## Abstract

We present a modification of the balanced proper orthogonal decomposition (balanced POD) algorithm for systems with simple impulse response tails. In this new method, we use dynamic mode decomposition (DMD) to estimate the slowly decaying eigenvectors that dominate the long-time behavior of the direct and adjoint impulse responses. This is done using a new, low-memory variant of the DMD algorithm, appropriate for large datasets. We then formulate analytic expressions for the contribution of these eigenvectors to the controllability and observability Gramians. These contributions can be accounted for in the balanced POD algorithm by simply appending the impulse response snapshot matrices (direct and adjoint, respectively) with particular linear combinations of the slow eigenvectors. Aside from these additions to the snapshot matrices, the algorithm remains unchanged. By treating the tails analytically, we eliminate the need to run long impulse response simulations, lowering storage requirements and speeding up ensuing computations. To demonstrate its effectiveness, we apply this method to two examples: the linearized, complex Ginzburg-Landau equation, and the two-dimensional fluid flow past a cylinder. As expected, reduced-order models computed using an analytic tail match or exceed the accuracy of those computed using the standard balanced POD procedure, at a fraction of the cost.

Original language | English (US) |
---|---|

Pages (from-to) | 5317-5333 |

Number of pages | 17 |

Journal | Journal of Computational Physics |

Volume | 231 |

Issue number | 16 |

DOIs | |

State | Published - Jun 20 2012 |

## All Science Journal Classification (ASJC) codes

- Numerical Analysis
- Modeling and Simulation
- Physics and Astronomy (miscellaneous)
- Physics and Astronomy(all)
- Computer Science Applications
- Computational Mathematics
- Applied Mathematics

## Keywords

- Balanced proper orthogonal decomposition
- Dynamic mode decomposition
- Empirical Gramian
- Impulse response
- Model reduction