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

Joachim Schöberl, Walter Zulehner

2007
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SIAM Journal on Matrix Analysis and Applications
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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
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... 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. with associated Lagrangian parameter p. As long as the matrix A is positive definite not only on ker B but on the whole space R n , the (negative) Schur complement S = BA −1 B T is well-defined. Then several approaches for an efficient solution procedure have been proposed. Most of them can be viewed as preconditioned Richardson methods for (1.1) typically accelerated by a Krylov subspace method, see Saad, van der Vorst [15] for a review of iterative methods for linear systems. The discussed preconditioners for K are 2-by-2 block matricesK depending on a preconditioner for approximating A and a preconditionerŜ which is either interpreted as approximation of the Schur complement S or as approximation of the so-called inexact Schur complement H = B −1 B T . Typical classes of such preconditioners are block diagonal preconditioners, see, e.g., Rusten and Winther [14], Silvester and Wathen [16], block triangular preconditioners (originating from the classical Uzawa method [1]), see, e.g., Elman, Golub [8], Bramble, Pasciak, Vassilev [5], symmetric indefinite preconditioners, see, e.g., Dyn, Ferguson [7], Bank, Welfert, Yserentant [2], and symmetric positive definite block (but not block-diagonal) preconditioners, see Vassilevski, Lazarov [18]. Depending on the properties of the preconditioned systems Krylov subspace methods either for symmetric indefinite or for non-symmetric systems like MINRES or GMRES were proposed. In Bramble and Pasciak [4] a block triangular preconditioner was used in order to obtain a preconditioned system which is symmetric and positive definite and, therefore, can be solved by the conjugate gradient method, which is usually considered as the best or at least the best-understood Krylov subspace method. The block triangular preconditioner in [4] requires a symmetric and positive definite approximation with < A. For a much more detailed discussion of available methods for saddle point problems we refer to the review article by Benzi, Golub and Liesen [3] . In this paper, however, we will focus on systems, where A is positive definite in a stable way (to be specified later) only on ker B, a typical situation for certain classes of optimization problems with PDE-constraints. One strategy to enforce the definiteness on the whole space R n is the augmented Lagrangian approach, where the matrix A and the vector f in (1.1) are replaced by a matrix of the form A W = A + B T W B and a vector f W = f + B T W g , respectively, with an appropriate matrix W , see e.g. Fortin and Glowinski [10]. This does not change the solution of the problem, and the new (1,1) block A W becomes positive definite if W is properly chosen, e.g, if it is positive definite, and all methods from above applied to the augmented system could be used, in principle. It is, however, a delicate issue to choose the matrix W in order to obtain good convergence properties, see the discussions in Golub and Greif [11], Golub, Greif and Varah [12] . Here we will take a different approach and discuss preconditionersK for the original system matrix K (without augmentation), which, nevertheless, work also well, in the case that A is positive definite only on the kernel of B. Under appropriate assumptions it will be shown that the preconditioned matrixK −1 K is even symmetric and positive definite in some appropriate scalar product. Therefore, conjugate gradient acceleration can be applied. In contrast to Bramble and Pasciak [4] this new technique requires a symmetric and positive definite approximation witĥ A > A, which is easier to achieve and can be applied also if A itself is only positive definite on the kernel of B. An important field of applications are PDE-constrained optimization problems, in particular, optimal control problems, see, e.g, Tröltzsch [17] . It is typical for optimal control problems that the cost functional contains an extra regularization parameter. If discretized by an appropriate finite element method, the resulting KKT system is of the form (1.1), where the matrices A and B depend on the underlying subdivision, say with mesh size h, and on the regularization parameter, say ν. For optimal control problems with elliptic state equations and distributed control a special symmetric indefinite preconditioner will be constructed and convergence rate estimates are given which are robust in h as well as in ν. The paper is organized as follows: In Section 2 the considered class of preconditioners is introduced and analyzed. Section 3 describes how the algebraic conditions for the preconditioners are linked to the conditions of the theorem of Brezzi for mixed variational problems, and a general framework for constructing the preconditioners is sketched. In Section 4 a problem from optimal control is discussed and preconditioners are constructed which are robust with respect to the mesh size as well as to the involved regularization parameter. Numerical experiments are presented in

doi:10.1137/060660977
fatcat:owz3gabdbngr5kv27lzboxx46q