uzawa type algorithms for nonsymmetric saddle point problems in this paper we consider iterative algorithms of uzawa type for solving linear nonsymmetric saddle point problems specifically we consider systems written as usual in block form where the upper left block is an invertible linear operator with positive definite symmetric part such saddle point problems arise for example in certain finite element and finite difference discretizations of navier stokes equations oseen equations and mixed finite element discretization of second order convection diffusion problems we consider two algorithms each of which utilizes a preconditioner for the operator in the upper left block convergence results for the algorithms are established in appropriate norms the convergence of one of the algorithms is shown assuming only that the preconditioner is spectrally equivalent to the inverse of the symmetric part of the operator the other algorithm is shown to converge provided that the preconditioner is a sufficiently accurate approximation of the inverse of the upper left block applications to the solution of steady state navier stokes equations are discussed and finally the results of numerical experiments involving the algorithms are presented

the numerical solution of equality constrained quadratic programming problems this paper proves that a large class of iterative schemes can be used to solve a certain constrained minimization problem the constrained minimization problem considered involves the minimization of a quadratic functional subject to linear equality constraints among this class of convergent iterative schemes are generalizations of the relaxed jacobi gauss seidel and symmetric gauss seidel schemes

a greedy strategy for coarse grid selection efficient solution of the very large linear systems that arise in numerical modeling of real world applications is often only possible through the use of multilevel techniques while highly optimized algorithms may be developed using knowledge about the origins of the matrix problem to be considered much recent interest has been in the development of purely algebraic approaches that may be applied in many situations without problem specific tuning here we consider an algebraic approach to finding the fine coarse partitions needed in multilevel approaches the algorithm is motivated by recent theoretical analysis of the performance of two common multilevel algorithms multilevel block factorization and algebraic multigrid while no guarantee on the rate of coarsening is given the splitting is shown to always yield an effective preconditioner in the two level sense numerical performance of two level and multilevel variants of this approach is demonstrated in combination with both algebraic multigrid and multilevel block factorizations and the advantages of each of these two algorithmic settings are explored

non resonance conditions for uniform observability in the problem of nonlinear output regulation abstract this paper deals with the design of an internal model based semiglobal output feedback regulator for possibly non minimum phase nonlinear systems taking advantage of the design tool proposed in isidori ieee trans automat control 45 10 2000 we show how the problem of output regulation can be reformulated into an output feedback stabilization problem of a suitably defined extended auxiliary system the output feedback stabilization of the extended auxiliary system is addressed in the second part of the paper where an observer based stabilizer is proposed the existence of the latter is characterized in terms of necessary and sufficient conditions which can be interpreted as nonlinear non resonance conditions between the modes of the exosystem and the zero dynamics of the controlled plant

linear algebra methods in a mixed approximation of magnetostatic problems in the solution of magnetostatic problems the use of a mixed formulation based on both the magnetic and magnetic displacement fields is particularly appropriate as it allows us to impose the physical conditions exactly and to maintain the continuity properties of the two fields together with an efficient treatment of boundary conditions the discretization by means of a proper finite element method yields a strongly structured algebraic linear system this paper is concerned with the solution of this large very sparse indefinite linear system in particular we present the implementation of some known direct and preconditioned iterative methods and discuss their performance on two dimensional 2 d and three dimensional 3 d specific models we show that the 2 d system can be efficiently handled by appropriate variants of these schemes while preliminary tests on the 3 d system give some insight in the understanding of the analysis that needs to be done

an iteration for indefinite systems and its application to the navier stokes equations for large sparse systems of linear equations iterative solution techniques are attractive in this paper we propose and examine the convergence of an iterative method for an important class of nonsymmetric and indefinite coefficient matrices based on the use of an indefinite and symmetric preconditioner we apply our technique to the linearized navier stokes equations the oseen equations

an optimal preconditioner for a class of saddle point problems with a penalty term iterative methods are considered for a class of saddle point problems with a penalty term arising from finite element discretizations of certain elliptic problems an optimal preconditioner which is independent of the discretization and the penalty parameter is constructed this approach is then used to design an iterative method with a convergence rate bounded independently of the lame parameters occurring in the equations of linear elasticity

