Symmetric Saddle Point Problems Research Articles

An improved lower bound on a positive stable block triangular preconditioner for saddle point problems

In this paper, a new lower bound on a positive stable block triangular preconditioner for saddle point problems is derived; it is superior to the corresponding result obtained by Cao [Z.-H. Cao, Positive stable block triangular preconditioners for symmetric saddle point problems, Appl. Numer. Math. 57 (2007) 899–910]. A numerical example is reported to confirm the presented result.

A note on the positive stable block triangular preconditioner for generalized saddle point problems

In this paper, we give a sharp eigenvalue bound for the positive stable block triangular preconditioned matrix presented in a recent paper by Cao [Z.-H. Cao, Positive stable block triangular preconditioners for symmetric saddle point problems, Appl. Numer. Math. 57 (2007) 899–910]. The intervals containing these eigenvalues of the preconditioned matrix remove the origin, which is benefit for further study. Numerical experiments of a model Stokes problem are presented to show the estimate and the effectiveness of the positive stable block triangular preconditioners.

Indefinite block triangular preconditioner for symmetric saddle point problems

In this paper, we consider an indefinite block triangular preconditioner for symmetric saddle point problems. The new eigenvalue distribution of the preconditioned matrix is derived and some corresponding results in Simoncini (Appl. Numer. Math. 49:63---80, 2004) and Wu et al. (Computing 84:183---208, 2009) are improved. Finally, numerical experiments of a model Stokes problem are reported.

A symmetric mixed finite element method for nearly incompressible elasticity based on biorthogonal systems

AbstractWe present a symmetric version of the nonsymmetric mixed finite element method presented in (Lamichhane, ANZIAM J 50 (2008), C324–C338) for nearly incompressible elasticity. The displacement–pressure formulation of linear elasticity is discretized using a Petrov–Galerkin discretization for the pressure equation in (Lamichhane, ANZIAM J 50 (2008), C324–C338) leading to a non‐symmetric saddle point problem. A new three‐field formulation is introduced to obtain a symmetric saddle point problem which allows us to use a biorthogonal system. Working with a biorthogonal system, we can statically condense out all auxiliary variables from the saddle point problem arriving at a symmetric and positive‐definite system based only on the displacement. We also derive a residual based error estimator for the mixed formulation of the problem. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012

Nonstandard Norms and Robust Estimates for Saddle Point Problems

In this paper we discuss how to find norms for parameter-dependent saddle point problems which lead to robust (i.e., parameter-independent) estimates of the solution in terms of the data. In a first step a characterization of such norms is given for a general class of symmetric saddle point problems. Then, for special cases, explicit formulas for these norms are derived. Finally, we will apply these results to distributed optimal control problems for elliptic equations and for the Stokes equations. The norms which lead to robust estimates turn out to differ from the standard norms typically used for these problems. This will lead to block diagonal preconditioners for the corresponding discretized problems with mesh-independent and robust convergence rates if used in preconditioned Krylov subspace methods.

On block preconditioners for saddle point problems with singular or indefinite (1, 1) block

We discuss a class of preconditioning methods for the iterative solution of symmetric algebraic saddle point problems, where the (1, 1) block matrix may be indefinite or singular. Such problems may arise, e.g. from discrete approximations of certain partial differential equations, such as the Maxwell time harmonic equations. We prove that, under mild assumptions on the underlying problem, a class of block preconditioners (including block diagonal, triangular and symmetric indefinite preconditioners) can be chosen in a way which guarantees that the convergence rate of the preconditioned conjugate residuals method is independent of the discretization mesh parameter. We provide examples of such preconditioners that do not require additional scaling. Copyright © 2010 John Wiley & Sons, Ltd.

Analysis of the nonlinear Uzawa algorithm for symmetric saddle point problems

In this paper we discuss the convergence behaviour of the nonlinear Uzawa algorithm for solving saddle point problems presented in a recent paper of Cao [Z.H. Cao, Fast Uzawa algorithm for generalized saddle point problems, Appl. Numer. Math. 46 (2003), pp. 157–171]. For a general case, the results on the convergence of the algorithm are given.

Generalized block triangular preconditioner for symmetric saddle point problems

In this paper, spectral properties and computational performance of a generalized block triangular preconditioner for symmetric saddle point problems are discussed in detail. We will provide estimates for the region containing both the nonreal and the real eigenvalues and generalize the results of Simoncini (Appl Numer Math 49:63–80, 2004) and Cao (Appl Numer Math 57:899–910, 2007). Finally, numerical experiments of the model Stokes problem are reported.

Schur complements on Hilbert spaces and saddle point systems

For any continuous bilinear form defined on a pair of Hilbert spaces satisfying the compatibility Ladyshenskaya–Babušca–Brezzi condition, symmetric Schur complement operators can be defined on each of the two Hilbert spaces. In this paper, we find bounds for the spectrum of the Schur operators only in terms of the compatibility and continuity constants. In light of the new spectral results for the Schur complements, we review the classical Babušca–Brezzi theory, find sharp stability estimates, and improve a convergence result for the inexact Uzawa algorithm. We prove that for any symmetric 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 1/3. As a consequence, we provide a new type of algorithm for discretizing saddle point problems, which combines the inexact Uzawa iterations with standard a posteriori error analysis and does not require the discrete stability conditions.

