The Choice of Smoothing Norm in Regularization -- A Key to Effectiveness.

摘要

Consider ill-posed problems of the form g(t) = integral from 0 to 1 of K(t,s)f(s)ds 0 < or = 1 (1) where g is given and f must be computed. The Tikhonov regularization procedure replaces (1) by a one-parameter family of minimization problems -- Minimize (Kf-g) absolute value squared + alpha Omega(f) -- where Omega is a smoothing norm chosen by the user. It is demonstrated by example that the choice of Omega is not simply a matter of convenience. This choice is shown to affect 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.