Parameter estimation for subgrid-scale models using Markov chain Monte Carlo approximate Bayesian computation

We demonstrate the use of approximate Bayesian computation (ABC) combined with an “improved” Markov chain Monte Carlo (IMCMC) method to estimate posterior distributions of model parameters in subgrid-scale (SGS) closures for large eddy simulations (LES) of turbulent flows. The ABC–IMCMC approach avoids the need to directly compute a likelihood function during parameter estimation, enabling greater flexibility compared to full Bayesian inversion methods. The method also naturally provides uncertainties in parameter estimates, avoiding the artificial certainty implied by many optimization methods for determining model parameters. In this study, we outline the salient features of the present ABC–IMCMC algorithm, including the use of an adaptive proposal and calibration step to accelerate the parameter estimation process. We demonstrate the approach by estimating parameters in two nonlinear SGS closure forms using three different types of reference data from direct numerical simulations of homogeneous isotropic turbulence. We show that the resulting parameter distributions give excellent agreement with reference probability density functions of the SGS stress and kinetic energy production rate in a priori tests, while also providing stable solutions in forward LES (i.e., a posteriori tests). The ABC–IMCMC method is thus shown to be an effective and efficient approach as compared to traditional ABC for estimating unknown parameters, including their uncertainties, in SGS closure models for LES of turbulent flows, and can be straightforwardly applied to other model forms and reference data.