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A Practical Examination of Some Numerical Methods for Linear Discrete Ill-Posed Problems
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In this paper, four well-known methods for the numerical solution of linear discrete ill-posed problems are investigated from a common point of view: namely, the type of algebraic expansion generated for the solu...Abstract:
Four well-known methods for the numerical solution of linear discrete ill-posed problems are investigated from a common point of view: namely, the type of algebraic expansion generated for the solu...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.
Journal ArticleDOI
The Collinearity Problem in Linear Regression. The Partial Least Squares (PLS) Approach to Generalized Inverses
TL;DR: In this article, the use of Partial Least Squares (PLS) for handling collinearities among the independent variables X in multiple regression is discussed, and successive estimates are obtained using the residuals from previous rank as a new dependent variable y.
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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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Algorithm 583: LSQR: Sparse Linear Equations and Least Squares Problems
TL;DR: This work was supported by Natural Sciences and Engineering Research Council of Canada Grant A8652, by the New Zealand Department of Scientific and Industrial Research, and by the Department of Energy under Contract DE-AT03-76ER72018.
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The truncated SVD as a method for regularization
TL;DR: In this article, the truncated singular value decomposition (SVD) is considered as a method for regularization of ill-posed linear least squares problems and compared with the usual regularized solution.
References
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Journal ArticleDOI
Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter
TL;DR: The generalized cross-validation (GCV) method as discussed by the authors is a generalized version of Allen's PRESS, which can be used in subset selection and singular value truncation, and even to choose from among mixtures of these methods.
Journal ArticleDOI
A Technique for the Numerical Solution of Certain Integral Equations of the First Kind
TL;DR: Here the authors will consider only nonsingular linear integral equations of the first kind, where the known functions h(x), K(x, y) and g(x) are assumed to be bounded and usually to be continuous.
Journal ArticleDOI
An Algorithm for Generalized Matrix Eigenvalue Problems.
Cleve B. Moler,G. W. Stewart +1 more
TL;DR: A new method, called the QZ algorithm, is presented for the solution of the matrix eigenvalue problem $Ax = \lambda Bx$ with general square matrices A and B with particular attention to the degeneracies which result when B is singular.
Journal ArticleDOI
On the Numerical Solution of Fredholm Integral Equations of the First Kind by the Inversion of the Linear System Produced by Quadrature
TL;DR: The purpose of the present note is to show how the same result can be obtained in a way which requires the inversion of only one matrix, whereas the method described by Phillips involves the inversions of two matrices.