Calibration of the stochastic multicloud model using bayesian inference

The stochastic multicloud model (SMCM) was recently developed [B. Khouider, J. Biello, and A. J. Majda, Commun. Math. Sci., 8 (2010), pp. 187-216] to represent the missing variability in general circulation models due to unresolved features of organized tropical convection. This research aims at finding a robust calibration methodology for the SMCM to estimate key model parameters from data. We formulate the calibration problem within a Bayesian framework to derive the posterior distribution over the model parameters. The main challenge here is due to the likelihood function which requires solving a large system of differential equations (the Kolmogorov equations) as many times as there are data points, which is prohibitive in terms of both computation time and storage requirements. The most attractive numerical techniques to compute the transient solutions to large Markov chains are based on matrix exponentials, but none is unconditionally acceptable for all classes of problems. We develop a parallel version of a preconditioning technique known as the uniformization method, using the PETSc (Portable, Extensible Toolkit for Scientific Computation) suite of sparse matrix-vector operations. The parallel uniformization method allows for fast and scalable approximations of large sparse matrix exponentials, without sacrificing accuracy. Sampling of the high-dimensional posterior distribution is achieved via the standard Markov chain Monte Carlo. The robustness of the calibration procedure is tested using synthetic data produced by a simple toy climate model. A sensitivity study to the length of the data time series and to the prior distribution is presented, and a sequential learning strategy is also tested.