Adaptive Eigenvalue Computation - Complexity Estimates

TL;DR

This paper develops an adaptive eigenvalue solver for symmetric operators, transforming the problem into a well-conditioned form on bcl_2, and demonstrates conditions for asymptotically optimal complexity.

Contribution

It introduces a fully adaptive eigenvalue computation method with complexity analysis based on problem transformation and dynamic error control.

Findings

The adaptive scheme reduces error at a fixed rate per iteration.

Under certain conditions, the method achieves asymptotically optimal complexity.

The problem transformation to bcl_2 makes the iterative process well-conditioned.

Abstract

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 $\ell_2$, 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.