Review of Fredholm Equations of the First Kind

Hoog, Frank de

doi:10.1007/978-94-009-9130-9_6

Cited by 18 publications

(12 citation statements)

References 13 publications

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“…and write the solution as a mean convergent expansion It can be shown [6] that the sequence of singular values {P.i} has zero as its only possible limit point (which means that the nonzero values #i--* 0 for i--, ao) and that the "smoother" the kernel, the faster the singular values decay to 0. Further, it is a general observation that the singular functions {u,} and {v,} oscillate with increasing frequency as i--+ m. Since the mapping of f, associated with K, can be written in terms of the SVE as…”

Section: The Significance Of the Mm-svd Algorithmmentioning

confidence: 99%

Computation of the singular value expansion

Hansen

1988

Computing

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--ZusammenfassungComputation of the Singular Value Expansion. A method for computing the singular values and singCar functions of real square-integrable kernels is presented. The analysis shows that a "good" discretization always yields a matrix whose singular value decomposition is closely related to the singular value expansion of the kernel. This relationship is important in connection with the solution of ill-posed problems since it shows that regularization of the algebraic problem, derived from an integral equation, is equivalent to regularization of the integral equation itself. AMS Subject Classifications." 65R20, 65F30.Key words." Fredholm integral equations of the first kind, singular value expansion, singular value decomposition, method of moments, regularization.Bereehnung der SinguNrwert-Entwieklung. Eine Methode znr Berechnung der singuI/iren Werte nnd der singulfiren Funktionen yon reellen, quadratisch-integrierbaren Kernen wird dargestellt. Die Analyse zeigt, dab eine ,,gute" Diskretisierung immer eine Matrix ergibt, deren Singul~irwert-Zerlegung mit der Singul/irwert-Entwicklung der Kerne eng verbunden ist. Dieser Zusammenhang ist wesentlich, wenn schlecht gestellte Probleme zu 16sen sind, well er zeigt, dab eine Regularisierung des von einer Integralgleichung hergeleiteten algebraischen Problems ~iquivalent ist zu einer Regularisierung der integralgleichung.

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Section: The Significance Of the Mm-svd Algorithmmentioning

confidence: 99%

Computation of the singular value expansion

Hansen

1988

Computing

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“…Consider now the choice of X for fixed p. When the data function is inexact, which is always the case in practical situations, the choice of X is often critical. In the numerical experiments of de Hoog [11] it is shown that the best choice of X can depend quite crucially on the actual errors present in the data rather than just their statistical properties. Any method of optimizing X should therefore take account of the data being analyzed.…”

Section: Ii/-/ii-(1-6)mentioning

confidence: 99%

Optimisation in the regularisation ill-posed problems

Davies

Anderssen

1986

J. Aust. Math. Soc. Series B, Appl. Math

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We survey the role played by optimization in the choice of parameters for Tikhonov regularization of first-kind integral equations. Asymptotic analyses are presented for a selection of practical optimizing methods applied to a model deconvolution problem. These methods include the discrepancy principle, cross-validation and maximum likelihood. The relationship between optimality and regularity is emphasized. New bounds on the constants appearing in asymptotic estimates are presented.

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“…Consequently, many methods are proposed which, on the evidence of a few specific cases, are claimed to be effective. But it is also known that the results are quite sensitive to the choice of method or even to the choice of certain parameters within a given algorithm (de Hoog [1]). Thus, any computed result must be regarded with a great deal of suspicion.…”

Section: Introductionmentioning

confidence: 99%

“…Specifically, equation (1) has to be wellposed, meaning that the solution is unique and depends continuously on the right hand side. In practice many problems do not satisfy this criterion.…”

Section: Introductionmentioning

confidence: 99%

Uncertainty in the solution of linear operator equations

Linz

1984

BIT

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Abstract.In this paper we introduce and explore the concept of uncertainty of an approximate solution of a linear operator equation. This uncertainty is a measure of the difference between a computed solution and other plausible answers. When the operator equation is wellposed, the uncertainty is closely related to the more traditional notion of the accuracy of an approximate solution. However, for illposed equations, the uncertainty concept gives a new way of looking at such problems and helps in assessing the meaning of the computed results.

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Review of Fredholm Equations of the First Kind

Cited by 18 publications

References 13 publications

Computation of the singular value expansion

Computation of the singular value expansion

Optimisation in the regularisation ill-posed problems

Uncertainty in the solution of linear operator equations

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