Constraint Preconditioning for Indefinite Linear Systems

By Carsten Keller, N. Gould, A. Wathen. SIAM Journal on Matrix Analysis and Applications, 2000

389 citations recorded by the source corpus.

Subfields: Atomic and Molecular Physics, and Optics, Computational Mechanics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/a702df1b482d1049cfdc556b051d7e37119eb590

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In Atomic and Molecular Physics, and Optics

In Computational Mechanics

In Computational Theory and Mathematics

Citations in the corpus

Interior point methods 25 years later (2012, background): “A plethora of preconditioners for symmetric indefinite systems arising in optimization have been proposed in the last decade [9, 10, 20, 28, 30, 64, 95].”
A Relaxed HSS Preconditioner for Saddle Point Problems from Meshfree Discretization (2013, background): “A variety of preconditioners have been proposed and studied in many papers, such as block diagonal and block triangular preconditioners [7,19,20,24], constraint preconditioners [8,25], HSS-based preconditioners [3, 6, 12, 13, 30], dimensional split preconditioners [15, 16], H−matrix preconditioners ”
Knitro: An Integrated Package for Nonlinear Optimization (2006, methodology): “We solve these problems using a projected conjugate gradient iteration [10, 20, 24, 26, 32], which is well suited for large problems and can handle the negative curvature case without the need for Hessian modifications.”
Neurobiology of Aging: Contents (2005, neutral): “As pointed out in [298], if either ZAZ or ZGZ are positive definite, then the indefinite preconditioner Pc (10.”
Novel preconditioners for the iterative solution to FE-discretized coupled consolidation equations (2007, positive): “To accelerate Krylov solvers in the solution of saddle point problems the so-called ‘‘Constraint Preconditioners’’ have been first introduced in constrained optimization [13].”
Primal‐dual active set methods for Allen–Cahn variational inequalities with nonlocal constraints (2010, background): “Similar to the scalar case one can show [78, 91, 110] that there are nN−r eigenvalues at 1, where r is the rank of the matrix C and nN is the size of the matrix B, or more precisely nN = (N + 1)J .”
Some Preconditioning Techniques for Saddle Point Problems (2008, background): “Given the equivalence to a symmetric positive definite problem, one might anticipate some special structure in the eigenvalues of the preconditioned matrix P−1A; what is perhaps not expected is that this matrix should generically be non-diagonalizable! As shown in [27] this is always the case, but t”
Block preconditioners for symmetric indefinite linear systems (2004, methodology): “To overcome the disadvantages mentioned above, in this paper, we propose to view the preconditioner (12) as some form of approximation to the inverse of A0 instead of the original motivation mentioned in Reference [14].”
A Preconditioner for Generalized Saddle Point Problems (2004, methodology): “Besides specialized sparse direct solvers [16, 17] we mention, among others, Uzawa-type schemes [11, 21, 24, 27, 62], block and approximate Schur complement preconditioners [4, 15, 20, 22, 41, 45, 46, 48, 51], splitting methods [18, 30, 31, 49, 57], indefinite preconditioning [23, 35, 39, 43, 48], i”