## Bayesian data assimilation for stochastic multiscale models of transport in porous media

We investigate Bayesian techniques that can be used to reconstruct field variables from partial observations. In particular, we target fields that exhibit spatial structures with a large spectrum of lengthscales. Contemporary methods typically describe the field on a grid and estimate structures which can be resolved by it. In contrast, we address the reconstruction of grid-resolved structures as well as estimation of statistical summaries of subgrid structures, which are smaller than the grid resolution. We perform this in two different ways (a) via a physical (phenomenological), parameterized subgrid model that summarizes the impact of the unresolved scales at the coarse level and (b) via multiscale finite elements, where specially designed prolongation and restriction operators establish the interscale link between the same problem defined on a coarse and fine mesh. The estimation problem is posed as a Bayesian inverse problem. Dimensionality reduction is performed by projecting the field to be inferred on a suitable orthogonal basis set, viz. the Karhunen-Loeve expansion of a multiGaussian. We first demonstrate our techniques on the reconstruction of a binary medium consisting of a matrix with embedded inclusions, which are too small to be grid-resolved. The reconstruction is performed using an adaptive Markov chain Monte Carlo method. We find that the posterior distributions of the inferred parameters are approximately Gaussian. We exploit this finding to reconstruct a permeability field with long, but narrow embedded fractures (which are too fine to be grid-resolved) using scalable ensemble Kalman filters; this also allows us to address larger grids. Ensemble Kalman filtering is then used to estimate the values of hydraulic conductivity and specific yield in a model of the High Plains Aquifer in Kansas. Strong conditioning of the spatial structure of the parameters and the non-linear aspects of the water table aquifer create difficulty for the ensemble Kalman filter. We conclude with a demonstration of the use of multiscale stochastic finite elements to reconstruct permeability fields. This method, though computationally intensive, is general and can be used for multiscale inference in cases where a subgrid model cannot be constructed.