## Abstract

We consider the wave equation with highly oscillatory initial data, where there is uncertainty in the wave speed, initial phase, and/or initial amplitude. To estimate quantities of interest related to the solution and their statistics, we combine a high-frequency method based on Gaussian beams with sparse stochastic collocation. Although the wave solution, uϵ, is highly oscillatory in both physical and stochastic spaces, we provide theoretical arguments for simplified problems and numerical evidence that quantities of interest based on local averages of |uϵ|2 are smooth, with derivatives in the stochastic space uniformly bounded in ϵ, where ϵ denotes the short wavelength. This observable related regularity makes the sparse stochastic collocation approach more efficient than Monte Carlo methods. We present numerical tests that demonstrate this advantage.

Original language | English (US) |
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Pages (from-to) | 1084-1110 |

Number of pages | 27 |

Journal | SIAM-ASA Journal on Uncertainty Quantification |

Volume | 4 |

Issue number | 1 |

DOIs | |

State | Published - 2016 |

Externally published | Yes |

### Bibliographical note

Funding Information:∗Received by the editors July 6, 2015; accepted for publication (in revised form) May 25, 2016; published electronically September 8, 2016. http://www.siam.org/journals/juq/4/M102923.html Funding: The work of the first author was partially supported by the Swedish Research Council under grant 2012-3808. The work of the fourth author was supported by the King Abdullah University of Science and Technology (KAUST) SRI Center for Uncertainty Quantification in Computational Science and Engineering. †Department of Mathematics and Swedish e-Science Research Center (SeRC), KTH, 100 44, Stockholm, Sweden (malenova@kth.se, olofr@nada.kth.se). ‡Department of Mathematics and Statistics, The University of New Mexico, Albuquerque, NM 87131 (motamed@math.unm.edu). §Department of Computer, Electrical and Mathematical Sciences & Engineering (CEMSE), King Abdullah University of Science and Technology (KAUST), UN 1500 Building 1, Al-Khawarizmi, Thuwal 23955-6900, Kingdom of Saudi Arabia (raul.tempone@kaust.edu.sa).

Publisher Copyright:

© 2016 Sharif Rahman.

## Keywords

- Asymptotic approximation
- Gaussian beam summation
- High frequency
- Sparse grids
- Stochastic collocation
- Stochastic regularity
- Uncertainty quantification
- Wave propagation

## ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Statistics, Probability and Uncertainty
- Discrete Mathematics and Combinatorics
- Applied Mathematics