Block-Diagonal Preconditioners for Indefinite Linear Algebraic Systems. Part I: Theory
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Eric de Sturler
We study block-diagonal preconditioners and fast iterative solvers for indefinite two-by-two block linear systems with zero (2,2) block. Our preconditioners are derived from a splitting of the (1,1) block, A = D E, where the matrix D can be efficiently inverted. Dierent splittings lead to dierent preconditioners, and we analyze properties of the preconditioned matrices, in particular their eigenvalue distributions. From the preconditioned linear system we derive a fixed point iteration as well as its so-called related system for solving the original block two-by-two problem. We study the convergence of the fixed point iteration and the eigenvalue distribution of the related system matrix. Using our analytical results we show that solving the original system by applying GMRES  to the related system is typically more efficient than solving it by applying GMRES to the preconditioned system. In addition, in the case of constrained problems the use of the xed point iteration or its related system leads to approximations that satisfy the constraints exactly after one iteration. Moreover, we show how scaling the original block two-by-two system can improve convergence dramatically. Our theoretical results are confirmed by numerical experiments on a constrained optimization application. Our approach is very general, as we make almost no assumptions on the given block two-by-two system. In particular, the system matrix might be nonsymmetric, and the (1,1) block A might be indefinite, even singular. This is the first paper in a two-part sequence. In the second paper we will study the use of our preconditioners in a broad variety of applications.