Abstract
A method for computing the singular values and singular 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.
Zusammenfassung
Eine Methode zur Berechnung der singulären Werte und der singulären Funktionen von reellen, quadratisch-integrierbaren Kernen wird dargestellt. Die Analyse zeigt, daß eine „gute” Diskretisierung immer eine Matrix ergibt, deren Singulärwert-Zerlegung mit der Singulärwert-Entwicklung der Kerne eng verbunden ist. Dieser Zusammenhang ist wesentlich, wenn schlecht gestellte Probleme zu lösen sind, weil er zeigt, daß eine Regularisierung des von einer Integralgleichung hergeleiteten algebraischen Problems äquivalent ist zu einer Regularisierung der Integralgleichung.
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The work was supported by the Danish Space Board.
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Hansen, P.C. Computation of the singular value expansion. Computing 40, 185–199 (1988). https://doi.org/10.1007/BF02251248
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DOI: https://doi.org/10.1007/BF02251248