  We consider the method of quasi-solutions (also referred to as Ivanov
regularization) for the regularization of linear ill-posed problems in
non-reflexive Banach spaces. Using the equivalence to a metric projection onto
the image of the forward operator, it is possible to show regularization
properties and to characterize parameter choice rules that lead to a convergent
regularization method, which includes the Morozov discrepancy principle.
Convergence rates in a suitably chosen Bregman distance can be obtained as
well. We also address the numerical computation of quasi-solutions to inverse
source problems for partial differential equations in $L^\infty(\Omega)$ using
a semismooth Newton method and a backtracking line search for the parameter
choice according to the discrepancy principle. Numerical examples illustrate
the behavior of quasi-solutions in this setting.
