Preconditioned Uzawa-type methods for finite-dimensional constrained saddle point problems (original) (raw)
An Inexact Uzawa Algorithm for Generalized Saddle-Point Problems and Its Convergence
We propose an inexact Uzawa algorithm with two variable relaxation parameters for solving the generalized saddle-point system. The saddle-point problems can be found in a wide class of applications, such as the augmented Lagrangian formulation of the constrained minimization, the mixed finite element method, the mortar domain decomposition method and the discretization of elliptic and parabolic interface problems. The two variable parameters can be updated at each iteration, requiring no a priori estimates on the spectrum of two preconditioned subsystems involved. The convergence and convergence rate of the algorithm are analysed. Both symmetric and nonsymmetric saddle-point systems are discussed, and numerical experiments are presented to demonstrate the robustness and effectiveness of the algorithm.
Uzawa type algorithms for nonsymmetric saddle point problems
Mathematics of Computation, 1999
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.
Analysis of the Inexact Uzawa Algorithm for Saddle Point Problems
SIAM Journal on Numerical Analysis, 1997
In this paper, we consider the so-called \inexact Uzawa" algorithm for iteratively solving block saddle point problems. Such saddle point problems arise, for example, in nite element and nite di erence discretizations of Stokes equations, the equations of elasticity and mixed nite element discretization of second order problems. We consider both the linear and nonlinear variants of the inexact Uzawa algorithm. We s h o w that the linear method always converges as long as the preconditioners de ning 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 nonlinear case, the inexact Uzawa algorithm is shown to converge provided that the nonlinear process approximating the inverse of the upper left hand block is of su cient accuracy. Bounds for the nonlinear iteration are given in terms of this accuracy parameter and the rate of convergence of the preconditioned Uzawa algorithm. Applications to the Stokes equations and mixed nite element discretization of second order elliptic problems are discussed and nally, the results of numerical experiments involving the algorithms are presented.
Iterative Methods for the Solution of Saddle Point Problem
2015
Some new iterative methods for numerical solution of mixed finite element approximation of Stokes problem are presented. The idea is the use of proper preconditioning for the conjugate gradient algorithm. A particular case gives a variant of the Arrow-Hurwicz method.
Nonlinear uzawa methods for solving nonsymmetric saddle point problems
Journal of Applied Mathematics and Computing, 2006
In [A new nonlinear Uzawa algorithm for generalized saddle point problems, Appl. Math. Comput., 175(2006), 1432-1454], a nonlinear Uzawa algorithm for solving symmetric saddle point problems iteratively, which was defined by two nonlinear approximate inverses, was considered. In this paper, we extend it to the nonsymmetric case. For the nonsymmetric case, its convergence result is deduced. Moreover, we compare the convergence rates of three nonlinear Uzawa methods and show that our method is more efficient than other nonlinear Uzawa methods in some cases. The results of numerical experiments are presented when we apply them to Navier-Stokes equations discretized by mixed finite elements.
Corrected Uzawa methods for solving large nonsymmetric saddle point problems
Applied Mathematics and Computation, 2006
In this paper we consider the solution of linear systems of large nonsymmetric saddle point problems by two nonlinear iterative methods, which are similar in some respects to the Uzawa-type methods and are called corrected Uzawa methods. Their convergence rates are analyzed. The results of numerical experiments are presented when we apply them to Navier-Stokes equations discretized by mixed finite elements.
SIAM Journal on Matrix Analysis and Applications, 2007
We consider large scale sparse linear systems in saddle point form. A natural property of such indefinite 2-by-2 block systems is the positivity of the (1,1) block on the kernel of the (2,1) block. Many solution methods, however, require that the positivity of the (1,1) block is satisfied everywhere. To enforce the positivity everywhere, an augmented Lagrangian approach is usually chosen. However, the adjustment of the involved parameters is a critical issue. We will present a different approach that is not based on such an explicit augmentation technique. For the considered class of symmetric and indefinite preconditioners, assumptions are presented that lead to symmetric and positive definite problems with respect to a particular scalar product. Therefore, conjugate gradient acceleration can be used. An important class of applications are optimal control problems. It is typical for such problems that the cost functional contains an extra regularization parameter. For control problems with elliptic state equations and distributed control a special indefinite preconditioner for the discretized problem is constructed which leads to convergence rates of the preconditioned conjugate gradient method that are not only independent of the mesh size but also independent of the regularization parameter. Numerical experiments are presented for illustrating the theoretical results.
Advances and Perspectives on Numerical Methods for Saddle Point Problems
2009
S (in alphabetical order) Constantin Bacuta Multilevel discretization of Saddle Point Problems without the discrete LBB condition (Joint work with Peter Monk) Using spectral results for Schur complement operators we prove a convergence result for the inexact Uzawa algorithm on general Hilbert spaces. We prove that for any symmetric and coercive saddle point problem, the inexact Uzawa algorithm converges, provided that the inexact process for inverting the residual at each step has the relative error smaller than a computable fixed threshold. As a consequence, we provide a new type of algorithms for discretizing saddle point problems, which implement the inexact Uzawa algorithm at the continuous level as a multilevel or adaptive algorithm. The discrete stability Ladyshenskaya-Babušca-Brezzi (LBB) condition might not be satisfied and the adaptivity is required only for solving symmetric and positive definite problems. The convergence result for the algorithm at the continuous level, c...