We consider a quasi-Newton method for minimization problems
arising in nonlinear, regularized distributed parameter
inverse problems. The regularization methods to be
considered are Tikhonov and Total Variation; the
latter may be viewed as an adaptive Tikhonov regularization.
The minimization method to be considered is a quasi-Newton
method which on each iteration results in a large, non-sparse
linear system in "**du"**, of the form

[ H(u) + A(u) ] du = -g(u).

The method to to be discussed combines
the inverse of **A(u)**
(or a fast approximation thereof) with a low-rank approximation
of **H(u)**
via the Sherman-Morrison formula,
and a Broyden-type update of the approximation **H(u)**.
Numerical examples will be presented.