of generalized local hermitian and skew hermitian splitting glhss iteration schemes for solving the non hermitian saddle point problems we derive conditions for guaranteeing the convergence of these iterative methods with different choices of the parameter matrices the generalized iterative methods lead to a series of existing and new iterative methods numerical experiments for a model stokes problem are provided further show that the new iteration methods are feasible and effective

a note on preconditioning for indefinite linear systems preconditioners are often conceived as approximate inverses for nonsingular indefinite matrices of saddle point or kkt form we show how preconditioners incorporating an exact schur complement lead to preconditioned matrices with exactly two or exactly three distinct eigenvalues thus approximations of the schur complement lead to preconditioners which can be very effective even though they are in no sense approximate inverses

an augmented lagrangian based approach to the oseen problem we describe an effective solver for the discrete oseen problem based on an augmented lagrangian formulation of the corresponding saddle point system the proposed method is a block triangular preconditioner used with a krylov subspace iteration like bicgstab the crucial ingredient is a novel multigrid approach for the 1 1 block which extends a technique introduced by schoberl for elasticity problems to nonsymmetric problems our analysis indicates that this approach results in fast convergence independent of the mesh size and largely insensitive to the viscosity we present experimental evidence for both isop2 p0 and isop2 p1 finite elements in support of our conclusions we also show results of a comparison with two state of the art preconditioners showing the competitiveness of our approach

on local hermitian and skew hermitian splitting iteration methods for generalized saddle point problems in this paper we first present a local hermitian and skew hermitian splitting lhss iteration method for solving a class of generalized saddle point problems the new method converges to the solution under suitable restrictions on the preconditioning matrix then we give a modified lhss mlhss iteration method and further extend it to the generalized saddle point problems obtaining the so called generalized mlhss gmlhss iteration method numerical experiments for a model navier stokes problem are given and the results show that the new methods outperform the classical uzawa method and the inexact parameterized uzawa method

effective preconditioning of uzawa type schemes for a generalized stokes problem the schur complement of a model problem is considered as a preconditioner for the uzawa type schemes for the generalized stokes problem the stokes problem with the additional term alpha boldmath u in the motion equation the implementation of the preconditioned method requires for each iteration only one extra solution of the poisson equation with neumann boundary conditions for a wide class of 2d and 3d domains a theorem on its convergence is proved in particular it is established that the method converges with a rate that is bounded by some constant independent of alpha some finite difference and finite element methods are discussed numerical results for finite difference mac scheme are provided

galois reconstruction of finite quantum groups let c be a small category and let f c malgf be a functor where malgf is the category of finite dimensional measured algebras over a field k or frobenius algebras we construct a universal hopf algebra aaut f such that f factorizes through a functor f c mcoalgf aaut f where mcoalgf aaut f is the category of finite dimensional measured aaut f comodule algebras this general reconstruction result allows us to recapture a finite dimensional hopf algebra a from the category mcoalgf a and the forgetful functor mcoalgf a malgf we have a aaut our universal construction is also done in a c algebra framework and we get compact quantum groups in the sense of woronowicz

compact matrix pseudogroups the compact matrix pseudogroup is a non commutative compact space endowed with a group structure the precise definition is given and a number of examples is presented among them we have compact group of matrices duals of discrete groups and twisted deformed su n groups the representation theory is developed it turns out that the tensor product of representations depends essentially on their order the existence and the uniqueness of the haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived the form of these relations differs from that in the group case this is due to the fact that the haar measure on pseudogroups is not central in general the corresponding modular properties are discussed the haar measures on the twistedsu 2 group and on the finite matrix pseudogroup are found

preconditioning for the steady state navier stokes equations with low viscosity we introduce a preconditioner for the linearized navier stokes equations that is effective when either the discretization mesh size or the viscosity approaches zero for constant coefficient problems with periodic boundary conditions we show that the preconditioning yields a system with a single eigenvalue equal to 1 so that performance is independent of both viscosity and mesh size for other boundary conditions we demonstrate empirically that convergence depends only mildly on these parameters and we give a partial analysis of this phenomenon we also show that some expensive subsidiary computations required by the new method can be replaced by inexpensive approximate versions of these tasks based on iteration with virtually no degradation of performance

