  The recently introduced harmonic resolvent framework is concerned with the
study of the input-output dynamics of nonlinear flows in the proximity of a
known time-periodic orbit. These dynamics are governed by the harmonic
resolvent operator, which is a linear operator in the frequency domain whose
singular value decomposition sheds light on the dominant input-output
structures of the flow. Although the harmonic resolvent is a mathematically
well-defined operator, the numerical computation of its singular value
decomposition requires inverting a matrix that becomes exactly singular as the
periodic orbit approaches an exact solution of the nonlinear governing
equations. The very poor condition properties of this matrix hinder the
convergence of classical Krylov solvers, even in the presence of
preconditioners, thereby increasing the computational cost required to perform
the harmonic resolvent analysis. In this paper we show that a suitable
augmentation of the (nearly) singular matrix removes the singularity, and we
provide a lower bound for the smallest singular value of the augmented matrix.
We also show that the desired decomposition of the harmonic resolvent can be
computed using the augmented matrix, whose improved condition properties lead
to a significant speedup in the convergence of classical iterative solvers. We
demonstrate this simple, yet effective, computational procedure on the
Kuramoto-Sivashinsky equation in the proximity of an unstable time-periodic
orbit.
