the construction of preconditioners for elliptic problems by substructuring i in earlier parts of this series of papers we constructed preconditioners for the discrete systems of equations arising from the numerical approximation of elliptic boundary value problems the resulting algorithms are well suited for implementation on computers with parallel architecture in this paper we will develop a technique which utilizes these earlier methods to derive even more efficient preconditioners the iterative algorithms using these new preconditioners converge to the solution of the discrete equations with a rate that is independent of the number of unknowns these preconditioners involve an incomplete chebyshev iteration for boundary interface conditions which results in a negligible increase in the amount of computational work theoretical estimates and the results of numerical experiments are given which demonstrate the effectiveness of the methods

third and fifth order newton gauss methods for solving nonlinear equations with n variables based on the mean value theorem of multivariable vectors function f x two new iterative schemes with third order and fifth order convergence are constructed respectively by using gauss quadrature formula for solving systems of nonlinear equations their error equations and asymptotic numerical convergence constants are obtained the two suggested methods are compared with the related methods for solving systems of nonlinear equations and boundary value problems of nonlinear odes in the numerical examples to demonstrate the efficiency and practicality

block monotone iterative methods for elliptic variational inequalities this paper discusses some important block iterative methods for solving variational inequalities with a nonlinear operator and proves their monotone convergence properties and comparison theorems these results extend c v pao s research work in numer math 72 1995 239 to variational inequalities

monotone multigrid methodsfor elliptic variational inequalities ii we derive globally convergent multigrid methods for discrete elliptic variational inequalities of the second kind as obtained from the approximation of related continuous problems by piecewise linear finite elements the coarse grid corrections are computed from certain obstacle problems the actual constraints are fixed by the preceding nonlinear fine grid smoothing this new approach allows the implementation as a classical v cycle and preserves the usual multigrid efficiency we give 1 o j 3 estimates for the asymptotic convergence rates the numerical results indicate a significant improvement as compared with previous multigrid approaches

on the linear convergence of descent methods for convex essentially smooth minimization consider the problem of minimizing over a polyhedral set the composition of an affine mapping with a strictly convex essentially smooth function a general result on the linear convergence of descent methods for solving this problem is presented by applying this result the linear convergence of both the gradient projection algorithm of goldstein and levitin and polyak and a matrix splitting algorithm using regular splitting is established the results do not require that the cost function be strongly convex or that the optimal solution set be bounded the key to the analysis lies in a new error bound for estimating the distance from a feasible point to the optimal solution set

rate of convergence for some constraint decomposition methods for nonlinear variational inequalities some general subspace correction algorithms are proposed for a convex optimization problem over a convex constraint subset one of the nontrivial applications of the algorithms is the solving of some obstacle problems by multilevel domain decomposition and multigrid methods for domain decomposition and multigrid methods the rate of convergence for the algorithms for obstacle problems is of the same order as the rate of convergence for jump coefficient linear elliptic problems in order to analyse the convergence rate we need to decompose a finite element function into a sum of functions from the subspaces and also satisfying some constraints a special nonlinear interpolation operator is introduced for decomposing the functions

on monotone and geometric convergence of schwarz methods for two sided obstacle problems convergence of schwarz methods is discussed for certain discretizations of two sided obstacle problems monotone and especially geometrical convergence are established if the stiffness matrix is a kind of m matrix and an h independent convergence rate is proved for uniformly overlapping decomposition

partially asynchronous parallel algorithms for network flow and other problems the problem of computing a fixed point of a nonexpansive function f is considered sufficient conditions are provided under which a parallel partially asynchronous implementation of the iteration x f x converges these results are then applied to i quadratic programming subject to box constraints ii strictly convex cost network flow optimization iii an agreement and a markov chain problem iv neural network optimization and v finding the least element of a polyhedral set determined by a weakly diagonally dominant leontief system finally simulation results illustrating the attainable speedup and the effects of asynchronism are presented

analysis of a damped nonlinear multilevel method in this paper we present a new algorithm that is obtained by introducing a damping parameter in the classical nonlinear multilevel method we analyse this damped nonlinear multilevel method in particular we prove global convergence and local efficiency for a suitable class of problems

on multi grid methods for variational inequalities we consider here a general class of algorithms for the numerical solution of variational inequalities a convergence proof is given and in particular a multi grid method is described numerical results are presented for the finite difference discretization of an obstacle problem for minimal surfaces

monotone multigrid methods for elliptic variational inequalities i we derive fast solvers for discrete elliptic variational inequalities of the first kind obstacle problems as resulting from the approximation of related continuous problems by piecewise linear finite elements using basic ideas of successive subspace correction we modify well known relaxation methods by extending the set of search directions extended underrelaxations are called monotone multigrid methods if they are quasioptimal in a certain sense by construction all monotone multigrid methods are globally convergent we take a closer look at two natural variants the standard monotone multigrid method and a truncated version for the considered model problems the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarse grid transport while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case

an additive schwarz method for variational inequalities this paper proposes an additive schwarz method for variational inequalities and their approximations by finite element methods the schwarz domain decomposition method is proved to converge with a geometric rate depending on the decomposition of the domain the result is based on an framework of convergence analysis established for general variational inequalities in hilbert spaces

adaptive multilevel methods for obstacle problems the authors consider the discretization of obstacle problems for second order elliptic differential operators by piecewise linear finite elements assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies the linear problems are solved iteratively by preconditioned conjugate gradient iterations the proposed preconditioners are treated theoretically as abstract additive schwarz methods and are implemented as truncated hierarchical basis preconditioners to allow for local mesh refinement semilocal and local a posteriors error estimates are derived providing lower and upper estimates for the discretization error the theoretical results are illustrated by numerical computations