fast iterative solution of stabilised stokes systems part ii using general block preconditioners mixed finite element approximation of the classical stokes problem describing slow viscous incompressible flow gives rise to symmetric indefinite systems for the discrete velocity and pressure variables iterative solution of such indefinite systems is feasible and is an attractive approach for large problems part i of this work described a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems a j wathen and d j silvester siam j numer anal 30 1993 pp 630 649 using simple arguments estimates for the eigenvalue distribution of the discrete stokes operator on which the convergence rate of the iteration depends are easily derived part i discussed the important case of diagonal preconditioning scaling this paper considers the more general class of block preconditioners where the partitioning into blocks corresponds to the natural partitioning into the velocity and pressure variables it is shown that provid

efficient preconditioning of the linearized navier stokes equations for incompressible flow we outline a new class of robust and efficient methods for solving subproblems that arise in the linearization and operator splitting of navier stokes equations we describe a very general strategy for preconditioning that has two basic building blocks a multigrid v cycle for the scalar convection diffusion operator and a multigrid v cycle for a pressure poisson operator we present numerical experiments illustrating that a simple implementation of our approach leads to an effective and robust solver strategy in that the convergence rate is independent of the grid robust with respect to the time step and only deteriorates very slowly as the reynolds number is increased

a flexible inner outer preconditioned gmres algorithm a variant of the gmres algorithm is presented that allows changes in the preconditioning at every step there are many possible applications of the new algorithm some of which are briefly discussed in particular a result of the flexibility of the new variant is that any iterative method can be used as a preconditioner for example the standard gmres algorithm itself can be used as a preconditioner as can cgnr or cgne the conjugate gradient method applied to the normal equations however the more appealing utilization of the method is in conjunction with relaxation techniques possibly multilevel techniques the possibility of changing preconditioners may be exploited to develop efficient iterative methods and to enhance robustness a few numerical experiments are reported to illustrate this fact

combination of augmented lagrangian technique and st preconditioner for saddle point problems abstract for symmetric indefinite linear systems we introduce a new triangular preconditioner based on symmetric and triangular st decomposition a new 1 1 block is obtained by augmented lagrangian technique the new st preconditioner is introduced by the combination of the new 1 1 block and symmetric and triangular st decomposition then a preconditioned system can be obtained by preconditioning technique which is superior to the original system in terms of condition number we study the spectral properties of preconditioned system such as eigenvalues the estimation of condition number and then give the quasi optimal parameter numerical examples are given to indicate that the new preconditioner has obvious efficiency advantages finally we conclude that the new st preconditioner is a better option to deal with large and sparse problems

block preconditioners based on approximate commutators this paper introduces a strategy for automatically generating a block preconditioner for solving the incompressible navier stokes equations we consider the pressure convection diffusion preconditioners proposed by kay loghin and wathen siam j sci comput 24 2002 pp 237 256 and silvester elman kay and wathen j comput appl math 128 2001 pp 261 279 numerous theoretical and numerical studies have demonstrated mesh independent convergence on several problems and the overall efficacy of this methodology a drawback however is that it requires the construction of a convection diffusion operator denoted f p projected onto the discrete pressure space this means that integration of this idea into a code that models incompressible flow requires a sophisticated understanding of the discretization and other implementation issues something often held only by the developers of the model as an alternative we consider automatic ways of computing f p based on purely algebraic considerations the new methods are closely related to the bfbt preconditioner of elman siam j sci comput 20 1999 pp 1299 1316 we use the fact that the preconditioner is derived from considerations of commutativity between the gradient and convection diffusion operators together with methods for computing sparse approximate inverses to generate the required matrix f p automatically we demonstrate that with this strategy the favorable convergence properties of the preconditioning methodology are retained

quantum symmetry groups of finite spaces we determine the quantum automorphism groups of finite spaces these are compact matrix quantum groups in the sense of woronowicz

a modified dimensional split preconditioner for generalized saddle point problems abstract for large and sparse saddle point linear systems arising from 2d linearized navier stokes equations benzi and guo recently studied a dimensional split ds preconditioner appl numer math 61 2011 66 76 by further applying it to generalized saddle point problems in this paper we present a modified dimensional split mds preconditioner this new preconditioner is based on a splitting of the generalized saddle point matrix resulting in an unconditional convergent fixed point iteration the basic iteration is accelerated by a krylov subspace method like restarted gmres the implementation of the mds preconditioner is discussed and a similar case is also analyzed finally numerical experiments of a model navier stokes problem are presented to illustrate the effectiveness of the mds preconditioner

