  The conceptual picture underlying resolvent analysis(RA) is that the
nonlinear term in the Navier-Stokes(NS) equations provides an intrinsic forcing
to the linear dynamics, a description inspired by control theory. The inverse
of the linear operator, defined as the resolvent, is interpreted as a transfer
function between the forcing and the velocity response. This inversion obscures
the physical interpretation of the governing equations and is prohibitive to
analytical manipulation, and for large systems leads to significant
computational cost and memory requirements. In this work we suggest an
alternative, inverse free, definition of the resolvent basis based on an
extension of the Courant-Fischer-Weyl min-max principle in which resolvent
modes are defined as stationary points of a constrained variational problem.
This leads to a straightforward approach to approximate the resolvent
(response) modes of complex flows as expansions in any basis. The proposed
method avoids matrix inversions and requires only the spectral decomposition of
a matrix of significantly reduced size as compared to the original system. To
illustrate this method and the advantages of the variational formulation we
present three examples. First, we consider streamwise constant fluctuations in
turbulent channel flow where an asymptotic analysis allows us to derive closed
form expressions for the optimal resolvent modes. Second, to illustrate the
cost saving potential, and investigate the limits, of the proposed method we
apply our method to both a 2-dimensional, 3-component equilibrium solution in
Couette flow and, finally, to a streamwise developing turbulent boundary layer.
For these larger systems we achieve a model reduction of up to two orders of
magnitude. Such savings have the potential to open RA to the investigation of
larger domains and more complex flow configurations.
