Algorithm 583: LSQR: Sparse Linear Equations and Least Squares Problems

Paige, Christopher C.; Saunders, Michael A.

doi:10.1145/355993.356000

Cited by 692 publications

(360 citation statements)

References 4 publications

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“…There are two closely related conjugate gradient based methods for least squares problems: CGLS [12] and LSQR 2 [76,77]. We focus our discussion on LSQR.…”

Section: Conjugate Gradient Methodsmentioning

confidence: 99%

See 1 more Smart Citation

Iterative Methods for Image Restoration

Berisha

Nagy

2014

Academic Press Library in Signal Processing: Volume 4 - Image, Video Processing and Analysis, Hardware, Audio, Acoustic and Spe

104

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Although image restoration methods based on spectral filtering techniques are very efficient, they can be applied only to problems with fairly simple spatially invariant blurring operators. Iterative methods, however, are much more flexible; they can be very efficient for spatially invariant as well as spatially variant blurs, they can incorporate a variety of regularization techniques and boundary conditions, and they can more easily incorporate additional constraints, such as nonnegativity. This chapter describes a variety of iterative methods used in image restoration, with a particular emphasis on efficiency, convergence behavior, and implementation. Discussion of MATLAB software implementing the methods is also provided.

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“…There are two closely related conjugate gradient based methods for least squares problems: CGLS [12] and LSQR 2 [76,77]. We focus our discussion on LSQR.…”

Section: Conjugate Gradient Methodsmentioning

confidence: 99%

“…It can be shown that f k converges to the solution of the least squares problem min g − Kf 2 . We omit specific implementation details, and refer to [76,77]. However, we do mention the following important points:…”

Section: Lsqr and Filteringmentioning

confidence: 99%

Iterative Methods for Image Restoration

Berisha

Nagy

2014

Academic Press Library in Signal Processing: Volume 4 - Image, Video Processing and Analysis, Hardware, Audio, Acoustic and Spe

104

View full text Add to dashboard Cite

show abstract

“…We solve for each frequency band using sparse matrix methods (LSQR; Paige and Saunders, 1982), applying first difference regularization to the attenuation terms and constrain the site terms to sum to zero. For the discretization, we assume cell size 0.1 X 0.1 degrees for all frequency bands.…”

Section: Methodology Overviewmentioning

confidence: 99%

Improved 2-D attenuation analysis for Northern Italy using a merged dataset from selected regional seismic networks

Morasca

Massa²,

Laprocina³

et al. 2010

J Seismol

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A merged, high-quality waveform data set from different seismic networks has been used to improve our understanding of lateral seismic attenuation for Northern Italy. In a previous study on the same region, Morasca et al. (2008) were able to resolve only a small area due to limited data coverage. For this reason the interpretation of the attenuation anomalies was difficult given the complexity of the region and the poor resolution of the available data. In order to better understand the lateral changes in the crustal structure and thickness of this region, we selected 770 earthquakes recorded by 54 stations for a total of almost 16000 waveforms derived from seismic networks operating totally or partially in Northern Italy. Direct S-wave and coda attenuation images were obtained using an amplitude ratio technique that eliminates source terms from the formulation. Both direct and early-coda amplitudes are used as input for the inversions and the results are compared.Results were obtained for various frequency bands ranging between 0.3-25.0 Hz, and in all cases show significant improvement with respect to the previous study since the resolved area has been extended and more crossing paths have been used to image smaller scale anomalies. Quality-factor estimates are consistent with the regional tectonic structure exhibiting a general trend of low attenuation under the Po Plain basin and higher values for the western Alps and northern Apennines. The interpretation of the results for the Eastern Alps is not simple, possibly because our resolution for this area is still not adequate to resolve small scale structures.Response to Reviewers: Dear Dr. ZahradnikBelow is our response to the two reviewers' comments and we have uploaded both the revision and another copy with highlighted changes. Both reviewers had very good suggestions and we tried to address each item. We feel fortunate to have received such thorough reviews, particularly reviewer #2 who had numerous comments that ultimately made the paper more clear. We emphasize that this current study is an extension of the 2008 Western Alps study by Morasca et al. (BSSA) but incorporating a much larger data set. The inversion methodology and assumptions on using the early coda is described in detail in the previous work. In addition, we cite in the current study other supporting papers that describe in great detail more of the method. Including this once again in this paper would seem redundant. In addition, we have highlighted changes in the revision as well. COMMENTS FOR THE AUTHOR:Reviewer #1: This paper presents a 2-D Attenuation study for Northern Italy. The authors use earthquakes recorded by several stations deployed in the region. This study extends geographically the knowledge of Q distribution especially north eastern of the studied zone. The presented results agree with previously reported. I found this manuscript to be well written and presenting a very complete study. I support its publication in JOSE. I have some minor comments:1.-Abstract shows ...

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“…It is known that, computationally speaking, GMRES is more expensive than other Krylov subspace methods, such as Bi-CGSTAB, [14], QMR [24] for general square matrices, or LSQR [19], [20] for anti-symmetric matrices. Nevertheless, it is widely used for solving linear systems derived from the discretization of partial differential equations, since theoretically the 2-norm of the residual vector is minimized inside the Krylov subspace at each step.…”

Section: Gmres(m)mentioning

confidence: 99%

A safeguard approach to detect stagnation of GMRES(m) with applications in Newton-Krylov methods

Gomes-Ruggiero¹,

Lopes²,

Toledo-Benavides³

2008

Comput.appl.math.

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Abstract.Restarting GMRES, a linear solver frequently used in numerical schemes, is known to suffer from stagnation. In this paper, a simple strategy is proposed to detect and avoid stagnation, without modifying the standard GMRES code. Numerical tests with the proposed modified GMRES(m) procedure for solving linear systems and also as part of an inexact Newton procedure, demonstrate the efficiency of this strategy.Mathematical subject classification: 65H10, 65F10.

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Algorithm 583: LSQR: Sparse Linear Equations and Least Squares Problems

Cited by 692 publications

References 4 publications

Iterative Methods for Image Restoration

Iterative Methods for Image Restoration

Improved 2-D attenuation analysis for Northern Italy using a merged dataset from selected regional seismic networks

A safeguard approach to detect stagnation of GMRES(m) with applications in Newton-Krylov methods

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