  This paper proposes a hierarchical, multi-resolution framework for the
identification of model parameters and their spatially variability from noisy
measurements of the response or output. Such parameters are frequently
encountered in PDE-based models and correspond to quantities such as density or
pressure fields, elasto-plastic moduli and internal variables in solid
mechanics, conductivity fields in heat diffusion problems, permeability fields
in fluid flow through porous media etc. The proposed model has all the
advantages of traditional Bayesian formulations such as the ability to produce
measures of confidence for the inferences made and providing not only
predictive estimates but also quantitative measures of the predictive
uncertainty. In contrast to existing approaches it utilizes a parsimonious,
non-parametric formulation that favors sparse representations and whose
complexity can be determined from the data. The proposed framework in
non-intrusive and makes use of a sequence of forward solvers operating at
various resolutions. As a result, inexpensive, coarse solvers are used to
identify the most salient features of the unknown field(s) which are
subsequently enriched by invoking solvers operating at finer resolutions. This
leads to significant computational savings particularly in problems involving
computationally demanding forward models but also improvements in accuracy. It
is based on a novel, adaptive scheme based on Sequential Monte Carlo sampling
which is embarrassingly parallelizable and circumvents issues with slow mixing
encountered in Markov Chain Monte Carlo schemes.
