efficient solvers for saddle point problems arising from domain decompositions with lagrange multipliers in this paper we are concerned with the domain decomposition method with lagrange multipliers for solving three dimensional elliptic problems with variable coefficients we shall first introduce a weighted saddle point problem resulting from this domain decomposition which can be solved by existing iterative methods then we will construct two simple preconditioners one for the system associated with the displacement variable and the other for the schur complement system associated with the multiplier variable that are applicable to various discretization schemes the new preconditioners possess such local properties that they can be implemented cheaply we will show that the condition number of the global preconditioned system grows only as the logarithm of the dimension of the local problem associated with an individual substructure and is independent of the large variations of the coefficients across the local interfaces

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

an augmented lagrangian method for identifying discontinuous parameters in elliptic systems the identification of discontinuous parameters in elliptic systems is formulated as a constrained minimization problem combining the output least squares and the equation error method the minimization problem is then proved to be equivalent to the saddle point problem of an augmented lagrangian the finite element method is used to discretize the saddle point problem and the convergence of the discretization is also proved finally an uzawa algorithm is suggested for solving the discrete saddle point problem and is shown to be globally convergent

on convergence and semi convergence of ssor like methods for augmented linear systems abstract in this paper we analyze the convergence and semi convergence of a class of ssor like methods with four real functions for augmented systems the class takes most existed ssor like methods as its special cases for nonsingular systems we obtain the minimum of convergence factors of all the ssor like methods in the class and study when it can be reached by the convergence factors of methods in the class by considering the equivalence of methods we show that most of the existed ssor like methods have the same minimum of convergence factors

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

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

the convergence factor of preconditioned algorithms of the arrow hurwicz type the convergence properties of the arrow hurwicz algorithm for solving mixed finite element problems are studied using a purely algebraic technique the convergence factor of the algorithm will be estimated the iteration parameters depending on the smallest and greatest eigenvalues of some generalized eigenvalue problems can be chosen optimally for special cases the convergence factor of the preconditioned arrow hurwicz algorithm is independent of the discretization parameter finally a quite general method that allows us to construct preconditioning operators of the arrow hurwicz algorithm for a wide class of problems is presented

long wave approximations for water waves in this paper we obtain new nonlinear systems describing the interaction of long water waves in both two and three dimensions these systems are symmetric and conservative rigorous convergence results are provided showing that solutions of the complete free surface euler equations tend to associated solutions of these systems as the amplitude becomes small and the wavelength large using this result as a tool a rigorous justification of all the two dimensional approximate systems recently put forward and analysed by bona chen and saut is obtained in the two dimensional context our methods also allows a significant improvement of the convergence estimate obtained by schneider and wayne in their justification of the decoupled korteweg de vries approximation of the two dimensional euler equations it also follows from our theory that coupled models provide a better description than the decoupled ones over short time scales results are obtained both on an unbounded domain for solutions that evanesce at infinity as well as for solutions that are spatially periodic

residual reduction algorithms for nonsymmetric saddle point problems in this paper we introduce and analyze uzawa algorithms for non symmetric saddle point systems convergence for the algorithms is established based on new spectral results about schur complements a new uzawa type algorithm with optimal relaxation parameters at each new iteration is introduced and analyzed in a general framework numerical results supporting the efficiency of the algorithms are presented for finite element discretization of steady state navier stokes equations

an inexact relaxed dpss preconditioner for saddle point problem based on the relaxed deteriorated positive definite and skew hermitian splitting dpss preconditioner in this paper we proposed a class of relaxed deteriorated positive definite and skew hermitian splitting rdpss preconditioner for solving the saddle point problem the proposed rdpss preconditioner is a technical modification of the deteriorated positive definite and skew hermitian splitting dpss preconditioner 36 the pss preconditioner is a straightforward application of the positive definite and skew hermitian splitting pss iteration method for solving non hermitian positive definite linear systems initially established by bai et al 37 numerical results have shown that the proposed rdpss preconditioner is advantageous over the existing dpss preconditioner

