### Abstract

We present a fully deterministic approach to a probabilistic interpretation of inverse problems in which unknown quantities are represented by random fields or processes, described by possibly non-Gaussian distributions. The description of the introduced random fields is given in a " white noise" framework, which enables us to solve the stochastic forward problem through Galerkin projection onto polynomial chaos. With the help of such a representation the probabilistic identification problem is cast in a polynomial chaos expansion setting and the Baye's linear form of updating. By introducing the Hermite algebra this becomes a direct, purely algebraic way of computing the posterior, which is comparatively inexpensive to evaluate. In addition, we show that the well-known Kalman filter is the low order part of this update. The proposed method is here tested on a stationary diffusion equation with prescribed source terms, characterised by an uncertain conductivity parameter which is then identified from limited and noisy data obtained by a measurement of the diffusing quantity.

Original language | English |
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Pages (from-to) | 5761-5787 |

Number of pages | 27 |

Journal | Journal of computational physics |

Volume | 231 |

Issue number | 17 |

DOIs | |

Publication status | Published - 1 Jul 2012 |

Externally published | Yes |

### Keywords

- Kalman filter
- Linear Bayesian update
- Minimum squared error estimate
- Minimum variance estimate
- Polynomial chaos expansion

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## Cite this

*Journal of computational physics*,

*231*(17), 5761-5787. https://doi.org/10.1016/j.jcp.2012.04.044