Sampling-free linear Bayesian update of polynomial chaos representations

Bojana V. Rosić*, Alexander Litvinenko, Oliver Pajonk, Hermann G. Matthies

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

54 Citations (Scopus)


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 languageEnglish
Pages (from-to)5761-5787
Number of pages27
JournalJournal of computational physics
Issue number17
Publication statusPublished - 1 Jul 2012
Externally publishedYes


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


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