Bayesian approach to the identification of parameters of differential equations

0565076 - ÚGN 2023 CZ eng D - Dizertace
Bérešová, Simona
Bayesian approach to the identification of parameters of differential equations.
Ústav geoniky AV ČR, v. v. i. Obhájeno: Fakulta elektrotechniky a informatiky, VŠB-TUO. 09.06.2022. - Ostrava: VŠB - Technická univerzita Ostrava, 2022. 122 s.
Institucionální podpora: RVO:68145535
Klíčová slova: Bayesian inversion * deflated conjugate gradients * delayed-acceptance Metropolis-Hastings * Markov chain Monte Carlo * material parameter identification * surrogate model
Obor OECD: Statistics and probability

The Bayesian approach to the solution of inverse problems provides us with the posterior distribution of unknown parameters. The topic of the thesis is the use of Markov chain Monte Carlo (MCMC) methods based on the Metropolis-Hastings algorithm for generating samples from the posterior distribution. The work focuses on inverse problems governed by computationally expensive forward models, especially numerical solutions of partial differential equations. The key subject is the acceleration of MCMC algorithms using surrogate models that are constructed and further updated during the sampling process. The sampling process often produces sequences of linear systems, for their solution, the deflated conjugate gradient (CG) method is used. Proposed sampling procedures were
implemented in Python with parallelization. The implemented package was used for the identification of material parameters in inverse problems from the field of geosciences. From the perspective of the author, the main contribution of this thesis consists in connecting knowledge from several research fields (probability theory, geotechnics, parallel computing, iterative methods) into the design of a framework for posterior sampling and the implementation of the resulting software package. The contributions also include the use of polynomial and radial basis function surrogate models, the incorporation of the deflated CG method, and the comparison of various sampling methods. The application of the Bayesian approach to chosen model problems is also a contribution to the field of computational geosciences. The thesis also provides a comprehensive view of the basic MH algorithm and its delayed-acceptance version on general measurable spaces including the discussion on the irreducibility conditions.
Trvalý link: https://hdl.handle.net/11104/0336621