In this work we present some recent results on multilevel (ML) algorithms for a class of inverse problems (IPs) with explicit non-negativity constraints imposed on the control. The IPs under scrutiny are formulated as regularized least-squares minimization problems
For a large number of applications, the explicit presence of
bound-constraints in the IP-formulation can be critical. For example,
in the inverse contamination problem [2], where the control
represents the initial concentration of an air pollutant,
non-negativity constraints not only render a physically meaningful
solution, but are essential for the correct recovery of spatially
localized solutions. The question addressed in the presented research
is whether the qualitative behavior established for the unconstrained
problem () can be reproduced in the case of the constrained
IP (
). The problem (
) is solved via the semi-smooth Newton
method described in [3] as an outer iteration with an
ML-preconditioned CG algorithm being used for solving the linear systems
arising at each outer iteration.