通过隐式类 Newton 惯性动力系统及其离散化求解极大共单调包含问题
Solving maximally comonotone inclusion problems via an implicit Newton-like inertial dynamical system and its discretization
AI总结：
针对 Hilbert 空间中的极大共单调包含问题，提出隐式类 Newton 惯性动力系统并离散化为快速惯性算法，证明轨迹收敛及估计结果。
摘要：
本文研究 Hilbert 空间中由极大共单调包含问题支配的隐式类 Newton 惯性动力系统。在适当条件下，我们不仅建立了沿系统轨迹的速度以及相关 Yosida 正则化算子取值的逐点估计和积分估计，还证明了轨迹弱收敛到该极大共单调算子的一个零点。此外，通过对所提出系统进行时间离散化，发展了一种新的惯性算法。分析表明，所得离散算法展现出与连续时间对应情形相匹配的快速收敛性质。最后，通过数值实验说明了这些理论结果。
英文摘要：
This paper deals with an implicit Newton-like inertial dynamical system governed by a maximally comonotone inclusion problem in a Hilbert space. Under suitable conditions, we establish not only pointwise estimates and integral estimates for the velocity and the value of the associated Yosida regularization operator along the trajectory of the system, but also the weak convergence of the trajectory to a zero of the maximally comonotone operator. Moreover, a new inertial algorithm is developed via a time discretization of the proposed system. Our analysis reveals that the resulting discrete algorithm exhibits fast convergence properties matching the ones of the continuous time counterpart. Finally, the theoretical results are illustrated by numerical experiments.
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