A linear regularization scheme for inverse problems with unbounded linear operators on Banach spaces

This paper extends the linear regularization scheme known as the approximate inverse to unbounded linear operators on Banach spaces. The principle of feature reconstruction is adapted from bounded operators to the unbounded scenario and, in addition, a new situation is examined where the data need to be pre-processed to fit into the mathematical model. In all these cases, invariance and regularization properties are surveyed and established for the example of fractional differentiation. Numerical results confirm the derived characteristics of the presented methods.