  Regularization is necessary for solving nonlinear ill-posed inverse problems
arising in different fields of geosciences. The base of a suitable
regularization is the prior expressed by the regularizer, which can be
non-adaptive or adaptive (data-driven). In this paper, we propose general
black-box regularization algorithms for solving nonlinear inverse problems such
as full-waveform inversion (FWI), which admit empirical priors that are
determined adaptively by sophisticated denoising algorithms. The nonlinear
inverse problem is solved by a proximal Newton method, which generalizes the
traditional Newton step in such a way to involve the gradients/subgradients of
a (possibly non-differentiable) regularization function through operator
splitting and proximal mappings. Furthermore, it requires to account for the
Hessian matrix in the regularized least-squares optimization problem. We
propose two different splitting algorithms for this task. In the first, we
compute the Newton search direction with an iterative method based upon the
first-order generalized iterative shrinkage-thresholding algorithm (ISTA), and
hence Newton-ISTA (NISTA). The iterations require only Hessian-vector products
to compute the gradient step of the quadratic approximation of the nonlinear
objective function. The second relies on the alternating direction method of
multipliers (ADMM), and hence Newton-ADMM (NADMM), where the least-square
optimization subproblem and the regularization subproblem in the composite are
decoupled through auxiliary variable and solved in an alternating mode. We
compare NISTA and NADMM numerically by solving full-waveform inversion with
BM3D regularizations. The tests show promising results obtained by both
algorithms. However, NADMM shows a faster convergence rate than Newton-ISTA
when using L-BFGS to solve the Newton system.