on a galerkin lagrange multiplier method for the stationary navier stokes equations a galerkin lagrange multiplier formulation is used for the numerical solution of the stationary navier stokes equations in order to avoid the construction of zero divergence elements the formulation is based on different approximating spaces for the velocity field and the pressure optimal rate of convergence estimates are derived moreover a galerkin newton scheme for the solution of the nonlinear equations is shown to be quadratically locally convergent another scheme is shown to be linearly globally convergent

preconditioners for generalized saddle point problems noindent we propose and examine block diagonal preconditioners and variants of indefinite preconditioners for block two by two generalized saddle point problems that is we consider the nonsymmetric nonsingular case where the 2 2 block is small in norm and we are particularly concerned with the case where the 1 2 block is different from the transposed 2 1 block we provide theoretical and experimental analyses of the convergence and eigenvalue distributions of the preconditioned matrices we also extend the results of de sturler and liesen siam j sci comput 26 2005 pp 1598 1619 to matrices with nonzero 2 2 block and to the use of approximate schur complements to demonstrate the effectiveness of these preconditioners we show convergence results spectra and eigenvalue bounds for two model navier stokes problems

kalman filtering with intermittent observations motivated by navigation and tracking applications within sensor networks we consider the problem of performing kalman filtering with intermittent observations when data travel along unreliable communication channels in a large wireless multihop sensor network the effect of communication delays and loss of information in the control loop cannot be neglected we address this problem starting from the discrete kalman filtering formulation and modeling the arrival of the observation as a random process we study the statistical convergence properties of the estimation error covariance showing the existence of a critical value for the arrival rate of the observations beyond which a transition to an unbounded state error covariance occurs we also give upper and lower bounds on this expected state error covariance

on solving block structured indefinite linear systems we consider 2 2 block indefinite linear systems whose 2 2 block is zero such systems arise in many applications we discuss two techniques that are based on modifying the 1 1 block in a way that makes the system easier to solve the main part of the paper focuses on an augmented lagrangian approach a technique that modifies the 1 1 block without changing the system size the choice of the parameter involved the spectrum of the linear system and its condition number are discussed and some analytical observations are provided a technique of deflating the 1 1 block is then introduced finally numerical experiments that validate the analysis are presented

a class of triangular splitting methods for saddle point problems in this paper we study a class of efficient iterative algorithms for the large sparse nonsingular saddle point problems based on the upper and lower triangular ult splitting of the coefficient matrix we call these algorithms ult methods first the ult algorithm is established and the characteristic of eigenvalues of the iteration matrix of these new methods is analyzed then we give the sufficient and necessary conditions for the convergence of these ult methods moreover the optimal iteration parameters and the corresponding convergence factors for some special cases of the ult methods are presented numerical experiments on a few model problems are presented to support the theoretical results and examine the numerical effectiveness of these new methods

block triangular preconditioners for nonsymmetric saddle point problems field of values analysis a preconditioned minimal residual method for nonsymmetric saddle point problems is analyzed the proposed preconditioner is of block triangular form the aim of this article is to show that a rigorous convergence analysis can be performed by using the field of values of the preconditioned linear system as an example a saddle point problem obtained from a mixed finite element discretization of the oseen equations is considered the convergence estimates obtained by using a field of values analysis are independent of the discretization parameter h several computational experiments supplement the theoretical results and illustrate the performance of the method

a generalization of preconditioned parameterized inexact uzawa method for indefinite saddle point problems the parameterized inexact uzawa methods have been used to solve some of the symmetric saddle point problems in this paper a new preconditioned parameterized inexact uzawa method is presented to solve indefinite saddle point problems after preconditioning theoretical analyses show that the iteration method converges under certain conditions so we propose three new algorithms based on these conditions numerical experiments are provided to show the effectiveness of the proposed preconditioner and all these algorithms have fantastic convergence rates by choosing optimal parameters

