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# Analysis and Computation of Acoustic and Elastic Wave Equations in Random Media

- Handle URI:
- http://hdl.handle.net/10754/623991
- Title:
- Analysis and Computation of Acoustic and Elastic Wave Equations in Random Media
- Authors:
- Abstract:
- We propose stochastic collocation methods for solving the second order acoustic and elastic wave equations in heterogeneous random media and subject to deterministic boundary and initial conditions [1, 4]. We assume that the medium consists of non-overlapping sub-domains with smooth interfaces. In each sub-domain, the materials coefficients are smooth and given or approximated by a finite number of random variable. One important example is wave propagation in multi-layered media with smooth interfaces. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems [2, 3], the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence is only algebraic. A fast spectral rate of convergence is still possible for some quantities of interest and for the wave solutions with particular types of data. We also show that the semi-discrete solution is analytic with respect to the random variables with the radius of analyticity proportional to the grid/mesh size h. We therefore obtain an exponential rate of convergence which deteriorates as the quantity h p gets smaller, with p representing the polynomial degree in the stochastic space. We have shown that analytical results and numerical examples are consistent and that the stochastic collocation method may be a valid alternative to the more traditional Monte Carlo method. Here we focus on the stochastic acoustic wave equation. Similar results are obtained for stochastic elastic equations.
- KAUST Department:
- Conference/Event name:
- Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2014)
- Issue Date:
- 6-Jan-2014
- Type:
- Poster

- Appears in Collections:
- Posters; Conference on Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2014)

# Full metadata record

DC Field | Value | Language |
---|---|---|

dc.contributor.author | Motamed, Mohammad | en |

dc.contributor.author | Nobile, Fabio | en |

dc.contributor.author | Tempone, Raul | en |

dc.date.accessioned | 2017-06-01T10:20:42Z | - |

dc.date.available | 2017-06-01T10:20:42Z | - |

dc.date.issued | 2014-01-06 | - |

dc.identifier.uri | http://hdl.handle.net/10754/623991 | - |

dc.description.abstract | We propose stochastic collocation methods for solving the second order acoustic and elastic wave equations in heterogeneous random media and subject to deterministic boundary and initial conditions [1, 4]. We assume that the medium consists of non-overlapping sub-domains with smooth interfaces. In each sub-domain, the materials coefficients are smooth and given or approximated by a finite number of random variable. One important example is wave propagation in multi-layered media with smooth interfaces. The numerical scheme consists of a finite difference or finite element method in the physical space and a collocation in the zeros of suitable tensor product orthogonal polynomials (Gauss points) in the probability space. We provide a rigorous convergence analysis and demonstrate different types of convergence of the probability error with respect to the number of collocation points under some regularity assumptions on the data. In particular, we show that, unlike in elliptic and parabolic problems [2, 3], the solution to hyperbolic problems is not in general analytic with respect to the random variables. Therefore, the rate of convergence is only algebraic. A fast spectral rate of convergence is still possible for some quantities of interest and for the wave solutions with particular types of data. We also show that the semi-discrete solution is analytic with respect to the random variables with the radius of analyticity proportional to the grid/mesh size h. We therefore obtain an exponential rate of convergence which deteriorates as the quantity h p gets smaller, with p representing the polynomial degree in the stochastic space. We have shown that analytical results and numerical examples are consistent and that the stochastic collocation method may be a valid alternative to the more traditional Monte Carlo method. Here we focus on the stochastic acoustic wave equation. Similar results are obtained for stochastic elastic equations. | en |

dc.subject | Low-Rank | en |

dc.title | Analysis and Computation of Acoustic and Elastic Wave Equations in Random Media | en |

dc.type | Poster | en |

dc.contributor.department | Computer, Electrical and Mathematical Sciences & Engineering (CEMSE) | en |

dc.conference.date | January 6-10, 2014 | en |

dc.conference.name | Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2014) | en |

dc.conference.location | KAUST | en |

dc.contributor.institution | University of New Mexico | en |

dc.contributor.institution | École Polytechnique Fédérale de Lausanne | en |

kaust.author | Tempone, Raul | en |

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