## Abstract

Rational Krylov subspace (RKS) techniques are well-established and powerful tools for projection-based model reduction of time-invariant dynamic systems. For hyperbolic wavefield problems, such techniques perform well in configurations where only a few modes contribute to the field. RKS methods, however, are fundamentally limited by the Nyquist-Shannon sampling rate, making them unsuitable for the approximation of wavefields in configuration characterized by large travel times and propagation distances, since wavefield responses in such configurations are highly oscillatory in the frequency-domain. To overcome this limitation, we propose to precondition the RKSs by factoring out the rapidly varying frequency-domain field oscillations. The remaining amplitude-functions are generally slowly varying functions of source position and spatial coordinate and allow for a significant compression of the approximation subspace. Our one-dimensional analysis together with numerical experiments for large-scale two-dimensional acoustic models shows superior approximation properties of preconditioned RKS compared with the standard RKS model-order reduction. The preconditioned RKS results in a reduction of the frequency sampling well below the Nyquist-Shannon rate, a weak dependence of the RKS size on the number of inputs and outputs for multiple-input/multiple-output problems, and, most importantly, in a significant coarsening of the finite-difference grid used to generate the RKS. A prototype implementation indicates that the preconditioned RKS algorithm is competitive in the modern high performance computing environment.

Original language | English |
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Pages (from-to) | 1486-1518 |

Number of pages | 33 |

Journal | Multiscale Modeling and Simulation |

Volume | 16 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2018 |

## Keywords

- geometrical optics
- model reduction
- Nyquist rate
- seismic exploration
- wave equation