Prediction Bands for ILL-Posed Problems

Abstract

Consider the linear ill-posed operator equation

$$Kx = z$$

(1.1)

where K: X→Yis an operator between two Hilbert spaces with the inner product denoted by

$$\left\langle {x,y} \right\rangle ,\,x,y\, \in \mathcal{X}\,or\,x,y\, \in \,\mathcal{Y}\,$$

The ill-posedness of the problem (see [1]) means that a small perturbation in the data zmay result in large changes in the solution to (1.1). This discontinuous dependence of the solution on the data requires regularization in order to approximately solve the ill-posed equation.