On the Stability of Null-Space Methods for KKT Systems

Fletcher, R.; Johnson, Tom

doi:10.1137/s0895479896297732

Cited by 25 publications

(19 citation statements)

References 7 publications

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“…Among them we mention null-space methods [1,20,24,40]; direct solvers [10,41]; the classical Uzawa algorithm [2] and the inexact Uzawa algorithm [14]; splitting schemes such as the one introduced in [12], which was later generalized to real positive matrices in [27]; preconditioned Krylov subspace solvers based on approximating the Schur complement or other methodologies [13,15,16,17,35,37,39,50]. See also [5,7,24,40,47,48] for surveys of existing methods and further references.…”

Section: Introductionmentioning

confidence: 99%

On Solving Block-Structured Indefinite Linear Systems

Golub¹,

Greif

2003

SIAM J. Sci. Comput.

147

115

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We consider 2 × 2 block indefinite linear systems whose (2, 2) block is zero. Such systems arise in many applications. We discuss two techniques that are based on modifying the (1, 1) block in a way that makes the system easier to solve. The main part of the paper focuses on an augmented Lagrangian approach: a technique that modifies the (1,1) block without changing the system size. The choice of the parameter involved, the spectrum of the linear system, and its condition number are discussed, and some analytical observations are provided. A technique of deflating the (1,1) block is then introduced. Finally, numerical experiments that validate the analysis are presented.

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Section: Introductionmentioning

confidence: 99%

On Solving Block-Structured Indefinite Linear Systems

Golub¹,

Greif

2003

SIAM J. Sci. Comput.

147

115

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“…One way of fixing Y in Equation is to extend B T to an n × n nonsingular matrix

([], \begin{matrix} array B^{T} & array V^{T} \end{matrix}) .

If we choose Y and Z satisfying

[\begin{matrix} B \\ V \end{matrix}] [\begin{matrix} Y & Z \end{matrix}] = I

then it is easy to see that B Z =0 and B Y = I . The factorization in Equation reduces to

A = [\begin{matrix} B^{T} & V^{T} & 0 \\ 0 & 0 & I \end{matrix}] [\begin{matrix} Y^{T} A Y & Y^{T} A Z & I \\ Z^{T} A Y & Z^{T} A Z & 0 \\ I & 0 & 0 \end{matrix}] [\begin{matrix} B & 0 \\ V & 0 \\ 0 & I \end{matrix}] .

This factorization was given by Fletcher et al,(, eq. 3.6) who described it as “readily observed (but not well known).”…”

Section: Null‐space Methods As a Factorizationmentioning

confidence: 99%

“…We may then write, without loss of generality,

B = ([], \begin{matrix} array B_{1} & array B_{2} \end{matrix})

, where

B_{1} \in R^{m \times m}

is nonsingular. If, as suggested by Fletcher et al, we make the choice of

V = ([], \begin{matrix} array 0 & array I \end{matrix})

, then

{[\begin{matrix} B \\ V \end{matrix}]}^{- 1} = {[\begin{matrix} B_{1} & B_{2} \\ 0 & I \end{matrix}]}^{- 1} = [\begin{matrix} B_{1}^{- 1} & - B_{1}^{- 1} B_{2} \\ 0 & I \end{matrix}] (= [\begin{matrix} Y & Z \end{matrix}]) .

This gives us the bases

Z_{f} = ([], \begin{matrix} array - B_{1}^{- 1} B_{2} \\ array I \end{matrix}), Y_{f} = ([], \begin{matrix} array B_{1}^{- 1} \\ array 0 \end{matrix}) .

This choice for Z is often called the fundamental basis,(, section 6) and we consequently label it Z f .…”

Section: Null‐space Methods As a Factorizationmentioning

confidence: 99%

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A comparative study of null‐space factorizations for sparse symmetric saddle point systems

Rees

Scott

2017

Numerical Linear Algebra App

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Summary Null‐space methods for solving saddle point systems of equations have long been used to transform an indefinite system into a symmetric positive definite one of smaller dimension. A number of independent works in the literature have identified that we can interpret a null‐space method as a matrix factorization. We review these findings, highlight links between them, and bring them into a unified framework. We also investigate the suitability of using null‐space factorizations to derive sparse direct methods and present numerical results for both practical and academic problems.

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“…For example, the matrix splitting iterative methods [7,11,18,36,40,45], Uzawa-type methods [12,17,21,22,25], HSS method and its variants [2,[4][5][6]8,9,28,29,38], Krylov subspace methods [1,10,[33][34][35]42], null space methods [24] and so on. When the saddle-point problem (1) is singular, there are also many relaxation iteration methods which have been established, e.g., the Uzawa-type methods [30,46,48,49] and the HSS-like methods [3,20,39].…”

Section: Introductionmentioning

confidence: 99%

Variants of the accelerated parameterized inexact Uzawa method for saddle-point problems

Liang

Zhang

2015

Bit Numer Math

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In this paper, based on the SOR and SSOR splittings of the (1,1) part of saddle-point coefficient matrix, some variants of the accelerated parameterized inexact Uzawa (VAPIU) method are proposed for solving nonsingular and singular saddlepoint problems. By choosing different parameter matrices, we derive some existing and new iterative methods. The corresponding convergence and semi-convergence of the VAPIU methods for solving nonsingular and singular saddle-point problems are studied in depth, respectively. The preconditioning strategies based on the VAPIU splittings of the coefficient matrices are presented. Numerical experiments are provided, which confirms that these new methods need less CPU times per iteration step comparing with some other methods for solving both nonsingular and singular saddle-point problems.

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On the Stability of Null-Space Methods for KKT Systems

Cited by 25 publications

References 7 publications

On Solving Block-Structured Indefinite Linear Systems

On Solving Block-Structured Indefinite Linear Systems

A comparative study of null‐space factorizations for sparse symmetric saddle point systems

Variants of the accelerated parameterized inexact Uzawa method for saddle-point problems

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