Modifying the QR-Decomposition to Constrained and Weighted Linear Least Squares

Gullisson, Mårten; Wedin, Per-Åke

doi:10.1137/0613079

Cited by 48 publications

(46 citation statements)

References 10 publications

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“…The number of arithmetic floating point operations required to solve (3.2) for all pixels in W i−1 is proportional to n i−1 and therefore modest. The solution of each least-squares problem can be computed, e.g., by determining a modified QRdecomposition of a 9 × 3 matrix based on modified Householder transformations; see Gulliksson and Wedin [12] for details on the latter. The modifications are required because of the weights in the least-squares problem.…”

Section: Algorithm 22 Multilevel Algorithmmentioning

confidence: 99%

Cascadic multilevel methods for fast nonsymmetric blur- and noise-removal

Morigi¹,

Reichel²,

Sgallari³

2010

Applied Numerical Mathematics

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Dedicated to Richard S. Varga on the occasion of his 80th birthday.Abstract. Image deblurring is a discrete ill-posed problem. This paper discusses cascadic multilevel methods designed for the restoration of images that have been contaminated by nonsymmetric blur and noise. Prolongation is carried out by nonlinear edge-preserving and noise-reducing operators, while restrictions are determined by weighted local least-squares approximation. The restoration problem is on each level solved by an iterative method, with the number of iterations determined by the discrepancy principle. The performance of several iterative methods is compared. Computed examples demonstrate the effectiveness of the image restoration methods proposed. The discrepancy principle requires that an estimate of the norm of the noise in the contaminated image be available. We illustrate how such an estimate can be computed with the aid of the nonlinear Perona-Malik diffusion equation.

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Section: Algorithm 22 Multilevel Algorithmmentioning

confidence: 99%

Cascadic multilevel methods for fast nonsymmetric blur- and noise-removal

Morigi¹,

Reichel²,

Sgallari³

2010

Applied Numerical Mathematics

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“…In [4] Forsgren also used the Binet-Cauchy formula and Cramer's rule to obtain the supremum of weighted pseudoinverses arising from solving the LSE problem [9] …”

Section: Corollary 21 Under the Notation And The Conditions In Theomentioning

confidence: 99%

“…When solving (1.2) by an interior method [1], [5], [6], [17], [22]- [24], one will obtain the following weighted least squares (WLS) problem min x∈R n W 1 2 (Xx − g) , (1.3) where W = W (τ ) ∈ P(X), τ > 0 is a parameter. Similarly, when solving the equality constrained least squares problem (LSE) [9] min x∈R n W 1 2 2 (Kx − g 2 ) subject to Lx = g 1 (1.4) by the weighting method, one will also obtain a WLS problem like (1.3). When τ → +∞, the minimum 2-norm solution of (1.3) will tend to the minimum 2-norm solution of (1.2) or (1.4).…”

Section: Introductionmentioning

confidence: 99%

Equivalent formulae for the supremum and stability of weighted pseudoinverses

Wei

1997

Math. Comp.

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Abstract. During recent decades, there have been a great number of research articles studying interior-point methods for solving problems in mathematical programming and constrained optimization. Stewart and O'Leary obtained an upper bound for scaled pseudoinverses supwhere P is a set of diagonal positive definite matrices. We improved their results to obtain the supremum of scaled pseudoinverses and derived the stability property of scaled pseudoinverses. Forsgren further generalized these results to derive the supremum of weighted pseudoinverses supP is a set of diagonally dominant positive semidefinite matrices, by using a signature decomposition of weighting matrices W and by applying the BinetCauchy formula and Cramer's rule for determinants. The results are also extended to equality constrained linear least squares problems. In this paper we extend Forsgren's results to a general complex matrix X to establish several equivalent formulae for sup, where P is a set of diagonally dominant positive semidefinite matrices, or a set of weighting matrices arising from solving equality constrained least squares problems. We also discuss the stability property of these weighted pseudoinverses.

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“…He obtains a backward error bound similar to our Theorem 4.3 and also shows that iterative refinement may be applied. Reid points out that the method is equivalent to a method of Gulliksson and Wedin [9], [10], which is expressed in the language of "M -invariant reflections. "…”

Section: Introduction Consider the Equality Constrained Least Squarementioning

confidence: 99%

Row-Wise Backward Stable Elimination Methods for the Equality Constrained Least Squares Problem

Cox¹,

Higham²

1999

SIAM J. Matrix Anal. & Appl.

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Abstract.It is well known that the solution of the equality constrained least squares (LSE) problem min Bx=d b − Ax 2 is the limit of the solution of the unconstrained weighted least squares problemas the weight µ tends to infinity, assuming that [T has full rank. We derive a method for the LSE problem by applying Householder QR factorization with column pivoting to this weighted problem and taking the limit analytically, with an appropriate rescaling of rows. The method obtained is a type of direct elimination method. We adapt existing error analysis for the unconstrained problem to obtain a row-wise backward error bound for the method. The bound shows that, provided row pivoting or row sorting is used, the method is well-suited to problems in which the rows of A and B vary widely in norm. As a by-product of our analysis, we derive a row-wise backward error bound of precisely the same form for the standard elimination method for solving the LSE problem. We illustrate our results with numerical tests.

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Modifying the QR-Decomposition to Constrained and Weighted Linear Least Squares

Cited by 48 publications

References 10 publications

Cascadic multilevel methods for fast nonsymmetric blur- and noise-removal

Cascadic multilevel methods for fast nonsymmetric blur- and noise-removal

Equivalent formulae for the supremum and stability of weighted pseudoinverses

Row-Wise Backward Stable Elimination Methods for the Equality Constrained Least Squares Problem

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