Abstract. Inexact Newton algorithms are commonlyused for solving large sparse nonlinear system of equations F (u ∗ ) = 0 arising, for example, from the discretization of partial differential equations. Even with global strategies such as linesearch or trust region, the methods often stagnate at local minima of �F �, especiallyfor problems with unbalanced nonlinearities, because the methods do not have built-in machineryto deal with the unbalanced nonlinearities. To find the same solution u ∗ , one maywant to solve instead an equivalent nonlinearlypreconditioned system F(u ∗ ) = 0 whose nonlinearities are more balanced. In this paper, we propose and studya nonlinear additive Schwarzbased parallel nonlinear preconditioner and show numericallythat the new method converges well even for some difficult problems, such as high Reynolds number flows, where a traditional inexact Newton method fails. Key words. nonlinear preconditioning, inexact Newton methods, Krylov subspace methods, nonlinear additive Schwarz, domain decomposition, nonlinear equations, parallel computing, incompressible

Abstract. 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. 1.

A general space decomposition technique is used to solve nonlinear convex minimization problems. The dierential of the minimization functional is required to satisfy some growth conditions that are weaker than Lipschitz continuity and strong monotonicity. Optimal rate of convergence is proved. If the dierential is Lipschitz continuous and strongly monotone, then the algorithms have uniform rate of convergence. The algorithms can be used for domain decomposition and multigrid type of techniques. Applications to linear elliptic and some nonlinear degenerated partial dierential equation are considered. 1 Introduction Domain decomposition and multigrid methods have been intensively studied for linear partial dierential equations. Recent research, see for example [32], reveals that domain decomposition and multigrid methods can be analysed using a same framework, see also [3], [13], [23], [17]. The present work uses this framework to analyse the convergence of two algorithms for convex...

Abstract. 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. 1.

. This work presents some space decomposition algorithms for a convex minimization problem. The algorithms has linear rate of convergence and the rate of convergence depends only on four constants. The space decomposition could be a multigrid or domain decomposition method. We explain the detailed procedure in implementing our algorithms for a two-level overlapping domain decomposition method and estimate the needed constants. Numerical tests are reported for linear as well as nonlinear elliptic problems. 1. Introduction In this work, we propose some general space decomposition algorithms for convex programming problems. Applications to a two level domain decomposition technique is discussed for linear and nonlinear elliptic equations. The proposed algorithms are given for a convex programming problem. Other applications of the algorithms include optimal control problems related to partial differential equations, eigenvalue problems, least-squares method for linear and nonlinear parti...

Abstract In this paper we study some nonoverlapping domain decomposition methods for solving a class of elliptic problems arising from composite materials and flows in porous media which contain many spatial scales. Our preconditioner differs from traditional domain decomposition preconditioners by using a coarse solver which is adaptive to small scale heterogeneous features. While the convergence rate of traditional domain decomposition algorithms using coarse solvers based on linear or polynomial interpolations may deteriorate in the presence of rapid small scale oscillations or high aspect ratios, our preconditioner is applicable to multiplescale problems without restrictive assumptions and seems to have a convergence rate nearly independent of the aspect ratio within the substructures. A rigorous convergence analysis based on the Schwarz framework is carried out, and we demonstrate the efficiency and robustness of the proposed preconditioner through numerical experiments which include problems with multiple-scale coefficients, as well problems with continuous scales.

. The Schwarz alternating method can be used to solve linear elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which result from solving a sequence of elliptic boundary value problems in each of the subdomains. This paper considers four Schwarz alternating methods for the N-dimensional, steady, viscous, incompressible Navier--Stokes equations, N 4. It is shown that the Schwarz sequences converge to the true solution provided that the Reynolds number is sufficiently small. Key words. domain decomposition, Schwarz alternating method, Navier--Stokes AMS subject classifications. 65N25, 65N55 PII. S1064827598347411 1.

Introduction In this work, we shall study the constrained minimization problem min v2K F (v) ; (1) where K is a nonempty closed convex set in a reflexive Banach space V in the strong topology and F : V 7! ! is a lower semicontinuous convex Gateau-differentiable function. Denote h\Delta; \Deltai the duality pairing of V and its dual space V 0 , i.e. the value of a linear functional at an element of V . We shall assume the differential of F satisfies hF 0 (w) \Gamma F 0 (v); w \Gamma vi kw \Gamma vk 2 V ; 8w;

. The Schwarz alternating method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each of the subdomains. This paper considers several Schwarz alternating methods for nonlinear elliptic problems. We show that Schwarz alternating methods can be embedded in the framework of common techniques such as Banach and Schauder fixed point methods and global inversion methods used to solve these nonlinear problems. Key words. domain decomposition, nonlinear elliptic PDE, Schwarz alternating method AMS subject classifications. 65N55, 65J15 PII. S1064827597327553 1. Introduction. The Schwarz alternating method was devised by H. A. Schwarz more than 100 years ago to solve linear boundary value problems. It has garnered interest recently because of its potential as an e#cient algorithm for pa...

this paper, we study the convergence rate of asynchronous block Jacobi and block Gauss-Seidel methods for finite or infinite dimensional convex minimization of 1 Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5007, Bergen, Norway. Email: Tai@mi.uib.no and URL: http://www.mi.uib.no/~tai. 2 Department of Mathematics, Box 354350, University of Washington, Seattle, WA 98195, U.S.A. Email: tseng@math.washington.edu and URL: http://www.math.washington.edu/~tseng. The work of the first author was supported by the Engineering and Physical Sciences Research Council, United Kingdom, Grant Reference GR/L98862. The work of the second author was supported by the National Science Foundation, Grant No. CCR9311621.