Incomplete Methods for Solving $A^T Ax = b$

Jennings, A.; Ajiz, Manzur Ali

doi:10.1137/0905067

Cited by 54 publications

(33 citation statements)

References 11 publications

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“…On the other hand, row-oriented incomplete Givens orthogonalization is not guaranteed to be breakdown-free: it may lead to zero diagonal entries in the incomplete R factor. In our experiments we found that breakdowns do occur in practice and that roworiented incomplete Givens codes need to be safeguarded against this type of failure (see, e.g., [5,13]). …”

Section: Previous Workmentioning

confidence: 98%

“…In Table 5.4 we show results obtained with an incomplete QR preconditioner based on modified Gram-Schmidt (denoted IMGS); this is essentially the incomplete Gram-Schmidt method in [13]. Our implementation is based on a column-oriented, right-looking algorithm.…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…Note that there is no need for theQ factor in the iterative phase of the algorithm; thereforeQ does not need to be saved. The first paper proposing to compute an incomplete orthogonal factorization of A for use as a preconditioner for the CGLS method is [13]. In it, the authors consider both methods based on Givens rotations and methods based on the Gram-Schmidt process.…”

Section: Previous Workmentioning

confidence: 99%

“…Moreover, for some of these methods intermediate storage requirements can be very high. As an example, we report on results obtained with the Gram-Schmidt-based incomplete QR method of Jennings and Ajiz [13]. We use a rather small square matrix, WEST0655, from the Matrix Market [18].…”

Section: Previous Workmentioning

confidence: 99%

See 3 more Smart Citations

A Robust Preconditioner with Low Memory Requirements for Large Sparse Least Squares Problems

Benzi

Tůma²

2003

SIAM J. Sci. Comput.

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Abstract. This paper describes a technique for constructing robust preconditioners for the CGLS method applied to the solution of large and sparse least squares problems. The algorithm computes an incomplete LDL T factorization of the normal equations matrix without the need to form the normal matrix itself. The preconditioner is reliable (pivot breakdowns cannot occur) and has low intermediate storage requirements. Numerical experiments illustrating the performance of the preconditioner are presented. A comparison with incomplete QR preconditioners is also included.

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Section: Previous Workmentioning

confidence: 98%

Section: Numerical Experimentsmentioning

confidence: 99%

Section: Previous Workmentioning

confidence: 99%

Section: Previous Workmentioning

confidence: 99%

See 2 more Smart Citations

A Robust Preconditioner with Low Memory Requirements for Large Sparse Least Squares Problems

Benzi

Tůma²

2003

SIAM J. Sci. Comput.

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“…It is well known that the incomplete QR decomposition of the matrix H can be determined by the incomplete modified Gram-Schmidt (IMGS) algorithm proposed by Jennings and Ajiz [10], and this algorithm never breaks down for the matrix with full column rank. However, it needs to computing directly and storing the column vectors of the matrix Q ∈ R mp×(3n−2) at each stage in the IMGS algorithm.…”

Section: Cimgs Preconditionermentioning

confidence: 99%

Iterative method for the least squares problem of a matrix equation with tridiagonal matrix constraint

Pan¹,

Yuan²

2012

ELA

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Abstract. The matrix-form LSQR method is presented in this paper for solving the least squares problem of the matrix equation AXB = C with tridiagonal matrix constraint. Based on a matrix-form bidiagonalization procedure, the least squares problem associated with the tridiagonal constrained matrix equation AXB = C reduces to a unconstrained least squares problem of linear system, which can be solved by using the classical LSQR algorithm. Furthermore, the preconditioned matrix-form LSQR method is adopted for solving the corresponding least squares problem.

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Preconditioners for Krylov subspace methods: An overview

Pearson

Pestana

2020

GAMM-Mitteilungen

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When simulating a mechanism from science or engineering, or an industrial process, one is frequently required to construct a mathematical model, and then resolve this model numerically. If accurate numerical solutions are necessary or desirable, this can involve solving large-scale systems of equations. One major class of solution methods is that of preconditioned iterative methods, involving preconditioners which are computationally cheap to apply while also capturing information contained in the linear system. In this article, we give a short survey of the field of preconditioning. We introduce a range of preconditioners for partial differential equations, followed by optimization problems, before discussing preconditioners constructed with less standard objectives in mind.

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Incomplete Methods for Solving $A^T Ax = b$

Cited by 54 publications

References 11 publications

A Robust Preconditioner with Low Memory Requirements for Large Sparse Least Squares Problems

A Robust Preconditioner with Low Memory Requirements for Large Sparse Least Squares Problems

Iterative method for the least squares problem of a matrix equation with tridiagonal matrix constraint

Preconditioners for Krylov subspace methods: An overview

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