  The linear functional strategy for the regularization of inverse problems is
considered. For selecting the regularization parameter therein, we propose the
heuristic quasi-optimality principle and some modifications including the
smoothness of the linear functionals. We prove convergence rates for the linear
functional strategy with these heuristic rules taking into account the
smoothness of the solution and the functionals and imposing a structural
condition on the noise. Furthermore, we study these noise conditions in both a
deterministic and stochastic setup and verify that for mildly-ill-posed
problems and Gaussian noise, these conditions are satisfied almost surely,
where on the contrary, in the severely-ill-posed case and in a similar setup,
the corresponding noise condition fails to hold. Moreover, we propose an
aggregation method for adaptively optimizing the parameter choice rule by
making use of improved rates for linear functionals. Numerical results indicate
that this method yields better results than the standard heuristic rule.
