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This paper presents an improved implicit sampling method for hierarchicalBayesian inverse problems. A widely used approach for sampling posteriordistribution is based on Markov chain Monte Carlo (MCMC). However, the samplesgenerated by MCMC are usually strongly correlated. This may lead to a smallsize of effective samples from a long Markov chain and the resultant posteriorestimate may be inaccurate. An implicit sampling method proposed in [11] cangenerate independent samples and capture some inherent non-Gaussian features ofthe posterior based on the weights of samples. In the implicit sampling method,the posterior samples are generated by constructing a map and distribute aroundthe MAP point. However, the weights of implicit sampling in previous works maycause excessive concentration of samples and lead to ensemble collapse. Toovercome this issue, we propose a new weight formulation and make resamplingbased on the new weights. In practice, some parameters in prior density areoften unknown and a hierarchical Bayesian inference is necessary for posteriorexploration. To this end, the hierarchical Bayesian formulation is used toestimate the MAP point and integrated in the implicit sampling framework.Compared to conventional implicit sampling, the proposed implicit samplingmethod can significantly improve the posterior estimator and the applicabilityfor high dimensional inverse problems. The improved implicit sampling method isapplied to the Bayesian inverse problems of multi-term time fractionaldiffusion models in heterogeneous media. To effectively capture theheterogeneity effect, we present a mixed generalized multiscale finite elementmethod (mixed GMsFEM) to solve the time fractional diffusion models in a coarsegrid, which can substantially speed up the Bayesian inversion.

An improved implicit sampling for Bayesian inverse problems of multi-term time fractional multiscale diffusion models