high order accurate solution of the incompressible navier stokes equations on massively parallel computers the emergence of petascale supercomputers requires us to develop today s simulation codes for incompressible flows by codes which are using numerical schemes and methods that are better able to exploit the offered computational power in that spirit we present a massively parallel high order navier stokes solver for large incompressible flow problems in three dimensions the governing equations are discretized with finite differences in space and a semi implicit time integration scheme this discretization leads to a large linear system of equations which is solved with a cascade of iterative solvers the iterative solver for the pressure uses a highly efficient commutation based preconditioner which is robust with respect to grid stretching the efficiency of the implementation is further enhanced by carefully setting the adaptive termination criteria for the different iterative solvers the computational work is distributed to different processing units by a geometric data decomposition in all three dimensions this decomposition scheme ensures a low communication overhead and excellent scaling capabilities the discretization is thoroughly validated first we verify the convergence orders of the spatial and temporal discretizations for a forced channel flow second we analyze the iterative solution technique by investigating the absolute accuracy of the implementation with respect to the different termination criteria third orr sommerfeld and squire eigenmodes for plane poiseuille flow are simulated and compared to analytical results fourth the practical applicability of the implementation is tested for transitional and turbulent channel flow the results are compared to solutions from a pseudospectral solver subsequently the performance of the commutation based preconditioner for the pressure iteration is demonstrated finally the excellent parallel scalability of the proposed method is demonstrated with a weak and a strong scaling test on up to o 10 4 processing units and o 10 1 1 grid points

fast nonsymmetric iterations and preconditioning for navier stokes equations discretization and linearization of the steady state navier stokes equations gives rise to a nonsymmetric indefinite linear system of equations in this paper we introduce preconditioning techniques for such systems with the property that the eigenvalues of the preconditioned matrices are bounded independently of the mesh size used in the discretization we confirm and supplement these analytic results with a series of numerical experiments indicating that krylov subspace iterative methods for nonsymmetric systems display rates of convergence that are independent of the mesh parameter in addition we show that preconditioning costs can be kept small by using iterative methods for some intermediate steps performed by the preconditioner

on hss based sequential two stage method for non hermitian saddle point problems abstract for large sparse saddle point problems with symmetric positive definite 1 1 block li et al studied an efficient iterative method see li et al 2011 25 by making use of the same preconditioning technique and a new matrix splitting based on the hermitian and skew hermitian splitting hss of the 1 1 block of the preconditioned non hermitian saddle point systems an efficient sequential two stage method is proposed for solving the non hermitian saddle point problems theoretical analysis shows the proposed iterative method is convergent and that the spectral radius of iterative matrix monotonically decreases and tends to 0 as the iterative parameter approaches infinity numerical experiments arising from naiver stokes problem are provided to show that the new iterative method is feasible effective and robust

least squares preconditioners for stabilized discretizations of the navier stokes equations this paper introduces two stabilization schemes for the least squares commutator lsc preconditioner developed by elman howle shadid shuttleworth and tuminaro siam j sci comput 27 2006 pp 1651 1668 for the incompressible navier stokes equations this preconditioning methodology is one of several choices that are effective for navier stokes equations and it has the advantage of being defined from strictly algebraic considerations it has previously been limited in its applicability to div stable discretizations of the navier stokes equations this paper shows how to extend the same methodology to stabilized low order mixed finite element approximation methods

an efficient solver for the incompressible navier stokes equations in rotation form we consider preconditioned iterative methods applied to discretizations of the linearized navier stokes equations in two and three dimensional bounded domains both unsteady and steady flows are considered the equations are linearized by picard iteration we make use of the rotation form of the momentum equations which has several advantages from the linear algebra point of view we focus on a preconditioning technique based on the hermitian skew hermitian splitting of the resulting nonsymmetric saddle point matrix we show that this technique can be implemented efficiently when the rotation form is used we study the performance of the solvers as a function of mesh size reynolds number time step and algorithm parameters our results indicate that fast convergence independent of problem parameters is achieved in many cases the preconditioner appears to be especially attractive in the case of low viscosity and for unsteady problems

semi convergence analysis of gmssor methods for singular saddle point problems recently some authors darvishi and hessari 2011 zhang et al 2012 discussed the generalized modified symmetric successive over relaxation gmssor method for saddle point problems in this paper we further study the gmssor method for solving singular saddle point problems we prove the semi convergence of the gmssor method without the assumption that the 1 1 block sub matrix a should be symmetric and analyze the spectral property of the corresponding preconditioned matrix numerical experiments are given to illustrate the efficiency of gmssor with appropriate parameters

