A fast two-point gradient method for solving non-smooth nonlinear ill-posed problems
A fast two-point gradient method for solving non-smooth nonlinear ill-posed problems
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DOI：
10.1016/j.cam.2020.113114
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发表时间：
2021-03
期刊：
影响因子：
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通讯作者：
Haie Long;B. Han;Li Li-Li
中科院分区：
文献类型：
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作者：
Haie Long;B. Han;Li Li-Li
提出并分析了一种求解非光滑不适定反问题的快速两点梯度(TPG)方法，其中正向算子仅是方向性的，而G？teaux是不可微的。这种方法被看作是无导数的Landweber迭代和Nester ov加速方案的一般情况的组合。由于前向映射在我们的例子中是不可微的，所以标准分析不适用于收敛分析。因此，在组合参数的某些假设下，我们还借助渐近稳定性的概念和广义切锥条件，给出了该方法的一个新的收敛分析。TPG方法的设计涉及组合参数的选择，文中对此进行了详细的讨论。此外，将TPG方法应用于非光滑半线性椭圆型偏微分方程组的逆源问题，其中Bouligand次微分可以用来代替不存在的G？teaux导数，并且相应的Bouligand两点梯度迭代是收敛的正则化格式。数值模拟结果表明，该方法优于Bouligand-Landweber迭代法。
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.