SIRT- and CG-type methods for the iterative solution of sparse linear least-squares problems
TLDR
In this article, the convergence and regularization properties of SIRT and CG-type methods for least-squares problems arising in tomography were compared and it was shown that virtually the same solutions as obtained by SIRT methods can be obtained by applying a CG-based method to a properly rescaled system, but with an amount of work proportional to the square root of the amount of SIR with SIRT.About:
This article is published in Linear Algebra and its Applications.The article was published on 1990-03-01 and is currently open access. It has received 105 citations till now. The article focuses on the topics: Linear least squares.read more
Citations
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Analysis of discrete ill-posed problems by means of the L-curve
TL;DR: The main purpose of this paper is to advocate the use of the graph associated with Tikhonov regularization in the numerical treatment of discrete ill-posed problems, and to demonstrate several important relations between regularized solutions and the graph.
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REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems
TL;DR: The package REGULARIZATION TOOLS consists of 54 Matlab routines for analysis and solution of discrete ill-posed problems, i.e., systems of linear equations whose coefficient matrix has the properties that its condition number is very large, and its singular values decay gradually to zero.
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AIR Tools - A MATLAB package of algebraic iterative reconstruction methods
TL;DR: A MATLAB package with implementations of several algebraic iterative reconstruction methods for discretizations of inverse problems based on the discrepancy principle, the monotone error rule, and the NCP criterion is presented.
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The discrete picard condition of discrete ill-posed problems
TL;DR: In this article, a necessary condition for obtaining good regularized solutions is that the Fourier coefficients of the right-hand side, when expressed in terms of the generalized SVD associated with the regularization problem, on the average decay to zero faster than the generalized singular values.
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3D imaging of nanomaterials by discrete tomography.
Kees Joost Batenburg,Sara Bals,Jan Sijbers,Christian Kübel,Paul A. Midgley,J. C. Hernandez,Ute Kaiser,Ezequiel Roberto Encina,Eduardo A. Coronado,G. Van Tendeloo +9 more
TL;DR: The basic principles of DART are described and it is shown that it can be applied successfully to three different types of samples, consisting of embedded ErSi(2) nanocrystals, a carbon nanotube grown from a catalyst particle and a single gold nanoparticle, respectively.
References
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Handbook of Mathematical Functions
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LSQR: An Algorithm for Sparse Linear Equations and Sparse Least Squares
TL;DR: Numerical tests are described comparing I~QR with several other conjugate-gradient algorithms, indicating that I ~QR is the most reliable algorithm when A is ill-conditioned.
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Iterative methods for the three-dimensional reconstruction of an object from projections
TL;DR: It is shown that in general ART produces erroneous reconstructions, and an alternative iterative method is proposed which will give correct reconstructions under certain conditions.
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Lower mantle heterogeneity, dynamic topography and the geoid
Bradford H. Hager,Robert W. Clayton,Mark A. Richards,R. P. Comer,R. P. Comer,Adam M. Dziewonski,Adam M. Dziewonski,Adam M. Dziewonski +7 more
TL;DR: In this paper, the total gravity field due to interior density contrasts and dynamic boundary topography predicts the longest-wavelength components of the geoid remarkably well, and neglecting dynamic surface deformation leads to geoid anomalies of opposite sign to those observed.