Digitale Medien
New York, NY [u.a.]
:
Wiley-Blackwell
Numerical Linear Algebra with Applications
1 (1994), S. 247-263
ISSN:
1070-5325
Schlagwort(e):
Engineering
;
Engineering General
Quelle:
Wiley InterScience Backfile Collection 1832-2000
Thema:
Mathematik
Notizen:
This paper presents a row relaxation method for solving the regularized ℓp least norm problem \documentclass{article}\pagestyle{empty}\begin{document}$$ {\rm minimize}P({\rm x}) = \frac{1}{2}\varepsilon \parallel {\rm x}\parallel _{\rm 2}^{\rm 2} + \parallel A{\rm x} - {\rm b}\parallel _p^p /p $$\end{document} where e and p are positive constants, 1〈p〈∞ The interest that we have in this problem lies in the observation that for small values of E the minimizer of P (X ) is a good substitute for a minimizer of the unregularized problem \documentclass{article}\pagestyle{empty}\begin{document}$$ {\rm minimize }U({\rm x}) = \parallel A{\rm x} - {\rm b}\parallel _p^p /p $$\end{document} It is shown that the dual of the regularized problem has the form \documentclass{article}\pagestyle{empty}\begin{document}$$ {\rm minimize }D({\rm y}) = {\rm b}^T {\rm y} - \frac{1}{2}\varepsilon \parallel A^T {\rm y/}\varepsilon \parallel _2^2 - \parallel {\rm y}\parallel _q^q /q $$\end{document} where q = p /(p - 1). Moreover, if y solves the dual problem then X = ATy/∊ solves the primal problem and P(ATy/∊) = D(y). Maximizing the dual objective function by changing one variable at a time results in a row relaxation method that resembles Kaczmarz's method. This feature makes the new method suitable for solving large sparse C, problems that arise in computerized tomography, geophysics, and groundwater hydrology. Numerical experiments illustrate the feasibility of our ideas.
Zusätzliches Material:
5 Tab.
Materialart:
Digitale Medien
URL:
http://dx.doi.org/10.1002/nla.1680010304
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