  This paper derives a new class of adaptive regularization parameter choice
strategies that can be effectively and efficiently applied when regularizing
large-scale linear inverse problems by combining standard Tikhonov
regularization and projection onto Krylov subspaces of increasing dimension
(computed by the Golub-Kahan bidiagonalization algorithm). The success of this
regularization approach heavily depends on the accurate tuning of two
parameters (namely, the Tikhonov parameter and the dimension of the projection
subspace): these are simultaneously set using new strategies that can be
regarded as special instances of bilevel optimization methods, which are solved
by using a new paradigm that interlaces the iterations performed to project the
Tikhonov problem (lower-level problem) with those performed to apply a given
parameter choice rule (higher-level problem). The discrepancy principle, the
GCV, the quasi-optimality criterion, and Regi\'{n}ska criterion can all be
adapted to work in this framework. The links between Gauss quadrature and
Golub-Kahan bidiagonalization are exploited to prove convergence results for
the discrepancy principle, and to give insight into the behavior of the other
considered regularization parameter choice rules. Several numerical tests
modeling inverse problems in imaging show that the new parameter choice
strategies lead to regularization methods that are reliable, and intrinsically
simpler and cheaper than other strategies already available in the literature.