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

a class of iterative methods for solving saddle point problems we consider the numerical solution of indefinite systems of linear equations arising in the calculation of saddle points we are mainly concerned with sparse systems of this type resulting from certain discretizations of partial differential equations we present an iterative method involving two levels of iteration similar in some respects to the uzawa algorithm we relate the rates of convergence of the outer and inner iterations proving that under natural hypotheses the outer iteration achieves the rate of convergence of the inner iteration the technique is applied to finite element approximations of the stokes equations

block triangular preconditioners for saddle point problems with a penalty term block triangular preconditioners for a class of saddle point problems with a penalty term are considered an important example is the mixed formulation of the pure displacement problem in linear elasticity it is shown that the spectrum of the preconditioned system is contained in a real positive interval and that the interval bounds can be made independent of the discretization and penalty parameters this fact is used to construct bounds of the convergence rate of the gmres method with respect to an energy norm numerical results are given for gmres and bi cgstab

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

the trace identity a powerful tool for constructing the hamiltonian structure of integrable systems a new approach to hamiltonian structures of integrable systems is proposed by making use of a trace identity for a variety of isospectral problems that can be unified to one model x u it is shown that both the related hierarchy of evolution equations and the hamiltonian structure can be obtained from the same solution of the equation vx u v

variational identities and applications to hamiltonian structures of soliton equations abstract this is an introductory report concerning our recent research on hamiltonian structures we will discuss variational identities associated with continuous and discrete spectral problems and their applications to hamiltonian structures of soliton equations our illustrative examples are the akns hierarchy and the volterra lattice hierarchy associated with semisimple lie algebras and two hierarchies of their integrable couplings associated with non semisimple lie algebras the resulting hamiltonian structures generate infinitely many commuting symmetries and conservation laws for the four soliton hierarchies the presented variational identities can be applied to hamiltonian structures of other soliton hierarchies

approximate inverse preconditioners via sparse sparse iterations the standard incomplete lu ilu preconditioners often fail for general sparse indefinite matrices because they give rise to unstable factors l and u in such cases it may be attractive to approximate the inverse of the matrix directly this paper focuses on approximate inverse preconditioners based on minimizing i am f where am is the preconditioned matrix an iterative descent type method is used to approximate each column of the inverse for this approach to be efficient the iteration must be done in sparse mode i e with sparse matrix by sparse vector operations numerical dropping is applied to maintain sparsity compared to previous methods this is a natural way to determine the sparsity pattern of the approximate inverse this paper describes newton global and column oriented algorithms and discusses options for initial guesses self preconditioning and dropping strategies some limited theoretical results on the properties and convergence of approximate inverses are derived numerical tests on problems from the harwell boeing collection and the fidap fluid dynamics analysis package show the strengths and limitations of approximate inverses finally some ideas and experiments with practical variations and applications are presented

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

analysis of the inexact uzawa algorithm for saddle point problems in this paper we consider the so called inexact uzawa algorithm for iteratively solving linear block saddle point problems such saddle point problems arise for example in finite element and finite difference discretizations of stokes equations the equations of elasticity and mixed finite element discretization of second order problems we consider both the linear and nonlinear variants of the inexact uzawa iteration we show that the linear method always converges as long as the preconditioners defining the algorithm are properly scaled bounds for the rate of convergence are provided in terms of the rate of convergence for the preconditioned uzawa algorithm and the reduction factor corresponding to the preconditioner for the upper left hand block in the case of nonlinear iteration the inexact uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of sufficient accuracy bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned linear uzawa algorithm applications to the stokes equations and mixed finite element discretization of second order elliptic problems are discussed and finally the results of numerical experiments involving the algorithms are presented

preconditioning techniques for nonsymmetric and indefinite linear systems abstract the standard preconditioning techniques for conjugate gradient methods often fail for matrices that are indefinite and or strongly nonsymmetric the most common alterative considered for these cases are either to use expensive direct solvers or to resort to one of many techniques based on the normal equations this paper examines several such alternatives and compares them in particular an incomplete lq factorization is proposed and some of its implementation details are described a number of experiments are reported to compare these methods

new extended lie algebra and the generalized integrable liouville hierarchy abstract starting from a new lie algebra g 1 the corresponding loop algebra g 1 is presented from which a liouville integrable hierarchy is given by using of variational identity it follows that an expanding lie algebra g 2 is obtained based on g 1 furthermore a related lax integrable hierarchy is presented by making use of its related loop algebra g 2 with the help of variational identity it is not difficult to prove that the hierarchy has hamiltonian structure and is liouville integrable it is also can be seen that the second hierarchy is the expanding model of the first one

an approach to generate superextensions of integrable systems an algorithm to generate integrable systems is extended to the super case some new examples of superextensions of integrable systems are illustrated we also generalize the trace identity due to tu to the super case and use it to establish hamiltonian structures of superextensions of integrable systems under consideration

