Aiming at optimization problems governed by partial differential equations (PDEs), local R-linear convergence of the Barzilai–Borwein (BB) method for a class of twice continuously Fréchet-differentiable functions is proven. Relying on this result, the mesh-independent principle for the BB-method is investigated. The applicability of the theoretical results is demonstrated for two different types of PDE-constrained optimization problems. Numerical experiments are given, which illustrate the theoretical results.

中文翻译：

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用于 PDE 约束优化的 Barzilai-Borwein 方法的收敛性和网格无关性

针对偏微分方程（PDE）控制的优化问题，证明了Barzilai-Borwein（BB）方法对一类连续两次Fréchet-微分函数的局部R-线性收敛。以此结果为基础，研究了 BB 方法的网格无关原理。理论结果的适用性证明了两种不同类型的 PDE 约束优化问题。给出了数值实验，说明了理论结果。