RVMD Reduced-Order Variational Mode Decomposition

By applying Parseval's theorem to transform the time-domain analytic-signal Hilbert bandwidth penalty into the positive-frequency domain (Equations 9, 17-18), RVMD minimizes F({phi_k, c_hat_k^+, omega_k}) via Block Coordinate Descent over the residual R_hat_k^+ = Q_hat^+ - sum_{i != k} phi_i (c_hat_i^+)^T (Equation 36), alternately updating the normalized spatial mode phi_k^{n+1} = Re{R_hat_k^+ (c_hat_k^+)^*} / ||Re{R_hat_k^+ (c_hat_k^+)^*}|| (Equation 39), the second-order Wiener-filtered temporal spectrum c_hat_k^{+,n+1} = G_hat_k^+ (R_hat_k^+)^T phi_k with g_hat_i^+ = 1 / [1 + 2*alpha*(omega_i^+ - omega_k)^2] (Equations 38, 40), and the center-of-gravity central frequency omega_k^{n+1} = ((c_hat_k^+)^H W^+ c_hat_k^+) / ((c_hat_k^+)^H c_hat_k^+) (Equation 41).

At BCD convergence, each RVMD spatial mode satisfies the Wiener-filtered two-point spatial correlation eigenvalue equation R_hat_k G_hat_k R_hat_k^H phi_k = lambda_k phi_k (Equation 47) with exact half-power filter bandwidth Delta = sqrt((2*sqrt(2) - 2)/alpha) (Equation 48), continuously bridging Proper Orthogonal Decomposition (POD, as alpha -> 0 and G_hat_k -> I, Equations 49-51) and single-frequency Discrete Fourier Transform modes (as alpha -> infty and Delta -> 0, Section 3.2).

Coupled with Hilbert spectral analysis (Equations 52-55), RVMD resolves non-stationary transient growth, shift-modes, and limit-cycle harmonics in a two-dimensional cylinder wake at Re = 100 and isolates intermittent screech-tone feedback modes in a three-dimensional turbulent supersonic rectangular jet at M_j = 1.55.

Because the joint optimization over {phi_k, c_hat_k, omega_k} is non-convex and does not enforce strict spatial or temporal orthogonality across modes for general alpha > 0, converged RVMD modes depend on the initial central-frequency distribution {omega_k^0} and penalty factor alpha and can share overlapping spectral energy when bandwidths intersect (Section 2.2, p. 11; Section 3.1, p. 14; Appendix A, pp. 29-31).