The Effective Choice of the Smoothing Norm in Regularization

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.