The effective choice of the smoothing norm in regularization

Abstract

We consider ill-posed problems of the form \[ g ( t ) = ∫ 0 1 K ( t , s ) f ( s ) d s , 0 ⩽ t ⩽ 1 , g(t) = \int _0^1 {K(t,s)f(s)ds,\quad 0 \leqslant t \leqslant 1,} \] 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 ( ‖ K f − g ‖ 2 + α Ω ( f ) ) ({\left \| {Kf - g} \right \|^2} + \alpha \Omega (f)) -where Ω \Omega is a smoothing norm chosen by the user. We demonstrate by example that the choice of Ω \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 Ω \Omega depends upon the character of the compactness of K and upon the smoothness of the desired solution.