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The recently introduced harmonic resolvent framework is concerned with thestudy of the input-output dynamics of nonlinear flows in the proximity of aknown time-periodic orbit. These dynamics are governed by the harmonicresolvent operator, which is a linear operator in the frequency domain whosesingular value decomposition sheds light on the dominant input-outputstructures of the flow. Although the harmonic resolvent is a mathematicallywell-defined operator, the numerical computation of its singular valuedecomposition requires inverting a matrix that becomes exactly singular as theperiodic orbit approaches an exact solution of the nonlinear governingequations. The very poor condition properties of this matrix hinder theconvergence of classical Krylov solvers, even in the presence ofpreconditioners, thereby increasing the computational cost required to performthe harmonic resolvent analysis. In this paper we show that a suitableaugmentation of the (nearly) singular matrix removes the singularity, and weprovide a lower bound for the smallest singular value of the augmented matrix.We also show that the desired decomposition of the harmonic resolvent can becomputed using the augmented matrix, whose improved condition properties leadto a significant speedup in the convergence of classical iterative solvers. Wedemonstrate this simple, yet effective, computational procedure on theKuramoto-Sivashinsky equation in the proximity of an unstable time-periodicorbit.

A computationally efficient approach for the removal of the phase shift singularity in harmonic resolvent analysis