Model Reduction for Fluids, Using Balanced Proper Orthogonal Decomposition

Abstract

Many of the tools of dynamical systems and control theory have gone largely unused for fluids, because the governing equations are so dynamically complex, both high-dimensional and nonlinear. Model reduction involves finding low-dimensional models that approximate the full high-dimensional dynamics. This paper compares three different methods of model reduction: proper orthogonal decomposition (POD), balanced truncation, and a method called balanced POD. Balanced truncation produces better reduced-order models than POD, but is not computationally tractable for very large systems. Balanced POD is a tractable method for computing approximate balanced truncations, that has computational cost similar to that of POD. The method presented here is a variation of existing methods using empirical Gramians, and the main contributions of the present paper are a version of the method of snapshots that allows one to compute balancing transformations directly, without separate reduction of the Gramians; and an output projection method, which allows tractable computation even when the number of outputs is large. The output projection method requires minimal additional computation, and has a priori error bounds that can guide the choice of rank of the projection. Connections between POD and balanced truncation are also illuminated: in particular, balanced truncation may be viewed as POD of a particular dataset, using the observability Gramian as an inner product. The three methods are illustrated on a numerical example, the linearized flow in a plane channel.

Alleviating Unsteady Aerodynamic Loads with Closed-Loop Flow Control

“…The proper orthogonal decomposition (POD) method and variants of POD are commonly used, but they can miss important flow physics [31]. The balanced POD method introduced by Rowley [32] can be better than POD in control applications, but it requires an adjoint solution to be computed, which makes the method unsuitable for experimental data. Ma et al [33] showed that the eigensystem realization algorithm [34] method obtained the same models as balanced POD, but an adjoint was not required.…”

Reduced Numerical Approximation of Reduced Fluid-Structure Interaction Problems With Applications in Hemodynamics

“…In the context of this work we only detail the specific choices performed in relation to the problem at hand, for more details about POD applied to fluid problems we address the reader to e.g., Rowley [29] and Willcox and Peraire [30].…”

A second golden age of aeroacoustics?

“…For this reason, reduced-order models based on proper orthogonal decomposition (POD) modes alone may require additional modes to reproduce important dynamics [151]. A reduced-order model can instead be formed by projecting the system operator onto the first few direct and adjoint-balanced truncation modes [152]. Balanced truncation modes are similar to POD modes in that they may be computed through a snapshot method.…”