Scalable Hierarchical Multilevel Sampling of Lognormal Fields Conditioned on Measured Data

We explore the problem of drawing posterior samples from a lognormal permeability field conditioned by noisy measurements at discrete locations. The underlying unconditioned samples are based on a scalable PDE-sampling technique that shows better scalability for large problems than the traditional Karhunen-Loeve sampling, while still allowing for consistent samples to be drawn on a hierarchy of spatial scales. Lognormal random fields produced in this scalable and hierarchical way are then conditioned to measured data by a randomized maximum likelihood approach to draw from a Bayesian posterior distribution. The algorithm to draw from the posterior distribution can be shown to be equivalent to a PDE-constrained optimization problem, which allows for some efficient computational solution techniques. Numerical results demonstrate the efficiency of the proposed methods. In particular, we are able to match statistics for a simple flow problem on the fine grid with high accuracy and at much lower cost on a scale of coarser grids.