Acceleration of the Power Method with Dynamic Mode Decomposition

An algorithm based on dynamic mode decomposition (DMD) for acceleration of the power method (PM) is presented. The PM is a simple technique for determining the dominant eigenmode of an operator A, and variants of the PM are widely used in reactor analysis. DMD is an algorithm for decomposing a time series of spatially dependent data and producing an explicit-in-time reconstruction for that data. By viewing successive PM iterates as snapshots of a time-varying system tending toward a steady state, DMD can be used to predict that steady state using (sometimes surprisingly small) iterates. The process of generating snapshots with the PM and extrapolating forward with DMD can be repeated. The resulting restarted, DMD-accelerated PM [or DMD-PM()] was applied to the two-dimensional International Atomic Energy Agency diffusion benchmark and compared to the unaccelerated PM and the Arnoldi method. Results indicate that DMD-PM() can reduce the number of power iterations required by a factor of approximately 5. However, the Arnoldi method always outperformed DMD-PM() for an equivalent number of matrix-vector products Av. In other words, DMD-PM() cannot compete with leading eigensolvers if one is not limited to snapshots produced by the PM. Contrarily, DMD-PM() can be readily applied as a postprocess to existing PM applications for which the Arnoldi method and similar methods are not directly applicable. A slight variation of the method was also found to produce reasonable approximations to the first and second harmonics without substantially affecting convergence of the dominant mode.