AI 中文总结
本文分析障碍问题的Bregman近端点方法，建立子问题的适定性，基于序列严格局部极小性推导H^1范数下的次线性及线性收敛速率，并证明其尖锐性。
AI 中文摘要
我们研究了障碍问题的Bregman近端点方法，障碍问题是一个在接触力学、最优设计和数学金融中出现的基本变分不等式。每一步Bregman近端步骤通过由Legendre函数生成的Bregman散度来正则化能量，从而产生一个半线性椭圆子问题。我们首先为这些子问题建立了适定性和严格可行性理论。然后我们分析了所得迭代在$H^1$-范数下的收敛性。我们的收敛性分析基于一个序列严格局部极小性不等式。我们抽象地展示了极小性的阶如何决定收敛速率。假设初始猜测位于精确解之上且满足单侧Bregman增长条件，我们建立了阶为$s \geq 2$的序列严格局部极小性，并推导出$H^1$-范数下的次线性收敛速率，形式为$O(k^{-\zeta})$，其中$\zeta\in(1/2,1]$取决于Legendre函数的选择。对于Shannon和Tsallis熵，我们证明了这些速率在均匀最坏情形意义下是尖锐的。在关于自由边界和障碍的额外假设下，我们使用Shannon和Spence熵建立了阶为$s = 1$的序列严格局部极小性。这产生了形式为$O(\varrho^k)$的线性收敛，其中$\varrho\in(0,1)$。
英文摘要
We study the Bregman proximal point method for the obstacle problem, a fundamental variational inequality arising in contact mechanics, optimal design, and mathematical finance. Each Bregman proximal step regularizes the energy through a Bregman divergence generated by a Legendre function, leading to a semilinear elliptic subproblem. We first establish a well-posedness and strict feasibility theory for these subproblems. We then analyze the convergence of the resulting iteration in the $H^1$-norm. Our convergence analysis is based on a sequential strict local minimality inequality. We show abstractly how the order of minimality determines the convergence rate. Assuming that the initial guess lies above the exact solution and a one-sided Bregman growth condition holds, we establish sequential strict local minimality of order $s \geq 2$ and derive sublinear convergence rates in the $H^1$-norm of the form $O(k^{-ζ})$, where $ζ\in(1/2,1]$ depends on the choice of Legendre function. For the Shannon and Tsallis entropies, we show that these rates are sharp in a uniform worst-case sense. Under additional assumptions on the free boundary and the obstacle, we establish sequential strict local minimality of order $s = 1$ with the Shannon and Spence entropies. This yields linear convergence of the form $O(\varrho^k)$ for some $\varrho\in(0,1)$.
