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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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The effective choice of the smoothing norm in regularization
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by Jane Cullum PDF
Math. Comp. 33 (1979), 149-170 Request permission

Abstract:

We consider ill-posed problems of the form \[ g(t) = \int _0^1 {K(t,s)f(s)ds,\quad 0 \leqslant t \leqslant 1,} \] where g and K are given, and we must compute f. The Tikhonov regularization procedure replaces (1) by a one-parameter family of minimization problems-Minimize $({\left \| {Kf - g} \right \|^2} + \alpha \Omega (f))$-where $\Omega$ is a smoothing norm chosen by the user. We demonstrate by example that the choice of $\Omega$ is not simply a matter of convenience. We then show how this choice affects the convergence rate, and the condition of the problems generated by the regularization. An appropriate choice for $\Omega$ depends upon the character of the compactness of K and upon the smoothness of the desired solution.
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Additional Information
  • © Copyright 1979 American Mathematical Society
  • Journal: Math. Comp. 33 (1979), 149-170
  • MSC: Primary 65R20; Secondary 41A25, 65D25
  • DOI: https://doi.org/10.1090/S0025-5718-1979-0514816-1
  • MathSciNet review: 514816