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90
Solving illconditioned and singular linear systems: A tutorial on regularization
 SIAM Rev
, 1998
"... Abstract. It is shown that the basic regularization procedures for finding meaningful approximate solutions of illconditioned or singular linear systems can be phrased and analyzed in terms of classical linear algebra that can be taught in any numerical analysis course. Apart from rewriting many kn ..."
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Cited by 83 (2 self)
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Abstract. It is shown that the basic regularization procedures for finding meaningful approximate solutions of illconditioned or singular linear systems can be phrased and analyzed in terms of classical linear algebra that can be taught in any numerical analysis course. Apart from rewriting many known results in a more elegant form, we also derive a new twoparameter family of merit functions for the determination of the regularization parameter. The traditional merit functions from generalized cross validation (GCV) and generalized maximum likelihood (GML) are recovered as special cases.
On stable numerical differentiation
 Mathem. of Computation
, 1968
"... Abstract. A new approach to the construction of finitedifference methods is presented. It is shown how the multipoint differentiators can generate regularizing algorithms with a stepsize h being a regularization parameter. The explicitly computable estimation constants are given. Also an iterative ..."
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Cited by 39 (22 self)
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Abstract. A new approach to the construction of finitedifference methods is presented. It is shown how the multipoint differentiators can generate regularizing algorithms with a stepsize h being a regularization parameter. The explicitly computable estimation constants are given. Also an iteratively regularized scheme for solving the numerical differentiation problem in the form of Volterra integral equation is developed. 1.
Accuracy assessments of aerosol optical properties retrieved from aerosol robotic network (aeronet) sun and sky radiance measurments
 JOURNAL OF GEOPHYSICAL RESEARCH
"... radiance measurements ..."
The LCurve and its Use in the Numerical Treatment of Inverse Problems
 in Computational Inverse Problems in Electrocardiology, ed. P. Johnston, Advances in Computational Bioengineering
, 2000
"... The Lcurve is a loglog plot of the norm of a regularized solution versus the norm of the corresponding residual norm. It is a convenient graphical tool for displaying the tradeoff between the size of a regularized solution and its fit to the given data, as the regularization parameter varies. The ..."
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Cited by 31 (2 self)
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The Lcurve is a loglog plot of the norm of a regularized solution versus the norm of the corresponding residual norm. It is a convenient graphical tool for displaying the tradeoff between the size of a regularized solution and its fit to the given data, as the regularization parameter varies. The Lcurve thus gives insight into the regularizing properties of the underlying regularization method, and it is an aid in choosing an appropriate regularization parameter for the given data. In this chapter we summarize the main properties of the Lcurve, and demonstrate by examples its usefulness and its limitations both as an analysis tool and as a method for choosing the regularization parameter. 1 Introduction Practically all regularization methods for computing stable solutions to inverse problems involve a tradeoff between the "size" of the regularized solution and the quality of the fit that it provides to the given data. What distinguishes the various regularization methods is how...
Unfolding methods in high energy physics experiments
 in Proceedings of the 1984 CERN School of Computing, CERN 8509 (1985) and DESY
"... Finite detector resolution and limited acceptance require to apply unfolding methods in high energy physics experiments. Information on the detector resolution is usually given by a set of Monte Carlo events. Based on the experience with a widely used unfolding program (RUN) a modified method has be ..."
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Cited by 21 (0 self)
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Finite detector resolution and limited acceptance require to apply unfolding methods in high energy physics experiments. Information on the detector resolution is usually given by a set of Monte Carlo events. Based on the experience with a widely used unfolding program (RUN) a modified method has been developed. The first step of the method is a maximum likelihood fit of the Monte Carlo distributions to the measured distribution in one, two or three dimensions; the finite statistic of the Monte Carlo events is taken into account by the use of Barlows method with a new method of solution. A clustering method is used before to combine bins in sparsely populated areas. In the second step a regularization is applied to the solution, which introduces only a small bias. The regularization parameter is determined from the data after a diagonalization and rotation procedure. 1. THE UNFOLDING PROBLEM
On The Choice Of Subspace For Iterative Methods For Linear Discrete IllPosed Problems
 Int. J. Appl. Math. Comput. Sci
, 2001
"... . Many iterative methods for the solution of linear discrete illposed problems with a large matrix require the computed approximate solutions to be orthogonal to the null space of the matrix. We show that it may be possible to determine a meaningful approximate solution with less computational work ..."
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Cited by 20 (14 self)
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. Many iterative methods for the solution of linear discrete illposed problems with a large matrix require the computed approximate solutions to be orthogonal to the null space of the matrix. We show that it may be possible to determine a meaningful approximate solution with less computational work when this requirement is not imposed. Key words. Minimal residual method, conjugate gradient method, linear illposed problems. 1. Introduction. This paper is concerned with the design of iterative methods for the computation of approximate solutions of linear systems of equations Ax = b, A # R mn , x # R n , b # R m , (1.1) with a large matrix A of illdetermined rank. Thus, A has many "tiny" singular values of di#erent orders of magnitude. In particular, A is severely illconditioned. Some of the singular values of A may be vanishing. We allow m # n or m < n. The righthand side vector b is not required to be in the range of A. Linear systems of equations of the fo...
