Implicit sampling is a weighted sampling method that is used in data assimilation to sequentially update state estimates of a stochastic model based on noisy and incomplete data. Here we apply implicit sampling to sample the posterior probability density of parameter estimation problems. The posterior probability combines prior information about the parameter with information from a numerical model, e.g., a partial differential equation (PDE), and noisy data. The result of our computations are parameters that lead to simulations that are compatible with the data. We demonstrate the usefulness of our implicit sampling algorithm with an example from subsurface flow. For an efficient implementation, we make use of multiple grids, BFGS optimization coupled to adjoint equations, and Karhunen–Loève expansions for dimensional reduction. Several difficulties of Markov chain Monte Carlo methods, e.g., estimation of burn-in times or correlations among the samples, are avoided because the implicit samples are independent.
"Parameter estimation by implicit sampling." Commun. Appl. Math. Comput. Sci. 10 (2) 205 - 225, 2015. https://doi.org/10.2140/camcos.2015.10.205