two modified block triangular splitting preconditioners for generalized saddle point problems abstract for the generalized saddle point problems firstly we introduce a modified generalized relaxed splitting mgrs preconditioner to accelerate the convergence rate of the krylov subspace methods based on a block triangular splitting of the saddle point matrix secondly we propose a modified block triangular splitting mbts this new preconditioner is easily implemented since it has simple block structure the spectral properties and the degrees of the minimal polynomials of the preconditioned matrices are discussed respectively moreover we apply the mgrs and the mbts preconditioners to three dimensional linearized navier stokes equations then we derive the quasi optimal parameters of the mgrs and the mbts preconditioners for two and three dimensional navier stokes equations respectively finally numerical experiments are illustrated to show the preconditioning effects of the two new preconditioners

a low order galerkin finite element method for the navier stokes equations of steady incompressible flow a stabilization issue and iterative methods a galerkin finite element method is considered to approximate the incompressible navier stokes equations together with iterative methods to solve a resulting system of algebraic equations this system couples velocity and pressure unknowns thus requiring a special technique for handling we consider the navier stokes equations in velocity kinematic pressure variables as well as in velocity bernoulli pressure variables the latter leads to the rotation form of nonlinear terms this form of the equations plays an important role in our studies a consistent stabilization method is considered from a new view point theory and numerical results in the paper address both the accuracy of the discrete solutions and the effectiveness of solvers and a mutual interplay between these issues when particular stabilization techniques are applied 2002 published by elsevier science b v

optimal parameters of the generalized symmetric sor method for augmented systems for the augmented system of linear equations zhang and lu 2008 recently studied the generalized symmetric sor method gssor with two parameters in this note the optimal parameters of the gssor method are obtained and numerical examples are given to illustrate the corresponding results

multigrid preconditioning for krylov methods for time harmonic maxwell s equations in three dimensions we consider the rapid simulation of three dimensional electromagnetic problems in geophysical parameter regimes where the conductivity may vary significantly and the range of frequencies is moderate toward developing a multigrid preconditioner we present a fourier analysis based on a finite volume discretization of a vector potential formulation of time harmonic maxwell s equations on a staggered grid in three dimensions we prove grid independent bounds on the eigenvalue and singular value ranges of the system obtained using a preconditioner based on exact inversion of the dominant diagonal blocks of the non hermitian coefficient matrix this result implies that a preconditioner that uses single multigrid cycles to effect inversion of the diagonal blocks also yields a preconditioned system with an ell 2 condition number bounded independent of the grid size we then present numerical examples for more realistic situations involving large variations in conductivity i e jump discontinuities block preconditioning with one multigrid cycle using dendy s boxmg solver is found to yield convergence in very few iterations apparently independent of the grid size the experiments show that the somewhat restrictive assumptions of the fourier analysis do not prohibit it from describing the essential local behavior of the preconditioned operator under consideration a very efficient practical solver is obtained

block lu preconditioners for symmetric and nonsymmetric saddle point problems in this paper a general purpose block lu preconditioner for saddle point problems is presented the main difference between the approach presented here and that of other studies is that an explicit accurate approximation of the schur complement matrix is efficiently computed this is used to compute a preconditioner to the schur complement matrix which in turn defines a preconditioner for the global system a number of different variants are developed and results are reported for a few linear systems arising from cfd applications

semi convergence analysis of uzawa methods for singular saddle point problems recently zheng bai and yang studied the parameterized uzawa method for solving singular saddle point problems b zheng z z bai x yang on semi convergence of parameterized uzawa methods for singular saddle point problems linear algebra appl 431 2009 808 817 in this paper we discuss the inexact uzawa method which covers the uzawa method the preconditioned uzawa method and the parameterized uzawa method to solve the singular saddle point problems we prove the semi convergence result under restrictions by verifying two necessary and sufficient conditions that is all elementary divisors associated with the eigenvalue 1 of its iterative matrix are linear and the pseudo spectral radius of the iterative matrix is less than 1 sufficient conditions for the semi convergence of several uzawa type methods are also provided in addition numerical examples are given to demonstrate the semi convergence of uzawa type methods