global and uniform convergence of subspace correction methods for some convex optimization problems this paper gives some global and uniform convergence estimates for a class of subspace correction based on space decomposition iterative methods applied to some unconstrained convex optimization problems some multigrid and domain decomposition methods are also discussed as special examples for solving some nonlinear elliptic boundary value problems

convergence estimates for product iterative methods with applications to domain decomposition in this paper we consider iterative methods for the solution of symmetric positive definite problems on a space which are defined in terms of products of operators defined with respect to a number of subspaces the simplest algorithm of this sort has an error reducing operator which is the product of orthogonal projections onto the complement of the subspaces new normreduction estimates for these iterative techniques will be presented in an abstract setting applications are given for overlapping schwarz algorithms with many subregions for finite element approximation of second order elliptic problems

convergence acceleration for continued fractions _ 1 a known method for convergence acceleration of limit periodic con tinued fractions k an a a is to replace the approximants 5 0 by modified approximants sn f where k a the present paper extends this idea to a larger class of converging continued fractions the modified approximants will then be 5 n where k a n is a converging continued fraction whose tails are all known and where a a n 0 as a measure for the improvement obtained by this method upper bounds for the ratio of the two truncation errors are found

on multilevel iterative methods for optimization problems this paper is concerned with multilevel iterative methods which combine a descent scheme with a hierarchy of auxiliary problems in lower dimensional subspaces the construction of auxiliary problems as well as applications to elasto plastic model and linear programming are described the auxiliary problem for the dual of a perturbed linear program is interpreted as a dual of perturbed aggregated linear program coercivity of the objective function over the feasible set is sufficient for the boundedness of the iterates equivalents of this condition are presented in special cases

convergence rate analysis of an asynchronous space decomposition method for convex minimization we analyze the convergence rate of an asynchronous space decomposition method for constrained convex minimization in a reflexive banach space this method includes as special cases parallel domain decomposition methods and multigrid methods for solving elliptic partial differential equations in particular the method generalizes the additive schwarz domain decomposition methods to allow for asynchronous updates it also generalizes the bpx multigrid method to allow for use as solvers instead of as preconditioners possibly with asynchronous updates and is applicable to nonlinear problems applications to an overlapping domain decomposition for obstacle problems are also studied the method of this work is also closely related to relaxation methods for nonlinear network flow accordingly we specialize our convergence rate results to the above methods the asynchronous method is implementable in a multiprocessor system allowing for communication and computation delays among the processors

multi grid solutions to the elastic plastic torsion problem in multiply connected domains the elastic plastic torsion problem for an elastic perfectly plastic cylinder with multiply connected cross section twisted around its longitudinal axis is formulated as an obstacle problem for an associated stress potential the obstacle being defined in terms of a generalized distance function based upon the reformulation of the obstacle problem as an equivalent linear complementarity problem the latter is discretized by means of finite difference techniques and a monotonically convergent iterative scheme for its numerical solution is developed at each step of the iteration the solution of a reduced system of discrete poisson equations is required which is done by applying multi grid techniques with respect to a hierarchy of grid point sets combined with a suitably chosen nested iteration process this results in a computationally very efficient algorithm for the approximate solution of the elastic plastic torsion problem

analysis of a multilevel iterative method for nonlinear finite element equations the multilevel iterative technique is a powerful technique for solving the systems of equations associated with discretized partial differential equations we describe how this technique can be combined with a globally convergent approximate newton method to solve nonlinear partial differential equations we show that asymptotically only one newton iteration per level is required thus the complexity for linear and nonlinear problems is essentially equal

a novel two grid method for semilinear elliptic equations a new finite element discretization technique based on two coarse and fine subspaces is presented for a semilinear elliptic boundary value problem the solution of a nonlinear system on the fine space is reduced to the solution of two small one linear and one nonlinear systems on the coarse space and a linear system on the fine space it is shown both theoretically and numerically that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation as a result the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation

rate of convergence of some space decomposition methods for linear and nonlinear problems convergence of a space decomposition method is proved for a class of convex programming problems a space decomposition refers to a method that decomposes a space into a sum of subspaces which could be a domain decomposition or a multilevel method when applied to partial differential equations two algorithms are proposed both can be used for linear as well as nonlinear elliptic problems and they reduce to the standard additive and multiplicative schwarz methods for linear elliptic problems

on the rate of convergence of a partially asynchronous gradient projection algorithm recently bertsekas and tsitsiklis proposed a partially asynchronous implementation of the gradient projection algorithm of goldstein and levitin and polyak for the problem of minimizing a differentiable function over a closed convex set in this paper the rate of convergence of this algorithm is analyzed it is shown that if the standard assumptions hold that is the solution set is nonempty and the gradient of the function is lipschitz continuous and i the isocost surfaces of the objective function restricted to the solution set are properly separated and ii a certain multifunction associated with the problem is locally upper lipschitzian then this algorithm attains a linear rate of convergence

a unified analysis of hoffman s bound via fenchel duality in 1952 hoffman showed that the distance from any point to the solution set of a linear system is bounded above by a constant times the norm of the residual subsequently this bound has been studied extensively and has found many practical uses in this paper we consider an extension of hoffman s bound to a partially infinite dimensional setting in which the norm defining the distance is replaced by a positively homogeneous convex function and the linear system is replaced by a convex inclusion of the form ax a in k where a is a continuous linear operator from some normed linear space into mathbf r m a is an element of the range space of a and k is a nonempty closed convex cone in mathbf r m when specialized to the finite dimensional case we unify and extend many existing results on hoffman s bound our analysis is based on the use of fenchel duality to express the distance as the supremum of a certain concave function over a bounded subset of the polar of k much of our analysis als