# A Hierarchical Bayesian Setting for an Inverse Problem in Linear Parabolic PDEs with Noisy Boundary Conditions

- Handle URI:
- http://hdl.handle.net/10754/624030
- Title:
- A Hierarchical Bayesian Setting for an Inverse Problem in Linear Parabolic PDEs with Noisy Boundary Conditions
- Authors:
- Abstract:
- In this work we develop a Bayesian setting to infer unknown parameters in initial-boundary value problems related to linear parabolic partial differential equations. We realistically assume that the boundary data are noisy, for a given prescribed initial condition. We show how to derive the joint likelihood function for the forward problem, given some measurements of the solution field subject to Gaussian noise. Given Gaussian priors for the time-dependent Dirichlet boundary values, we analytically marginalize the joint likelihood using the linearity of the equation. Our hierarchical Bayesian approach is fully implemented in an example that involves the heat equation. In this example, the thermal diffusivity is the unknown parameter. We assume that the thermal diffusivity parameter can be modeled a priori through a lognormal random variable or by means of a space-dependent stationary lognormal random field. Synthetic data are used to test the inference. We exploit the behavior of the non-normalized log posterior distribution of the thermal diffusivity. Then, we use the Laplace method to obtain an approximated Gaussian posterior and therefore avoid costly Markov Chain Monte Carlo computations. Expected information gains and predictive posterior densities for observable quantities are numerically estimated using Laplace approximation for different experimental setups.
- KAUST Department:
- Citation:
- Ruggeri F, Sawlan Z, Scavino M, Tempone R (2017) A Hierarchical Bayesian Setting for an Inverse Problem in Linear Parabolic PDEs with Noisy Boundary Conditions. Bayesian Analysis 12: 407–433. Available: http://dx.doi.org/10.1214/16-BA1007.
- Publisher:
- Journal:
- Issue Date:
- 12-May-2016
- DOI:
- 10.1214/16-BA1007
- Type:
- Article
- ISSN:
- 1936-0975
- Sponsors:
- Part of this work was carried out while F. Ruggeri and M. Scavino were Visiting Professors at KAUST. Z. Sawlan, M. Scavino and R. Tempone are members of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering.
- Additional Links:
- http://projecteuclid.org/euclid.ba/1463078272

- Appears in Collections:
- Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

# Full metadata record

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

dc.contributor.author | Ruggeri, Fabrizio | en |

dc.contributor.author | Sawlan, Zaid A | en |

dc.contributor.author | Scavino, Marco | en |

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

dc.date.accessioned | 2017-06-05T06:02:23Z | - |

dc.date.available | 2017-06-05T06:02:23Z | - |

dc.date.issued | 2016-05-12 | en |

dc.identifier.citation | Ruggeri F, Sawlan Z, Scavino M, Tempone R (2017) A Hierarchical Bayesian Setting for an Inverse Problem in Linear Parabolic PDEs with Noisy Boundary Conditions. Bayesian Analysis 12: 407–433. Available: http://dx.doi.org/10.1214/16-BA1007. | en |

dc.identifier.issn | 1936-0975 | en |

dc.identifier.doi | 10.1214/16-BA1007 | en |

dc.identifier.uri | http://hdl.handle.net/10754/624030 | - |

dc.description.abstract | In this work we develop a Bayesian setting to infer unknown parameters in initial-boundary value problems related to linear parabolic partial differential equations. We realistically assume that the boundary data are noisy, for a given prescribed initial condition. We show how to derive the joint likelihood function for the forward problem, given some measurements of the solution field subject to Gaussian noise. Given Gaussian priors for the time-dependent Dirichlet boundary values, we analytically marginalize the joint likelihood using the linearity of the equation. Our hierarchical Bayesian approach is fully implemented in an example that involves the heat equation. In this example, the thermal diffusivity is the unknown parameter. We assume that the thermal diffusivity parameter can be modeled a priori through a lognormal random variable or by means of a space-dependent stationary lognormal random field. Synthetic data are used to test the inference. We exploit the behavior of the non-normalized log posterior distribution of the thermal diffusivity. Then, we use the Laplace method to obtain an approximated Gaussian posterior and therefore avoid costly Markov Chain Monte Carlo computations. Expected information gains and predictive posterior densities for observable quantities are numerically estimated using Laplace approximation for different experimental setups. | en |

dc.description.sponsorship | Part of this work was carried out while F. Ruggeri and M. Scavino were Visiting Professors at KAUST. Z. Sawlan, M. Scavino and R. Tempone are members of the KAUST SRI Center for Uncertainty Quantification in Computational Science and Engineering. | en |

dc.publisher | Institute of Mathematical Statistics | en |

dc.relation.url | http://projecteuclid.org/euclid.ba/1463078272 | en |

dc.rights | Archived with thanks to Bayesian Analysis.Creative Commons Attribution 4.0 International License. | en |

dc.rights.uri | http://creativecommons.org/licenses/by/4.0/ | en |

dc.subject | Bayesian inference | en |

dc.subject | Heat equation | en |

dc.subject | Linear parabolic PDEs | en |

dc.subject | Noisy boundary parameters | en |

dc.subject | Thermal diffusivity | en |

dc.title | en | |

dc.type | Article | en |

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

dc.identifier.journal | Bayesian Analysis | en |

dc.eprint.version | Publisher's Version/PDF | en |

dc.contributor.institution | CNR - IMATI, Consiglio Nazionale delle Ricerche, Milano, , Italy | en |

dc.contributor.institution | Instituto de Estadística (IESTA), Universidad de la República, Montevideo, , Uruguay | en |

kaust.author | Sawlan, Zaid A | en |

kaust.author | Scavino, Marco | en |

kaust.author | Tempone, Raul | en |

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