Quantifying information content for ill-posed problems

Abstract

The notion of the 'effective rank' for the discretization of an ill-posed operator equation Kf=g is introduced as a means of quantifying information content for the problem. For operators K with a singular value decomposition effective rank is a computable quantity which depends on the singular values of K, the regularization method being applied, and an error amplification ratio, which relates error in the solution to noise in the data. Examples are presented in which the effective rank is computed for the backward heat equation and for second differentiation of data.