Linear model reduction using spectral proper orthogonal decomposition

TL;DR

This work introduces Spectral Solution Operator Projection (SSOP), a trajectory-focused model-reduction method that uses Spectral POD (SPOD) modes to encode entire state trajectories over a time interval. By projecting the time-domain solution of a linear time-invariant system onto SPOD modes, SSOP computes static coefficients that yield highly accurate reconstructions with competitive computational cost compared to POD-Galerkin and balanced truncation. The authors derive a frequency-domain formulation, develop data-driven and data-free operator approximations to keep online costs small, and demonstrate two benchmark problems where SSOP achieves orders-of-magnitude lower error than traditional time-domain ROMs. A data-free resolvent variant further broadens applicability, and results indicate substantial practical impact for real-time or optimization-driven simulations, with clear pathways to nonlinear extensions. SPOD-based space-time representations thus offer a powerful, efficient alternative to conventional space-only methods, particularly for long trajectories and systems with structured forcing.

Abstract

Most model reduction methods reduce the state dimension and then temporally evolve a set of coefficients that encode the state in the reduced representation. In this paper, we instead employ an efficient representation of the entire trajectory of the state over some time interval of interest and then solve for the static coefficients that encode the trajectory on the interval. We use spectral proper orthogonal decomposition (SPOD) modes, which are provably optimal for representing long trajectories and substantially outperform any representation of the trajectory in a purely spatial basis (e.g., POD). We develop a method to solve for the SPOD coefficients that encode the trajectories for forced linear dynamical systems given the forcing and initial condition, thereby obtaining the accurate prediction of the dynamics afforded by the SPOD representation of the trajectory. The method, which we refer to as spectral solution operator projection (SSOP), is derived by projecting the general time-domain solution for a linear time-invariant system onto the SPOD modes. We demonstrate the new method using two examples: a linearized Ginzburg-Landau equation and an advection-diffusion problem. In both cases, the error of the proposed method is orders of magnitude lower than that of POD-Galerkin projection and balanced truncation. The method is also fast, with CPU time comparable to or lower than both benchmarks in our examples. Finally, we describe a data-free space-time method that is a derivative of the proposed method and show that it is also more accurate than balanced truncation in most cases.

Figures (17)

• Figure 1: The proposed model reduction approach (bottom panel) compared against a standard space-only linear model reduction (top panel). To represent a trajectory, the space-only basis vectors are multiplied by time-dependent coefficients; the SPOD modes with their oscillatory time dependence are multiplied by static coefficients. In the space-only case, the coefficients are obtained by integrating a set of ODEs forward in time, whereas in the SPOD case, the coefficients are obtained by solving a linear algebraic system.
• Figure 2: Number of degrees of freedom (DOFs) required to achieve $98\%$ representation accuracy of trajectories as a function of the length of the time interval of the trajectory $[0,T]$. For POD, one must specify all the mode coefficients at every time step, whereas spectral and space-time POD modes are themselves time-dependent. Thus, by leveraging spatiotemporal correlations, fewer DOFs are needed to represent a trajectory to a given accuracy by specifying the SPOD or space-time POD coefficients. As the time interval becomes long, the SPOD and space-time POD modes become equally efficient at representing trajectories. The data come from the Ginzburg-Landau system introduced in Section \ref{['sec:results']}, and the details on the comparison of POD and SPOD with the same number of degrees of freedom can be found there.
• Figure 3: Ginzburg-Landau state and forcing trajectories: (a) the state resulting from the white forcing in (c); (b) the state resulting from the Gaussian forcing in (d). Both forcings have the same spatial correlation, but the short temporal correlation in the white forcing leads to more jagged structures in the corresponding state. Both trajectories consist of waves traveling in the positive $x$-direction that are amplified in a region near $x=0$.
• Figure 4: SPOD mode energies: (a-b, left axis) energy $\lambda$ of the retained and unretained modes. The top red curve is the energy of the first SPOD mode as a function of frequency $\omega$. The lower red and blue curves are the energies of the lower mode numbers, as functions of frequency. The retained modes (red) are the overall highest-energy modes, and the threshold (dashed) is determined as the energy of the $N_\omega r = 10240$-th most energetic mode; (a-b, right axis) number of modes that clear the threshold as a function of frequency. (c-d): the fraction of excluded energy (see \ref{['eq:excluded_energy']}) as a function of $r$, the average number of modes per frequency.
• Figure 5: FOM trajectory (a) and ROM predictions thereof (b-d) along with the errors for the Ginzburg-Landau system. The error fields shown are the absolute value of the difference of the FOM and ROM trajectories. The peak error value (and the upper limit on the error color scale) is $87 \%$ of the peak absolute value of the state.