Comparison of Reduced- and Full-space Algorithms for PDE-constrained Optimization

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PDE-constrained optimization problems are often solved using reduced-space quasi- Newton algorithms. Quasi-Newton methods are eective for problems with relatively few degrees of freedom, but their performance degrades as the problem size grows. In this paper, we compare two inexact-Newton algorithms that avoid the algorithmic scaling is- sues of quasi-Newton methods. The two inexact-Newton algorithms are distinguished by reduced-space and full-space implementations. Numerical experiments demonstrate that the full-space (or one-shot) inexact-Newton algorithm is typically the most ecient ap- proach; however, the reduced-space algorithm is an attractive compromise, because it requires less intrusion into existing solvers than the full-space approach while retaining excellent algorithmic scaling. We also highlight the importance of using inexact-Hessian- vector products in the reduced-space.