We propose and analyze a fast two-point gradient (TPG) method for solving non-smooth ill-posed inverse problems where the forward operator is merely directionally but not Gâteaux differentiable. This method is seen as a combination of a derivative-free Landweber iteration and a general case of Nesterov’s acceleration scheme. Since the forward mapping is not Gâteaux differentiable in our case, the standard analysis is not applicable to the convergence analysis. Under certain assumptions on the combination parameters, we therefore provide a new convergence analysis of the proposed method also with the help of the concept of asymptotic stability and a generalized tangential cone condition. The design of the TPG method involves the choices of the combination parameters which are carefully discussed. Moreover, the TPG method is applied to an inverse source problem for a non-smooth semilinear elliptic PDE where a Bouligand subdifferential can be used in place of the non-existing Gâteaux derivative, and the corresponding Bouligand two-point gradient iteration is shown to be a convergent regularization scheme. Numerical simulations are presented to illustrate the advantages over the Bouligand–Landweber iteration.

我们提出并分析了一种快速的两点梯度（TPG）方法，用于解决非光滑不适定逆问题，其中前向算符只是方向性的，而Gâteaux则不是可微的。该方法被视为无导数的Landweber迭代与Nesterov加速方案的一般情况的组合。由于在我们的情况下正向映射无法与Gâteaux区分，因此标准分析不适用于收敛分析。因此，在某些关于组合参数的假设下，我们还借助渐近稳定性和广义切向锥条件的概念，为提出的方法提供了一种新的收敛性分析。TPG方法的设计涉及对组合参数的选择，并进行了详细讨论。此外，TPG方法应用于非光滑半线性椭圆PDE的逆源问题，其中可以使用Bouligand次微分代替不存在的Gâteaux导数，并且相应的Bouligand两点梯度迭代被证明是收敛的正则化方案。进行了数值模拟，以说明在Bouligand–Landweber迭代方面的优势。