MODEL REDUCTION FOR FLUIDS, USING BALANCED PROPER ORTHOGONAL DECOMPOSITION

Research output: Chapter in Book/Report/Conference proceedingChapter

9 Scopus citations

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.

Original languageEnglish (US)
Title of host publicationModeling and Computations in Dynamical Systems
Subtitle of host publicationIn Commemoration of the 100th Anniversary of the Birth of John von Neumann
PublisherWorld Scientific Publishing Co.
Pages301-317
Number of pages17
ISBN (Electronic)9789812774569
ISBN (Print)9812565965
DOIs
StatePublished - Jan 1 2006

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Keywords

  • balanced truncation
  • Model reduction
  • proper orthogonal decomposition
  • snapshots

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