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.

中文翻译：

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使用自适应正则化通过近端牛顿法进行全波形反演

正则化对于解决地球科学不同领域中出现的非线性不适定逆问题是必要的。合适的正则化的基础是正则化器表达的先验，它可以是非自适应的或自适应的（数据驱动的）。在本文中，我们提出了用于解决非线性逆问题的通用黑盒正则化算法，例如全波形反演 (FWI)，该算法允许通过复杂的去噪算法自适应地确定经验先验。非线性逆问题通过近端牛顿法解决，该方法通过算子分裂和近端映射将传统的牛顿步骤推广到涉及（可能不可微的）正则化函数的梯度/次梯度。此外，它需要考虑正则化最小二乘优化问题中的 Hessian 矩阵。我们为此任务提出了两种不同的分割算法。首先，我们使用基于一阶广义迭代收缩阈值算法（ISTA）的迭代方法计算牛顿搜索方向，因此牛顿-ISTA（NISTA）。迭代只需要 Hessian 向量积来计算非线性目标函数的二次近似的梯度步长。第二种依赖于乘法器的交替方向方法（ADMM），因此牛顿-ADMM（NADMM），其中最小二乘优化子问题和复合中的正则化子问题通过辅助变量解耦并以交替模式求解。我们通过使用 BM3D 正则化求解全波形反演，在数值上比较 NISTA 和 NADMM。测试显示两种算法都获得了有希望的结果。但是，在使用 L-BFGS 求解牛顿系统时，NADMM 显示出比 Newton-ISTA 更快的收敛速度。