Limiting accuracy of segregated solution methods for nonsymmetric saddle point problems

Nonsymmetric saddle point problems arise in a wide variety of applications in computational science and engineering. The aim of this paper is to discuss the numerical behavior of several nonsymmetric iterative methods applied for solving the saddle point systems via the Schur complement reduction or the null-space projection approach. Krylov subspace methods often produce the iterates which fluctuate rather strongly. Here we address the question whether large intermediate approximate solutions reduce the final accuracy of these two-level (inner–outer) iteration algorithms. We extend our previous analysis obtained for symmetric saddle point problems and distinguish between three mathematically equivalent back-substitution schemes which lead to a different numerical behavior when applied in finite precision arithmetic. Theoretical results are then illustrated on a simple model example.

Matching-based preprocessing algorithms to the solution of saddle-point problems in large-scale nonconvex interior-point optimization

Interior-point methods are among the most efficient approaches for solving large-scale nonlinear programming problems. At the core of these methods, highly ill-conditioned symmetric saddle-point problems have to be solved. We present combinatorial methods to preprocess these matrices in order to establish more favorable numerical properties for the subsequent factorization. Our approach is based on symmetric weighted matchings and is used in a sparse direct LDL T factorization method where the pivoting is restricted to static supernode data structures. In addition, we will dynamically expand the supernode data structure in cases where additional fill-in helps to select better numerical pivot elements. This technique can be seen as an alternative to the more traditional threshold pivoting techniques. We demonstrate the competitiveness of this approach within an interior-point method on a large set of test problems from the CUTE and COPS sets, as well as large optimal control problems based on partial differential equations. The largest nonlinear optimization problem solved has more than 12 million variables and 6 million constraints.

Positive stable block triangular preconditioners for symmetric saddle point problems

In this paper we discuss the spectral properties and the computational performance of a positive stable block triangular preconditioner for the solution of the general symmetric saddle point problem. We will show that the eigenvalues of the preconditioned matrix are all real and provide estimates for the interval containing these real eigenvalues. Numerical experiments of a model Stokes problem are presented.

Fast corrected Uzawa methods for solving symmetric saddle point problems

Fast corrected Uzawa methods for solving symmetric saddle point problems

Optimization of the generalized method of Hermitian and skew-Hermitian splitting iterations for solving symmetric saddle-point problems

An algorithm for solving a nonsingular symmetric system of linear equations with a saddle point is examined. This algorithm has two constant iteration parameters and is an extension of the algorithm of Hermitian and skew-Hermitian splitting iterations (the HSS algorithm). Analytical formulas are derived for the optimal values of the iteration parameters. The formulation of the optimization problem is a classical one for the saddle-point problems. The results obtained are sharp.

Nonlinear uzawa methods for solving nonsymmetric saddle point problems

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.

Analysis of iterative algorithms of Uzawa type for saddle point problems*1

In this paper, we consider the convergence criteria for iterative algorithms of Uzawa type for solving linear saddle point problems. Theoretically weaker convergence criteria than before are established for the general case and these are used to deduce conditions for convergence of two special cases: the exact Uzawa algorithm and the linear one-step method. The conclusions given here hold for both symmetric and nonsymmetric saddle point problems. These new sufficient conditions are compared with some known results and illustrated by two examples. Numerical experiments to verify the conclusions in this paper for the preconditioned exact Uzawa algorithm are provided.

Analysis of iterative algorithms of Uzawa type for saddle point problems

In this paper, we consider the convergence criteria for iterative algorithms of Uzawa type for solving linear saddle point problems. Theoretically weaker convergence criteria than before are established for the general case and these are used to deduce conditions for convergence of two special cases: the exact Uzawa algorithm and the linear one-step method. The conclusions given here hold for both symmetric and nonsymmetric saddle point problems. These new sufficient conditions are compared with some known results and illustrated by two examples. Numerical experiments to verify the conclusions in this paper for the preconditioned exact Uzawa algorithm are provided.

Block triangular preconditioners for symmetric saddle-point problems

In this paper we study the spectral properties and the computational performance of a block triangular preconditioner for the solution of the general symmetric saddle-point problem. We provide estimates for the region containing both the nonreal and the real eigenvalues. Moreover, we show that an indefinite inner product can be employed to devise an efficient short-term recurrence Krylov subspace solver to be used with the analyzed preconditioner. Numerical experiments on a variety of application problems are also reported.

Block LU Preconditioners for Symmetric and Nonsymmetric Saddle Point Problems

In this paper, a general-purpose block LU preconditioner for saddle point problems is presented. The main difference between the approach presented here and that of other studies is that an explicit, accurate approximation of the Schur complement matrix is efficiently computed. This is used to compute a preconditioner to the Schur complement matrix, which in turn defines a preconditioner for the global system. A number of different variants are developed and results are reported for a few linear systems arising from CFD applications.

On Schwarz-type Smoothers for Saddle Point Problems

In this paper we consider additive Schwarz-type iteration methods for saddle point problems as smoothers in a multigrid method. Each iteration step of the additive Schwarz method requires the solutions of several small local saddle point problems. This method can be viewed as an additive version of a (multiplicative) Vanka-type iteration, well-known as a smoother for multigrid methods in computational fluid dynamics. It is shown that, under suitable conditions, the iteration can be interpreted as a symmetric inexact Uzawa method. In the case of symmetric saddle point problems the smoothing property, an important part in a multigrid convergence proof, is analyzed for symmetric inexact Uzawa methods including the special case of the additive Schwarz-type iterations. As an example the theory is applied to the Crouzeix-Raviart mixed finite element for the Stokes equations and some numerical experiments are presented.