ABSTRACT In this paper, we establish an initial theory regarding the second-order asymptotical regularization (SOAR) method for the stable approximate solution of ill-posed linear operator equations in Hilbert spaces, which are models for linear inverse problems with applications in the natural sciences, imaging and engineering. We show the regularizing properties of the new method, as well as the corresponding convergence rates. We prove that, under the appropriate source conditions and by using Morozov's conventional discrepancy principle, SOAR exhibits the same power-type convergence rate as the classical version of asymptotical regularization (Showalter's method). Moreover, we propose a new total energy discrepancy principle for choosing the terminating time of the dynamical solution from SOAR, which corresponds to the unique root of a monotonically non-increasing function and allows us to also show an order optimal convergence rate for SOAR. A damped symplectic iterative regularizing algorithm is developed for the realization of SOAR. Several numerical examples are given to show the accuracy and the acceleration effect of the proposed method. A comparison with other state-of-the-art methods are provided as well.

中文翻译：

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关于线性不适定逆问题的二阶渐近正则化

摘要 在本文中，我们建立了关于希尔伯特空间中不适定线性算子方程稳定近似解的二阶渐近正则化 (SOAR) 方法的初始理论，该方程是线性逆问题的模型，在自然科学中的应用，成像和工程。我们展示了新方法的正则化特性，以及相应的收敛速度。我们证明，在适当的源条件下，通过使用 Morozov 的常规差异原理，SOAR 表现出与渐近正则化的经典版本（Showalter 方法）相同的幂型收敛速度。此外，我们提出了一个新的总能量差异原则，用于选择来自 SOAR 的动态解决方案的终止时间，它对应于单调非递增函数的唯一根，并允许我们显示 SOAR 的阶次最优收敛速度。为实现SOAR，开发了一种阻尼辛迭代正则化算法。给出了几个数值例子来说明所提出方法的准确性和加速效果。还提供了与其他最先进方法的比较。