  The Multiscale Hierarchical Decomposition Method (MHDM) was introduced as an
iterative method for total variation regularization, with the aim of recovering
details at various scales from images corrupted by additive or multiplicative
noise. Given its success beyond image restoration, we extend the MHDM iterates
in order to solve larger classes of linear ill-posed problems in Banach spaces.
Thus, we define the MHDM for more general convex or even non-convex penalties,
and provide convergence results for the data fidelity term. We also propose a
flexible version of the method using adaptive convex functionals for
regularization, and show an interesting multiscale decomposition of the data.
This decomposition result is highlighted for the Bregman iteration method that
can be expressed as an adaptive MHDM. Furthermore, we state necessary and
sufficient conditions when the MHDM iteration agrees with the variational
Tikhonov regularization, which is the case, for instance, for one-dimensional
total variation denoising. Finally, we investigate several particular instances
and perform numerical experiments that point out the robust behavior of the
MHDM.
