The effective choice of the smoothing norm in regularization

Author:
Jane Cullum

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

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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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Article copyright:
© Copyright 1979
American Mathematical Society