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Due to simplicity, computational cheapness, and efficiency, the Barzilai andBorwein (BB) gradient method has received a significant amount of attention indifferent fields of optimization. In the first part of this paper, based onspectral analysis, R-linear global convergence for the BB-method is proven forstrictly convex quadratic problems posed in infinite-dimensional Hilbertspaces. Then this result is strengthened to R-linear local convergence for aclass of twice continuously Fr\'echet-differentiable functions. In the secondpart, aiming at problems governed by partial differential equations (PDE), themesh-independent principle is investigated for the BB-method. The applicabilityof these results is demonstrated for three different types of PDE-constrainedoptimization problems. Numerical experiments illustrate the theoreticalresults.

Analysis and Performance of the Barzilai-Borwein Step-Size Rules for Optimization Problems in Hilbert Spaces