Pseudospectral behavior of the linear operator and its influence on modal amplitudes in the dynamic mode decomposition

We investigate the pseudospectral behavior of the linear operator obtained from dynamic mode decomposition (DMD) and the extent to which it can effect the amplitudes of the DMD modes. While the amplitude is generally used to guide the selection of dominant modes, we observe that, when performing exact DMD with a large number of POD bases (for which the linear operator is projected onto), modes with physical structures do not align with high amplitudes. Meanwhile, performing DMD via explicitly forming the companion matrix from the snapshots does not suffer from such a misalignment, given that the data matrix has full rank. This draws us to examine the pseudospectra of the DMD linear operators obtained from both approaches. We find that, in exact DMD, increasing the number of POD bases also increases the nonnormality of the reduced-order linear operator. This increasing nonnormality can be characterized by the pseudospectral level as well as the eigenvalue condition number, computed by taking the inner product between the direct and adjoint eigenvectors. Meanwhile, the companion matrix exhibits much lower nonnormality, making it more robust against the perturbations due to numerical error. We demonstrate these using a large dataset of a turbulent wake flow over a NACA 0012 airfoil. The difference in the levels of nonormality of the linear operators leaves a cautionary note to the selection of DMD algorithms, as the nonnormality may also play a crucial role in the data-driven resolvent analysis.