  We exploit the similarities between Tikhonov regularization and Bayesian
hierarchical models to propose a regularization scheme that acts like a
distributed Tikhonov regularization where the amount of regularization varies
from component to component. In the standard formulation, Tikhonov
regularization compensates for the inherent ill-conditioning of linear inverse
problems by augmenting the data fidelity term measuring the mismatch between
the data and the model output with a scaled penalty functional. The selection
of the scaling is the core problem in Tikhonov regularization. If an estimate
of the amount of noise in the data is available, a popular way is to use the
Morozov discrepancy principle, stating that the scaling parameter should be
chosen so as to guarantee that the norm of the data fitting error is
approximately equal to the norm of the noise in the data. A too small value of
the regularization parameter would yield a solution that fits to the noise
while a too large value would lead to an excessive penalization of the
solution. In many applications, it would be preferable to apply distributed
regularization, replacing the regularization scalar by a vector valued
parameter, allowing different regularization for different components of the
unknown, or for groups of them. A distributed Tikhonov-inspired regularization
is particularly well suited when the data have significantly different
sensitivity to different components, or to promote sparsity of the solution.
The numerical scheme that we propose, while exploiting the Bayesian
interpretation of the inverse problem and identifying the Tikhonov
regularization with the Maximum A Posteriori (MAP) estimation, requires no
statistical tools. A combination of numerical linear algebra and optimization
tools makes the scheme computationally efficient and suitable for problems
where the matrix is not explicitly available.
