Fourier Model Reduction for Large-Scale Linear Systems

Fourier model reduction (FMR) is a new method for obtaining stable, accurate, low-order models of very large linear systems. The technique draws on traditional control and dynamical system concepts and utilizes them in a way which is computationally very efficient. Discrete-time Fourier coefficients of the large system are calculated and used to construct a reduced-order model that preserves stability properties of the original system. Many coefficients can be calculated, which results in a very accurate representation of the system dynamics, but only a single factorization of the large system is required. The resulting system can be further reduced using explicit formulae for balanced truncation.

The method has been applied to a number of computational fluid dynamic systems, including unsteady motion of a two-dimensional subsonic airfoil and unsteady flow in a supersonic diffuser. In both cases, FMR is found to work extremely well. In comparison with other widely used techniques, such as the proper orthogonal decomposition and Arnoldi method, FMR is computationally more efficient, preserves the stability of the original system, uses both input and output information, and, for smooth transfer functions, is valid over a wide range of frequencies.

Publications
Willcox, K. and Megretski, A., “Fourier Series for Accurate, Stable, Reduced-Order Models in Large-Scale Applications,” SIAM Journal for Scientific Computing, Vol. 26, No. 3, 2005, pp. 944-962.