A Multiple Input Image Restoration Approach
 Journal of Visual Communication and Image Representation
, 1990
"... this paper image restoration applications, where multiple distorted versions of the same original image are available, are considered. A general adaptive restoration algorithm is derived on the basis of a set theoretic regularization technique. The adaptivity of the algorithm is introduced in two wa ..."
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Cited by 17 (5 self)
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this paper image restoration applications, where multiple distorted versions of the same original image are available, are considered. A general adaptive restoration algorithm is derived on the basis of a set theoretic regularization technique. The adaptivity of the algorithm is introduced in two ways: (a) by a constraint ,operator which incorporates properties of the response of the hu man visual system into the restoration process and (b) by a weight matrix which assigns greater importance for the deconvolution process to areas of high spatial activity than to areas of low spatial activity. Different degrees of trust are assigned to the various distorted images depending on the amounts of noise. The proposed algorithm is general and can be used for any type of linear distortion and constraint operators. It can also be used to restore signals other than images. Experimental results obtained by an iterative implementation of the proposed algorithms are pre sented. c 1990 Academic Press, Inc
An Affine Scaling Algorithm For Minimizing Total Variation In Image Enhancement
, 1994
"... . A computational algorithm is proposed for image enhancement based on total variation minimization with constraints. This constrained minimization problem is introduced by Rudin et al [13, 14, 15] to enhance blurred and noisy images. Our computational algorithm solves the constrained minimization p ..."
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Cited by 16 (1 self)
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. A computational algorithm is proposed for image enhancement based on total variation minimization with constraints. This constrained minimization problem is introduced by Rudin et al [13, 14, 15] to enhance blurred and noisy images. Our computational algorithm solves the constrained minimization problem directly by adapting the affine scaling method for the unconstrained l 1 problem [3]. The resulting computational scheme, when viewed as an image enhancement process, has the feature that it can be used in an interactive manner in situations where knowledge of the noise level is either unavailable or unreliable. This computational algorithm can be implemented with a conjugate gradient method. It is further demonstrated that the iterative enhancement process is efficient. Key Words. image enhancement, image reconstruction, deconvolution, minimal total variation, affine scaling algorithm, projected gradient method Department of Computer Science and Advanced Computing Research Institut...
A LargeScale TrustRegion Approach to the Regularization of Discrete IllPosed Problems
 RICE UNIVERSITY
, 1998
"... We consider the problem of computing the solution of largescale discrete illposed problems when there is noise in the data. These problems arise in important areas such as seismic inversion, medical imaging and signal processing. We pose the problem as a quadratically constrained least squares pro ..."
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Cited by 12 (4 self)
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We consider the problem of computing the solution of largescale discrete illposed problems when there is noise in the data. These problems arise in important areas such as seismic inversion, medical imaging and signal processing. We pose the problem as a quadratically constrained least squares problem and develop a method for the solution of such problem. Our method does not require factorization of the coefficient matrix, it has very low storage requirements and handles the high degree of singularities arising in discrete illposed problems. We present numerical results on test problems and an application of the method to a practical problem with real data.
Macromolecular SizeandShape Distributions by Sedimentation Velocity Analytical Ultracentrifugation
, 2006
"... ABSTRACT Sedimentation velocity analytical ultracentrifugation is an important tool in the characterization of macromolecules and nanoparticles in solution. The sedimentation coefficient distribution c(s) of Lamm equation solutions is based on the approximation of a single, weightaverage frictional ..."
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Cited by 12 (1 self)
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ABSTRACT Sedimentation velocity analytical ultracentrifugation is an important tool in the characterization of macromolecules and nanoparticles in solution. The sedimentation coefficient distribution c(s) of Lamm equation solutions is based on the approximation of a single, weightaverage frictional coefficient of all particles, determined from the experimental data, which scales the diffusion coefficient to the sedimentation coefficient consistent with the traditional s; M 2/3 power law. It provides a high hydrodynamic resolution, where diffusional broadening of the sedimentation boundaries is deconvoluted from the sedimentation coefficient distribution. The approximation of a single weightaverage frictional ratio is favored by several experimental factors, and usually gives good results for chemically not too dissimilar macromolecules, such as mixtures of folded proteins. In this communication, we examine an extension to a twodimensional distribution of sedimentation coefficient and frictional ratio, c(s,fr), which is representative of a more general set of sizeandshape distributions, including massStokes radius distributions, c(M,R S), and sedimentation coefficientmolar mass distributions c(s,M). We show that this can be used to determine average molar masses of macromolecules and characterize macromolecular distributions, without the approximation of any scaling relationship between hydrodynamic and thermodynamic parameters.