  In this paper we present a globally convergent algorithm for the computation
of a minimizer of the Tikhonov functional with sparsity promoting penalty term
for nonlinear forward operators in Banach space. The dual TIGRA method uses a
gradient descent iteration in the dual space at decreasing values of the
regularization parameter $\alpha_j$, where the approximation obtained with
$\alpha_j$ serves as the starting value for the dual iteration with parameter
$\alpha_{j+1}$. With the discrepancy principle as a global stopping rule the
method further yields an automatic parameter choice. We prove convergence of
the algorithm under suitable step-size selection and stopping rules and
illustrate our theoretic results with numerical experiments for the nonlinear
autoconvolution problem.
