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Published July 10, 2021 | Submitted
Journal Article Open

Data-driven resolvent analysis


Resolvent analysis identifies the most responsive forcings and most receptive states of a dynamical system, in an input–output sense, based on its governing equations. Interest in the method has continued to grow during the past decade due to its potential to reveal structures in turbulent flows, to guide sensor/actuator placement and for flow control applications. However, resolvent analysis requires access to high-fidelity numerical solvers to produce the linearized dynamics operator. In this work, we develop a purely data-driven algorithm to perform resolvent analysis to obtain the leading forcing and response modes, without recourse to the governing equations, but instead based on snapshots of the transient evolution of linearly stable flows. The formulation of our method follows from two established facts: (i) dynamic mode decomposition can approximate eigenvalues and eigenvectors of the underlying operator governing the evolution of a system from measurement data, and (ii) a projection of the resolvent operator onto an invariant subspace can be built from this learned eigendecomposition. We demonstrate the method on numerical data of the linearized complex Ginzburg–Landau equation and of three-dimensional transitional channel flow, and discuss data requirements. Presently, the method is suitable for the analysis of laminar equilibria, and its application to turbulent flows would require disambiguation between the linear and nonlinear dynamics driving the flow. The ability to perform resolvent analysis in a completely equation-free and adjoint-free manner will play a significant role in lowering the barrier of entry to resolvent research and applications.

Additional Information

© The Author(s), 2021. Published by Cambridge University Press. Received 15 October 2020; revised 2 March 2021; accepted 11 April 2021. We gratefully acknowledge L. Cordier, as well as the anonymous referees, for helpful comments and suggestions. This work was supported by the PRIME programme of the German Academic Exchange Service (DAAD) with funds from the German Federal Ministry of Education and Research (BMBF) and by the U.S. Army Research Office (ARO W911NF-17-1-0306). The authors report no conflict of interest.

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August 20, 2023
October 23, 2023