Walter Zulehner

Analysis of iterative methods for saddle point problems: a unified approach

In this paper two classes of iterative methods for saddle point problems are considered: inexact Uzawa algorithms and a class of methods with symmetric preconditioners. In both cases the iteration matrix can be transformed to a symmetric matrix by block diagonal matrices, a simple but essential observation which allows one to estimate the convergence rate of both classes by studying associated eigenvalue problems. The obtained estimates apply for a wider range of situations and are partially sharper than the known estimates in literature. A few numerical tests are given which confirm the sharpness of the estimates.

Symmetric Indefinite Preconditioners for Saddle Point Problems with Applications to PDE-Constrained Optimization Problems

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.

A class of smoothers for saddle point problems

Abstract In this paper smoothing properties are shown for a class of iterative methods for saddle point problems with smoothing rates of the order 1/m, where m is the number of smoothing steps. This generalizes recent results by Braess and Sarazin, who could prove this rates for methods where, in the context of the Stokes problem, the pressure correction equation is solved exactly, which is not needed here.

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 Schwarz-type Smoothers for Saddle Point Problems

Summary.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.

Inexact Data-Sparse Boundary Element Tearing and Interconnecting Methods

The boundary element tearing and interconnecting (BETI) methods have recently been introduced as boundary element counterparts of the well-established finite element tearing and interconnecting (FETI) methods. In this paper we present inexact data-sparse versions of the BETI methods which avoid the elimination of the primal unknowns and dense matrices. However, instead of symmetric and positive definite systems, we finally have to solve twofold saddle point problems. The proposed iterative solvers and preconditioners result in almost optimal solvers whose complexity is proportional to the number of unknowns on the skeleton up to some polylogarithmical factor. Moreover, the solvers are robust with respect to large coefficient jumps.

A simple homotopy method for determining all isolated solutions to polynomial systems

A new homotopy method for solving systems of polynomial equations is presented. The homotopy equation is extremely simple: It is linear with respect to the homotopy parameter and only one auxiliary parameter is needed to regularize the problem. Within some limits, an arbitrary starting problem can be chosen, as long as its solution set is known. No restrictions on the polynomial systems are made. A few numerical tests are reported which show the influence of the auxiliary parameter, resp. the starting problem, upon the computa- tional cost of the method.

Convergence analysis of multigrid methods with collective point smoothers for optimal control problems

In this paper we consider multigrid methods for solving saddle point problems. The choice of an appropriate smoothing strategy is a key issue in this case. Here we focus on the widely used class of collective point smoothers. These methods are constructed by a point-wise grouping of the unknowns leading to, e.g., collective Richardson, Jacobi or Gauss-Seidel relaxation methods. Their smoothing properties are well-understood for scalar problems in the symmetric and positive definite case. In this work the analysis of these methods is extended to a special class of saddle point problems, namely to the optimality system of optimal control problems. For elliptic distributed control problems we show that the convergence rates of multigrid methods with collective point smoothers are bounded independent of the grid size and the regularization (or cost) parameter.

Stability estimates and structural spectral properties of saddle point problems

For a general class of saddle point problems sharp estimates for Babuška’s inf-sup stability constants are derived in terms of the constants in Brezzi’s theory. In the finite-dimensional Hermitian case more detailed spectral properties of preconditioned saddle point matrices are presented, which are helpful for the convergence analysis of common Krylov subspace methods. The theoretical results are applied to two model problems from optimal control with time-periodic state equations. Numerical experiments with the preconditioned minimal residual method are reported.

A Robust Multigrid Method for Elliptic Optimal Control Problems

We consider the discretized optimality system of a special class of elliptic optimal control problems and propose an all-at-once multigrid method for solving this discretized system. Under standard assumptions the convergence of the multigrid method and the robustness of the convergence rates with respect to the involved parameter are shown. Numerical experiments are presented for illustrating the theoretical results.