discrete time lqg controls with control dependent noise this paper presents some studies on partially observed linear quadratic gaussian lqg models where the stochastic disturbances depend on both the states and the controls and the measurements are bilinear in the noise and the states controls while the separation theorem of standard lqg design does not apply suboptimal linear state estimate feedback controllers are derived based on certain linearizations the controllers are useful for nonlinear stochastic systems where the linearized models include terms bilinear in the noise and states controls and are significantly more accurate than if the bilinear terms are set to zero the controllers are calculated by solving a generalized discrete time riccati equation which in turn has properties relating to well posedness of the associated lqg problem

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

on the nonlinear inexact uzawa algorithm for saddle point problems this paper discusses convergence behavior of the nonlinear inexact uzawa algorithm for solving saddle point problems presented in a recent paper of bramble pasciak and vassilev siam j numer anal 34 1997 pp 1072 1092 it is shown that this algorithm converges under a condition weaker than that stated in their paper

a discrete weighted helmholtz decomposition and its application we propose a discrete weighted helmholtz decomposition for edge element functions the decomposition is orthogonal in a weighted l 2 inner product and stable uniformly with respect to the jumps in the discontinuous weight function as an application the new helmholtz decomposition is applied to demonstrate the quasi optimality of a preconditioned edge element system for solving a saddle point maxwell system in non homogeneous media by a non overlapping domain decomposition preconditioner i e the condition number grows only as the logarithm of the dimension of the local subproblem associated with an individual subdomain and more importantly it is independent of the jumps of the physical coefficients across the interfaces between any two subdomains of different media numerical experiments are presented to validate the effectiveness of the non overlapping domain decomposition preconditioner

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

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

an efficient linear solver for nonlinear parameter identification problems in this paper we study some efficient numerical methods for parameter identifications in elliptic systems the proposed numerical methods are conducted iteratively and each iteration involves only solving positive definite linear algebraic systems although the original inverse problems are ill posed and highly nonlinear the positive definite systems can be naturally preconditioned with their corresponding block diagonal matrices numerical experiments are presented to illustrate the efficiency of the proposed algorithms

two new variants of nonlinear inexact uzawa algorithms for saddle point problems in this paper we consider some nonlinear inexact uzawa methods for iteratively solving linear saddle point problems by means of a new technique we first give an essential improvement on the convergence results of bramble paschiak vassilev for a known nonlinear inexact uzawa algorithm then we propose two new algorithms which can be viewed as a combination of the known nonlinear inexact uzawa method with the classical steepest descent method and conjugate gradient method respectively the two new algorithms converge under very practical conditions and do not require any apriori estimates on the minimal and maximal eigenvalues of the preconditioned systems involved including the preconditioned schur complement numerical results of the algorithms applied for the stokes problem and a purely linear system of algebraic equations are presented to show the efficiency of the algorithms

global and superlinear convergence of inexact uzawa methods for saddle point problems with nondifferentiable mappings this paper investigates inexact uzawa methods for nonlinear saddle point problems we prove that the inexact uzawa method converges globally and superlinearly even if the derivative of the nonlinear mapping does not exist we show that the newton type decomposition method for saddle point problems is a special case of a newton uzawa method we discuss applications of inexact uzawa methods to separable convex programming problems and coupling of finite elements boundary elements for nonlinear interface problems

on preconditioned uzawa methods and sor methods for saddle point problems this paper studies convergence analysis of a preconditioned inexact uzawa method for nondifferentiable saddle point problems the sor newton method and the sor bfgs method are special cases of this method we relax the bramble pasciak vassilev condition on preconditioners for convergence of the inexact uzawa method for linear saddle point problems the relaxed condition is used to determine the relaxation parameters in the sor newton method and the sor bfgs method furthermore we study global convergence of the multistep inexact uzawa method for nondifferentiable saddle point problems