Full waveform inversion by proximal Newton method using adaptive regularization

SUMMARY Regularization is necessary for solving non-linear 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), smooth or non-smooth, variational-based or not. Nevertheless, tailoring a suitable and easy-to-implement prior for describing geophysical models is a non-trivial task. In this paper, we propose two generic optimization algorithms to implement arbitrary regularization in non-linear inverse problems such as full-waveform inversion (FWI), where the regularization task is recast as a denoising problem. We assess these optimization algorithms with the plug-and-play block matching (BM3D) regularization algorithm, which determines empirical priors adaptively without any optimization formulation. The non-linear inverse problem is solved with 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 non-linear objective function. The second relies on the alternating direction method of multipliers (ADMM), and hence Newton-ADMM (NADMM), where the least-squares optimization subproblem and the regularization subproblem in the composite objective function are decoupled through auxiliary variable and solved in an alternating mode. The least-squares subproblem can be solved with exact, inexact, or quasi-Newton methods. We compare NISTA and NADMM numerically by solving FWI with BM3D regularization. The tests show promising results obtained by both algorithms. However, NADMM shows a faster convergence rate than NISTA when using L-BFGS to solve the Newton system.