an extended finite element method applied to levitated droplet problems we consider a standard model for incompressible two phase flows in which a localized force at the interface describes the effect of surface tension if a level set method is applied then the approximation of the interface is in general not aligned with the triangulation this causes severe difficulties w r t the discretization and often results in large spurious velocities in this paper we reconsider a modified extended finite element method xfem which in previous papers has been investigated for relatively simple two phase flow model problems and apply it to a physically realistic levitated droplet problem the results show that due to the extension of the standard fe space one obtains much better results in particular for large interface tension coefficients furthermore a certain cut off technique results in better efficiency without sacrificing accuracy copyright 2010 john wiley sons ltd

a preconditioning technique for indefinite systems resulting from mixed approximations of elliptic problems this paper provides a preconditioned iterative technique for the solution of saddle point problems these problems typically arise in the numerical approximation of partial differential equations by lagrange multiplier techniques and or mixed methods the saddle point problem is reformulated as a symmetric positive definite system which is then solved by conjugate gradient iteration applications to the equations of elasticity and stokes are discussed and the results of numerical experiments are given

shift splitting preconditioners for saddle point problems in this paper we first present a shift splitting preconditioner for saddle point problems the preconditioner is based on a shift splitting of the saddle point matrix resulting in an unconditional convergent fixed point iteration based on the idea of the splitting we further propose a local shift splitting preconditioner some properties of the local shift splitting preconditioned matrix are studied these preconditioners extend those studied by bai yin and su for solving non hermitian positive definite linear systems bai et al 2006 finally numerical experiments of a model stokes problem are presented to show the effectiveness of the proposed preconditioners

two block triangular preconditioners for asymmetric saddle point problems in this paper two block triangular preconditioners for the asymmetric saddle point problems with singular 1 1 block are presented the spectral characteristics of the preconditioned matrices are discussed in detail theoretical analysis shows that all the eigenvalues of the preconditioned matrices are strongly clustered numerical experiments are reported to the efficiency of the proposed preconditioners

fourier analysis of iterative methods for elliptic boundary value problems his paper presents a fourier method for analyzing stationary iterative methods and preconditioners for discretized elliptic boundary value problems as in the von neumann stability analysis of hyperbolic and parabolic problems the approach is easier to apply reveals more details about convergence properties than about standard techniques and can be applied in a systematic way to a wide class of numerical methods although the analysis is applicable only to periodic problems the results essentially reproduce those of classical convergence and condition number analysis for problems with other boundary conditions such as the dirichlet problem in addition they give suggestive new evidence of the strengths and weaknesses of methods such as incomplete factorization preconditioners in the dirichlet case

comparison of performance of iterative methods for singular and nonsingular saddle point linear systems arising from navier stokes equations abstract we compare the convergence performance of iterative methods for solving singular and nonsingular linear systems arising from discretization and linearization of the steady state navier stokes equations with a combination of analytic and empirical results we study the effects of singularity and nonsingularity on convergence we will show that the convergence performance for solving a singular linear system is better than that for solving a nonsingular linear system

analysis of the relaxed deteriorated pss preconditioner for singular saddle point linear systems abstract the relaxed deteriorated pss rdpss preconditioner proposed by cao et al 13 is further generalized to solve singular saddle point linear systems properties of the rdpss splitting and convergence of the corresponding iteration method are studied by using the moore penrose inverse the rdpss iteration method and the rdpss preconditioned gmres are both proved to converge to the least squares solution of the singular saddle point linear system the rdpss preconditioned matrix is also analyzed results about eigenvalue distributions are derived moreover the rdpss preconditioner is generalized to a class of more general preconditioners the corresponding convergence and eigenvalue distribution results are analyzed numerical experiments are presented to illustrate the effectiveness of the rdpss preconditioner and its variants to accelerate gmres for solving singular saddle point linear systems

a preconditioner for linear systems arising from interior point optimization methods we explore a preconditioning technique applied to the problem of solving linear systems arising from primal dual interior point algorithms in linear and quadratic programming the preconditioner has the attractive property of improved eigenvalue clustering with increased ill conditioning of the 1 1 block of the saddle point matrix it fits well into the optimization framework since the interior point iterates yield increasingly ill conditioned linear systems as the solution is approached we analyze the spectral characteristics of the preconditioner utilizing projections onto the null space of the constraint matrix and demonstrate performance on problems from the netlib and cuter test suites the numerical experiments include results based on inexact inner iterations