numerical accuracy of a pade type non reflecting boundary condition for the finite element solution of acoustic scattering problems at high frequency the present text deals with the numerical solution of two dimensional high frequency acoustic scattering problems using a new high order and asymptotic pade type artificial boundary condition the pade type condition is easy to implement in a galerkin least squares iterative finite element solver for arbitrarily convex shaped boundaries the method accuracy is investigated for different model problems and for the scattering problem by a submarine shaped scatterer as a result relatively small computational domains optimized according to the shape of the scatterer can be considered while yielding accurate computations for high frequencies copyright 2005 john wiley sons ltd

a separation principle for a class of non uco systems this paper introduces a new approach to output feedback stabilization of single input single output systems which unlike other techniques found in the literature does not use quasilinear high gain observers and control input saturation to achieve separation between the state feedback and observer designs rather we show that by using nonlinear high gain observers working in state coordinates together with a dynamic projection algorithm the same kind of separation principle is achieved for a larger class of systems which are not uniformly completely observable by working in state coordinates this approach avoids using knowledge of the inverse of the observability mapping to estimate the state of the plant which is otherwise needed when using high gain observers to estimate the output time derivatives

a stability analysis of incomplete lu factorizations abstract the combination of iterative methods with preconditionings based on incomplete lu factorizations constitutes an effective class of methods for solving the sparse linear systems arising from the discretization of elliptic partial differential equations in this paper we show that there are some settings in which the incomplete lu preconditioners are not effective and we demonstrate that their poor performance is due to numerical instability our analysis consists of an analytic and numerical study of a sample two dimensional non self adjoint elliptic problem discretized by several finite difference schemes author

a note on the problem of semiglobal practical stabilization of uncertain nonlinear systems via dynamic output feedback it is known that if a system can be robustly globally asymptotically stabilized by means of a feedback that is driven by functions that are uniformly completely observable uco then this system can be practically semiglobally stabilized by means of possibly dynamic output feedback this papers discusses a significant structural hypothesis under which the existence of a dynamic feedback driven by uco functions is guaranteed the class of systems which satisfy this hypothesis includes any stabilizable and detectable linear system and any relative degree one nonlinear system which is stabilizable by dynamic output feedback in particular the hypothesis does not require the system to be minimum phase

integrable systems of derivative nonlinear schrodinger type and their multi hamiltonian structure a spectral problem and the associated hierarchy of schrodinger type equations are proposed it is shown that the hierarchy is integrable in liouville s sense and possesses multi hamiltonian structure it is found that several kinds of important equation such as the kaup newell kn equation the chen lee liu cll equation the gerdjikov ivanov gi equation the modified korteweg de vries equation and the sharma tasso olever equation are members in the hierarchy as its special reductions moreover kn cll and gi equations are described by using a unified generalized derivative schrodinger equation involving a parameter and their hamiltonian structure and lax pairs are also given by unified and explicit formulae

an approach for constructing nonisospectral hierarchies of evolution equations for a given isospectral lambda t 0 hierarchy of evolution equations the author proposes a simple method of constructing its corresponding nonisospectral lambda t lambda n n or 0 hierarchy of evolution equations closely related to tau symmetries it is crucial to find an initial lax operator w0 and an initial vector field g0 satisfying the key equation w0 l l g0 i in which l i are spectral and identity operators respectively as examples the author presents the corresponding nonisospectral hierarchies of equations and displays the fundamental relations generating symmetry algebras for the kdv hierarchy the akns hierarchy and a new integrable hierarchy

hamiltonian and quasi hamiltonian structures associated with semi direct sums of lie algebras the trace variational identity is generalized to zero curvature equations associated with non semi simple lie algebras or equivalently lie algebras possessing degenerate killing forms an application of the resulting generalized variational identity to a class of semi direct sums of lie algebras in the akns case furnishes hamiltonian and quasi hamiltonian structures of the associated integrable couplings three examples of integrable couplings for the akns hierarchy are presented one hamiltonian and two quasi hamiltonian

fast iterative solution of stabilised stokes systems part i using simple diagonal 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 the use of stabilisation methods for convenient but unstable mixed elements introduces stabilisation parameters we show how these can be chosen to obtain rapid iterative convergence we propose a conjugate gradient like method the method of preconditioned conjugate residuals which is applicable to symmetric indefinite problems describe the effects of stabilisation on the algebraic structure of the discrete stokes operator and derive estimates of the eigenvalue spectrum of this operator on which the convergence rate of the iteration depends here we discuss the simple case of diagonal preconditioning our results apply to both locally and globally stabilised mixed elements as well as to elements which are inherently stable we demonstrate that convergence rates comparable to that achieved using the diagonally scaled conjugate gradient method applied to the discrete laplacian are approachable for the stokes problem