Michael Reichel (ari)
Es spricht Prof. Dr. Björn Sprungk (TU Bergakademie Freiberg)
zum Thema: Sampling methods for the Bayesian approach to inverse problems
Abstract:
The Bayesian approach to inverse problems allows quite elegantly to quantify the uncertainty in data-based reconstructions of an unknown ground truth and, in particular, yields to well-posedness. However, the price to pay comes in terms of the computationally challenging task of evaluating and sampling the resulting solution -- which is the so called posterior, a probability measure in high- or infinite-dimensional function space. In this talk we motivate the Bayesian approach to inverse problems from an uncertainty quantification point of view, briefly discuss its locally Lipschitz well-posedness, and later focus on Markov chain Monte Carlo methods for sampling and integration with respect to the posterior distribution. Here we present our contributions to Metropolis-Hastings algorithms in function spaces, discuss convergence in terms of geometric ergodicity, and present numerical experiments illustrating its dimension-indepedent performance which is, moreover, robust to the level of observational noise in the data.
If time permits, we discuss in the last part of the talk recent results on combining Metropolis-Hastings with interacting particle sampling methods based on Euler-Maruyama discretizations of stochastic differential equations of McKean-Vlasov type.
Dienstag, 1. Juli 2025, 15:00 Uhr, Curie Hörsaal
(Kaffee ab 14:30 Uhr im Raum C 325)
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