  Bayesian approaches are one of the primary methodologies to tackle an inverse
problem in high dimensions. Such an inverse problem arises in hydrology to
infer the permeability field given flow data in a porous media. It is common
practice to decompose the unknown field into some basis and infer the
decomposition parameters instead of directly inferring the unknown. Given the
multiscale nature of permeability fields, wavelets are a natural choice for
parameterizing them. This study uses a Bayesian approach to incorporate the
statistical sparsity that characterizes discrete wavelet coefficients. First,
we impose a prior distribution incorporating the hierarchical structure of the
wavelet coefficient and smoothness of reconstruction via scale-dependent
hyperparameters. Then, Sequential Monte Carlo (SMC) method adaptively explores
the posterior density on different scales, followed by model selection based on
Bayes Factors. Finally, the permeability field is reconstructed from the
coefficients using a multiresolution approach based on second-generation
wavelets. Here, observations from the pressure sensor grid network are computed
via Multilevel Adaptive Wavelet Collocation Method (AWCM). Results highlight
the importance of prior modeling on parameter estimation in the inverse
problem.
