Hydrogeological inverse problems consist in finding physical parameter fields (e.g. hydraulic conductivity, porosity, etc.) explaining state variables (e.g. hydraulic head, solute concentration, etc.) measured at some observation points or indirect observations, such as geophysics. In a bayesian framework, prior information about the expected characteristics of the subsurface can be taken into account. The general aim of this project is to develop efficient Monte Carlo (MC) methods based on Multiple-Point Statistics (MPS) for solving inverse problems. The goals are to characterize geological reservoirs by accounting for all available information provided by direct or indirect observations, and to evaluate parameter uncertainty. The choice of this framework is motivated by two main arguments: 1) Monte Carlo statistical methods are widely developed, mathematically rigorous, and provide algorithms based on simulations, useful for solving high-dimensional problems for which analytic solutions cannot be formulated; 2) MPS algorithms allow for very flexible geostatistical simulations providing highly realistic models that take complex geological knowledge and direct observations into account.

So far, no efficient MC methods based on MPS is avalaible. To obtain an efficient Markov chain Monte Carlo (McMC) algorithm, we intend to first develop new sampling strategies for the proposal distribution in Markov chains, i.e. new random processes for generating a candidate from the current model in the chain. To do so, we plan to exploit our MPS techniques (in particular the Direct Sampling) together with sensitivities of the state variables (at the observation locations) to the parameter field. This will be done to localize areas to be preferentially re-simulated due to the enhanced possibility to get a better fit at the obervation points, i.e. an increased likelihood that the observed data comes from the proposed model. We intend to use an adjoint approach to efficiently compute these sensitivities.

Then, we plan to rigorously integrate the developed strategies in McMC algorithms for optimization and for sampling the posterior distribution. For the latter case, we intend to adapt (non-symmetric) proposal distributions and define acceptance criteria that ensure that mathematical properties, such as the reversibility of the chain, are verified. Furthermore, to accelerate the convergence, we envision to implement advanced McMC schemes based on multiple parallel chains and multiple-try approaches, while maintaining the required properties of the sampling algorithms.

Synthetic problems will be considered first, to focus on the development of the new methods, while keeping the computational burden reasonable, and to allow comparisons to a rejection sampler. Finally, the methods will be applied at the Emme field site in Switzerland.