In this paper we consider ill-posed inverse problems, both linear and nonlinear, by a heavy ball method in which a strongly convex regularization function is incorporated to detect the feature of the sought solution. We develop ideas on how to adaptively choose the step-sizes and the momentum coefficients to achieve acceleration over the Landweber-type method. We then analyze the method and establish its regularization property when it is terminated by the discrepancy principle. Various numerical results are reported which demonstrate the superior performance of our method over the Landweber-type method by reducing substantially the required number of iterations and the computational time.

翻译：本文考虑通过重球法求解线性和非线性不适定逆问题，其中引入强凸正则化函数以探测待求解的特征。我们提出如何自适应选择步长和动量系数以实现比Landweber型方法更快收敛的思路。进一步分析该方法，并建立其在基于偏差原则终止时的正则化性质。数值实验结果表明，与Landweber型方法相比，本方法在显著减少所需迭代次数和计算时间方面展现出更优性能。