Acceleration of the Power Method with Dynamic Mode Decomposition

Abstract

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) $n$ 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($n$)] 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($n$) can reduce the number of power iterations required by a factor of approximately 5. However, the Arnoldi method always outperformed DMD-PM($n$) for an equivalent number of matrix-vector products Av. In other words, DMD-PM($n$) cannot compete with leading eigensolvers if one is not limited to snapshots produced by the PM. Contrarily, DMD-PM($n$) 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.

Keywords:

• Power method
• dynamic mode decomposition, acceleration

Acknowledgments

The material presented is based on work supported in part by a Faculty Development Grant awarded to Kansas State University by the U.S. Nuclear Regulatory Commission under award number NRC-HQ-60-17-G-0010.