Approximating the resolvent operator with hierarchically semi-separable matrix representation and randomized sketching*

Resolvent analysis has become an indispensable technique to understand the dynamics of turbulent flows.  Computational challenges with resolvent analysis remain in tackling large-scale problems arising from high Reynolds numbers and multi-dimensional external flows.  Here, we consider the use of a hierarchically semi-separable (HSS) matrix representation for the linear operator with randomized sketching to extend the applicability of resolvent analysis. The HSS representation considers a binary index tree to extract a hierarchical series of diagonal and off-diagonal blocks from the linear operator.  The off-diagonal blocks can be compressed with their low-rank approximations, resulting in a `sketched' linear operator inside-out. To construct the resolvent operator for different frequencies, only the diagonal blocks at the lowest hierarchy level need to be modified.  Consequently, the matrix inversion can be achieved via a series of rank-k updates by tracking the hierarchy bottom-up, significantly reducing the resources needed for the matrix inverse. We demonstrate the use of this approach on a 2D channel flow and a turbulent airfoil wake.