This paper considers the numerical solution of inverse problems with an L1 data fitting term, whichis challenging due to the lack of differentiability of the objective functional. Utilizing convex duality,the problem is reformulated as minimizing a smooth functional with pointwise constraints,which can be efficiently solved using a semismooth Newton method. In order to achieve superlinearconvergence, the dual problem requires additional regularization. For both the primal and the dualproblems, the choice of the regularization parameters is crucial. We propose adaptive strategiesfor choosing these parameters. The regularization parameter in the primal formulation is chosenaccording to a balancing principle derived from the model function approach, whereas the one in thedual formulation is determined by a path-following strategy based on the structure of the optimalityconditions. Several numerical experiments confirm the efficiency and robustness of the proposedmethod and adaptive strategy.