The effective choice of the smoothing norm in regularization

We consider ill-posed problems of the form (1) g(t) = jQKit, s)fis)ds, 0 < t « 1, where g and K are given, and we must compute /. The Tikhonov regularizaron procedure replaces (1) by a one-parameter family of minimization problems— 2 Minimize i\\Kf g\\ + aft(/))—where ft is a smoothing norm chosen by the user. We demonstrate by example that the choice of ft 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 ft depends upon the character of the compactness of K and upon the smoothness of the desired solution.