Abstract: In this talk, we present an improved implicit sampling method for hierarchical Bayesian
inverse problems. A widely used approach for sampling posterior distribution is based on Markov
chain Monte Carlo (MCMC). However, the samples generated by MCMC are usually strongly
correlated. This may lead to a small size of effective samples from a long Markov chain and the
resultant posterior estimate may be inaccurate. An implicit sampling method proposed before
can generate independent samples and capture some inherent non-Gaussian features of the
posterior based on the weights of samples.. However, the weights of implicit sampling in previous
works may cause excessive concentration of samples and lead to ensemble collapse. To overcome
this issue, we propose a new weight formulation and make resampling based on the new weights.
The hierarchical Bayesian formulation is used to estimate the MAP point and integrated in the
implicit sampling framework. Compared to conventional implicit sampling, the proposed implicit
sampling method can significantly improve the posterior estimator and the applicability for high
dimensional inverse problems. The improved implicit sampling method is applied to the Bayesian
inverse problems of multi-term time fractional diffusion models in heterogeneous media. To
effectively capture the heterogeneity effect, we present a mixed generalized multiscale finite
element method (mixed GMsFEM) to solve the time fractional diffusion models in a coarse grid,
which can substantially speed up the Bayesian inversion.
keywords: Bayesian inversion, improved implicit sampling, mixed GMsFEM, multi-term time
fractional diffusion models