On a new notion of regularizer
A. G. Ramm
TL;DR
The paper addresses ill-posed inverse problems by introducing a new notion of regularizer that bounds the worst-case deviation over all feasible $v$ in a compact set ${\mathcal K}$, rather than a single true solution. It provides a constructive framework (via a penalized objective $F_\delta(v)$) to obtain regularizers in this new sense and proves convergence leveraging the modulus of continuity of the inverse on the image of ${\mathcal K}$. The stable numerical differentiation example shows that for $a>1$ one can achieve vanishing rates $\eta(\delta) \to 0$, while for $a=1$ the new notion is unattainable, highlighting the practical limits. Overall, the work connects theoretical regularization with data-driven feasible sets and offers a robust method for constructing regularizers in nonlinear, possibly noninjective settings.
Abstract
A new understanding of the notion of regularizer is proposed. It is argued that this new notion is more realistic than the old one and better fits the practical computational needs. An example of the regularizer in the new sense is given. A method for constructing regularizers in the new sense is proposed and justified.