Adaptive eigenvalue computation: complexity estimates

This paper is concerned with the design and analysis of a fully adaptive eigenvalue solver for linear symmetric operators. After transforming the original problem into an equivalent one formulated on ℓ ₂, the space of square summable sequences, the problem becomes sufficiently well conditioned so that a gradient type iteration can be shown to reduce the error by some fixed factor per step. It then remains to realize these (ideal) iterations within suitable dynamically updated error tolerances. It is shown under which circumstances the adaptive scheme exhibits in some sense asymptotically optimal complexity. This research was supported in part by the Leibniz Programme of the DFG, by the SFB 401 funded by DFG, the DFG Priority Program SPP1145 and by the EU NEST project BigDFT.