  We study multi-parameter Tikhonov regularization, i.e., with multiple
penalties. Such models are useful when the sought-for solution exhibits several
distinct features simultaneously. Two choice rules, i.e., discrepancy principle
and balancing principle, are studied for choosing an appropriate
(vector-valued) regularization parameter, and some theoretical results are
presented. In particular, the consistency of the discrepancy principle as well
as convergence rate are established, and an a posteriori error estimate for the
balancing principle is established. Also two fixed point algorithms are
proposed for computing the regularization parameter by the latter rule.
Numerical results for several nonsmooth multi-parameter models are presented,
which show clearly their superior performance over their single-parameter
counterparts.
