Results 1  10
of
41
Nonlinearly preconditioned inexact Newton algorithms
 SIAM J. Sci. Comput
, 2000
"... 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 loc ..."
Abstract

Cited by 51 (18 self)
 Add to MetaCart
(Show Context)
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 builtin 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
Convergence rate analysis of an asynchronous space decomposition method for convex minimization
, 1998
"... 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 equatio ..."
Abstract

Cited by 29 (11 self)
 Add to MetaCart
(Show Context)
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.
Global and uniform convergence of subspace correction methods for some convex optimization problems
 Math. Comp
, 2001
"... 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 examp ..."
Abstract

Cited by 28 (10 self)
 Add to MetaCart
(Show Context)
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.
Global Convergence of Subspace Correction Methods for Convex Optimization Problems
, 1998
"... 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 th ..."
Abstract

Cited by 25 (6 self)
 Add to MetaCart
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...
Efficient Domain Decomposition Methods for Elliptic Problems Arising from Flows in Heterogeneous Porous Media
 COMPUTING AND VISUALIZATION IN SCIENCE
"... In this paper we study domain decomposition methods for solving some elliptic problem arising from flows in heterogeneous porous media. Due to the multiple scale nature of the elliptic coefficients arising from the heterogeneous formations, the construction of efficient domain decomposition method ..."
Abstract

Cited by 18 (0 self)
 Add to MetaCart
(Show Context)
In this paper we study domain decomposition methods for solving some elliptic problem arising from flows in heterogeneous porous media. Due to the multiple scale nature of the elliptic coefficients arising from the heterogeneous formations, the construction of efficient domain decomposition methods for these problems requires a coarse solver which is adaptive to the ne scale features, [4]. We propose the use of a multiscale coarse solver based on a finite volumefinite element formulation. The resulting domain decomposition methods seem to induce a convergence rate nearly independent of the aspect ratio of the extreme permeability values within the substructures. A rigorous convergence analysis based on the Schwarz framework is carried out, and we demonstrate the efficiency and robustness of the preconditioner through numerical experiments which include problems with multiple scale coecients, as well as problems with continuous scales.
Rate of Convergence for some constraint decomposition methods for nonlinear variational inequalities
 NUMER. MATH. (2003) 93: 755–786
, 2003
"... ..."
Multigrid methods for obstacle problems
"... Abstract. In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
(Show Context)
Abstract. In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which is closely related to truncated multigrid. The numerical properties of algorithms are carefully assessed by means of a degenerate problem and a problem with a complicated coincidence set. 1.
On Schwarz Alternating Methods For Nonlinear Elliptic PDEs
 SIAM J. Sci. Comput
, 2000
"... . 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 o ..."
Abstract

Cited by 12 (5 self)
 Add to MetaCart
(Show Context)
. 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...
On Linear Monotone Iteration And Schwarz Methods For Nonlinear Elliptic PDEs
"... . 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 o ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
(Show Context)
. 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. In this paper, proofs of convergence of some Schwarz Alternating Methods for nonlinear elliptic problems which are known to have solutions by the monotone method (also known as the method of subsolutions and supersolutions) are given. In particular, an additive Schwarz method for scalar as well some coupled nonlinear PDEs are shown to converge to some solution on finitely many subdomains, even when multiple solutions are possible. In the coupled system case, each subdomain PDE is linear, decoupled and can be solved concurrently with other subdomain PDEs. These results are applicable to several models in population biology. Key words. domain decomposition, nonlinear elliptic PDE, Schwarz alternating method, monotone methods, subsolution, supersolution AMS subject classifications. 65N55, 65J15 1.
Applications Of A Space Decomposition Method To Linear And Nonlinear Elliptic Problems
 NUMER. METH. FOR PART. DIFF. EQUAT
, 1998
"... 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 p ..."
Abstract

Cited by 9 (5 self)
 Add to MetaCart
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 twolevel overlapping domain decomposition method and estimate the needed constants. Numerical tests are reported for linear as well as nonlinear elliptic problems.