A randomized time-domain algorithm for efficiently computing resolvent modes

Abstract

© 2021, American Institute of Aeronautics and Astronautics Inc.. All rights reserved.We introduce a new algorithm for computing resolvent modes of large systems based on randomized singular value decomposition (RSVD) combined with a time-marching method. The most expensive steps of the RSVD algorithm in the context of resolvent analysis, which constitute a bottleneck in its application to large systems, are replaced by leveraging the time-domain equations that have given rise to the resolvent operator. Specifically, the actions of the resolvent operator and its adjoint on a vector are obtained by equivalent direct and adjoint marching operations in the time domain. Our algorithm exploits streaming calculations to alleviate memory issues emerging for large systems, and we develop strategies to minimize the time-stepping cost while maintaining a desired level of accuracy. We validated our proposed algorithm by comparing the resolvent modes and gains of a Ginzburg-Landau model problem to those obtained from RSVD. Then, we use an axisymmetric jet and a three-dimensional extension thereof to assess and demonstrate the accuracy, cost, and memory efficiency of our new algorithm when applied to a high-dimensional system. In the three-dimensional case, we achieve orders-of-magnitude reduction in both CPU and memory usage compared to a direct application of RSVD.