on block preconditioners for nonsymmetric saddle point problems we discuss a class of preconditioning methods for an iterative solution of algebraic nonsymmetric saddle point problems arising from a mixed finite element discretization of partial differential equations in particular the navier stokes equation we prove that block diagonal and block triangular preconditioners based on symmetric positive definite blocks guarantee that the convergence rate of the method is independent of the mesh parameter h

output feedback stabilization of networked control systems with packet dropouts in this paper we discuss the modeling and control of a class of networked control systems ncss with packet dropouts for the cases that there may be packet dropouts in both the backward and the forward channels in the communication network and that the network induced delays are shorter than one sampling period the closed loop ncs is modeled as a discrete time switched system with four subsystems by using the asynchronous dynamical systems approach and the average dwell time method sufficient conditions for the exponential stability of the closed loop ncs are presented in terms of nonlinear matrix inequalities and the relation between the packet dropout rate and the stability of the closed loop ncs is explicitly established a procedure involving an iterative algorithm is proposed to design the observer based output feedback controllers lastly an illustrative example is given to demonstrate the effectiveness of the proposed results

lqg control for mimo systems over multiple erasure channels with perfect acknowledgment this technical note concerns control applications over lossy data networks sensor data is transmitted to an estimation control unit over a network and control commands are issued to subsystems over the same network sensor and control packets may be randomly lost according to a bernoulli process in this context the discrete time linear quadratic gaussian lqg optimal control problem is considered in schenato a complete analysis was carried out for the case that sensor measurements and control inputs are delivered into a single packet to the estimator and to the actuators respectively here a nontrivial generalization for mimo systems is presented under the assumption that each sensor and each actuator exchange data with the control unit in an independent way by using their own data packet no aggregation in such a framework it is shown that the separation principle still holds in the case where packet arrivals are acknowledged by the receiver moreover the optimal lqg control is a linear function of the state that explicitly depends on the loss probabilities of the actuator channels such a dependence is not present in the single channel case considered in mean square in the infinite horizon case stability conditions on the packet arrival probabilities are provided in terms of linear matrix inequalities lmis

parameterized generalized shift splitting preconditioners for nonsymmetric saddle point problems abstract to solve nonsymmetric saddle point problems the parameterized generalized shift splitting pgss preconditioner is presented and analyzed the corresponding pgss iteration method can be applied not only to the nonsingular saddle point problems but also to the singular ones the convergence and semi convergence of the pgss iteration method are discussed carefully meanwhile the spectral properties of the preconditioned matrix and the strategy of the choices of the parameters are given numerical experiments further demonstrate that the pgss iteration method and the pgss preconditioner are efficient and have better performance than some existing iteration methods and newly proposed preconditioners respectively for solving both the nonsingular and singular nonsymmetric saddle point problems

two augmentation preconditioners for nonsymmetric and indefinite saddle point linear systems with singular 1 1 blocks abstract in this paper two preconditioners based on augmentation are introduced for the solution of large saddle point type systems with singular 1 1 blocks we study the spectral characteristics of the preconditioners show that all eigenvalues of preconditioned matrices are strongly clustered finally numerical experiments are also reported for illustrating the efficiency of the presented preconditioners

brief paper optimal linear estimation for systems with multiple packet dropouts this paper is concerned with the optimal linear estimation problem for linear discrete time stochastic systems with multiple packet dropouts based on a packet dropout model the optimal linear estimators including filter predictor and smoother are developed via an innovation analysis approach the estimators are computed recursively in terms of the solution of a riccati difference equation of dimension equal to the order of the system state plus that of the measurement output the steady state estimators are also investigated a sufficient condition for the convergence of the optimal linear estimators is given simulation results show the effectiveness of the proposed optimal linear estimators

iterative methods for cyclically reduced non self adjoint linear systems we study iterative methods for solving linear systems of the type arising from two cyclic discretizations of non self adjoint two dimensional ellip tic partial differential equations a prototype is the convection diffusion equa tion the methods consist of applying one step of cyclic reduction resulting in a reduced system of half the order of the original discrete problem com bined with a reordering and a block iterative technique for solving the reduced system for constant coefficient problems we present analytic bounds on the spectral radii of the iteration matrices in terms of cell reynolds numbers that show the methods to be rapidly convergent in addition we describe numerical experiments that supplement the analysis and that indicate that the methods compare favorably with methods for solving the unreduced system

