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Communication Dans Un Congrès Année : 2018

A Tridiagonalization Method for Symmetric and Quasi-Definite Saddle-Point Systems (CM 2018, Colorado, USA)

Résumé

We consider the solution of symmetric saddle-point systems with symmetric and positive definite leading block. Such systems arise in numerous applications, including optimization, fluid dynamics, and data assimilation. In the large-scale case, or the case where the blocks are only available as operators, it is common to employ a Krylov method. Prime candidates are the methods MINRES and SYMMLQ of Paige and Saunders, both of which were designed with general symmetric indefinite systems in mind, but neither of which exploits the specific block structure of the system. The main idea stems from the simple observation that the solution the system with right-hand side (b,c) may be written as the sum of the solutions of the same system with right-hand sides (b,0) and (0,c). In turn, those two systems are the optimality conditions of a least-squares and a least-norm problem. We propose an approach that meshes an iterative method for least-squares problems with one for least-norm problems in such a way that both problems can be solved in one pass. Each iteration of the proposed procedure has the same cost and almost the same storage requirements as one iteration of MINRES or SYMMLQ but only roughly half as many iterations are required. Our approach is based on an orthogonal tridiagonalization process initially proposed by Saunders, Simon and Yip in 1988 for square, but not necessarily symmetric, matrices. The process suggests two approaches of the Krylov kind for solving a linear system: one of the minimum-residual kind in the vein of MINRES and named USYMQR, and one of the minimum-norm kind in the vein of SYMMLQ and named USYMLQ. The tridiagonalization process reduces to the symmetric Lanczos process in the symmetric case but differs from the Golub-Kahan biorthogonalization process in that it must be initialized with two vectors. As a consequence, both USYMQR and USYMLQ can be used in conjunction to solve a system and an adjoint system at the same time. We present preconditioned and regularized variants that apply to symmetric and quasi-definite linear systems and illustrate the performance of our implementation on systems coming from optimization and fluid flow. We also present variants of the tridiagonalization procedure specifically designed to address rank-deficient problems.
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hal-03023422 , version 1 (25-11-2020)

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  • HAL Id : hal-03023422 , version 1

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Alfredo Buttari, Dominique Orban, Daniel Ruiz, David Titley-Peloquin. A Tridiagonalization Method for Symmetric and Quasi-Definite Saddle-Point Systems (CM 2018, Colorado, USA). 15th Copper Mountain Conference On Iterative Methods (CM 2018), Front Range Scientific Computations, Inc, Boulder, Colorado; University of Colorado, Sandia National Labs; Emory University; Society for Industrial and Applied Mathematics: SIAM Activity Group on Linear Algebra, Mar 2018, Copper Mountain, Colorado, United States. ⟨hal-03023422⟩
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