moving mesh finite element methods for the incompressible navier stokes equations this work presents the first effort in designing a moving mesh algorithm to solve the incompressible navier stokes equations in the primitive variables formulation the main difficulty in developing this moving mesh scheme is how to keep it divergence free for the velocity field at each time level the proposed numerical scheme extends a recent moving grid method based on harmonic mapping r li t tang and p w zhang j comput phys 170 2001 pp 562 588 which decouples the pde solver and the mesh moving algorithm this approach requires interpolating the solution on the newly generated mesh designing a divergence free preserving interpolation algorithm is the first goal of this work selecting suitable monitor functions is important and is found challenging for the incompressible flow simulations which is the second goal of this study the performance of the moving mesh scheme is tested on the standard periodic double shear layer problem no spurious vorticity patterns appear when even fairly coarse grids are used

performance and analysis of saddle point preconditioners for the discrete steady state navier stokes equations we examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady state navier stokes equations with a combination of analytic and empirical results we study the effects of fundamental parameters on convergence we demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers the structure of these distributions is independent of the discretization mesh size but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller these characteristics are directly correlated with the convergence properties of iterative solvers

new choices of preconditioning matrices for generalized inexact parameterized iterative methods for large sparse saddle point problems chen and jiang recently studied a class of generalized inexact parameterized iterative methods see f chen y l jiang a generalization of the inexact parameterized uzawa methods for saddle point problems appl math comput 206 2008 765 771 in this paper the methods are modified and some choices of preconditioning matrices are given these preconditioning matrices have advantages in solving large sparse linear system numerical experiments of a model stokes problem are presented

pressure schur complement preconditioners for the discrete oseen problem we consider several preconditioners for the pressure schur complement of the discrete steady oseen problem two of the preconditioners are well known from the literature and the other is new supplemented with an appropriate approximate solve for an auxiliary velocity subproblem these approaches give rise to a family of the block preconditioners for the matrix of the discrete oseen system in the paper we critically review possible advantages and difficulties of using various schur complement preconditioners we recall existing eigenvalue bounds for the preconditioned schur complement and prove such for the newly proposed preconditioner these bounds hold both for lbb stable and stabilized finite elements results of numerical experiments for several model two dimensional and three dimensional problems are presented in the experiments we use lbb stable finite element methods on uniform triangular and tetrahedral meshes one particular conclusion is that in spite of essential improvement in comparison with simple scaled mass matrix preconditioners in certain cases none of the considered approaches provides satisfactory convergence rates in the case of small viscosity coefficients and a sufficiently complex e g circulating advection vector field

on element by element schur complement approximations abstract we discuss a methodology to construct sparse approximations of schur complements of two by two block matrices arising in finite element discretizations of partial differential equations earlier results from 2 are extended to more general symmetric positive definite matrices of two by two block form the applicability of the method for general symmetric and nonsymmetric matrices is analysed the paper demonstrates the applicability of the presented method providing extensive numerical experiments

an adaptive multigrid technique for the incompressible navier stokes equations an automatic adaptive refinement technique has been coupled to the multigrid approach to produce an efficient and stable solution strategy for solving the steady state incompressible navier stokes equations solutions have been obtained for the driven cavity and flow over a backward facing step for reynolds numbers up to 5000 and 800 respectively the refinement criterion is based on the local truncation error the solution error is monitored and automatic refinement can continue until it is reduced to a satisfactory level for driven cavity flow at re 1000 the adaptive refinement approach reduced the computer memory and cpu time to 20 and 40 of the requirements of the pure multigrid method the primitive variable formulation of the navier stokes equations is used so the method can be extended easily to three dimensions application to other nonlinear elliptic problems is equally possible

a modified product preconditioner for indefinite and asymmetric generalized saddle point matrices in this paper we present a new modified product preconditioner mp for a class of large sparse linear systems with indefinite and asymmetric matrices the eigenvalue distribution and form of the eigenvectors of the presented new preconditioned matrix and its minimal polynomial are investigated some numerical experiments illustrate that the proposed new preconditioner performs better than block diagonal preconditioner block triangular preconditioner constraint preconditioner and product preconditioner