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Bayesian Methods in Multiscale Modeling

Bayesian Methods in Multiscale Modeling

Multiscale modeling is an inherently statistical problem, involving the parsing and propagation of data between different levels of simulation. The inevitable filtering out of information at the higher scales introduces uncertainty. The utility of multiscale models of physical systems greatly increases when this uncertainty can be quantified.

An important new method for quantifying the uncertainty in models of physical systems using both ab initio and experimental data has been developed in collaboration with physical chemists and statisticians at Los Alamos National Lab, in the context of work with the Department of Energy's Carbon Capture Simulation Initiative. The method operates through model calibration — in its simplest form, a fitting of model parameters to experimental data — using a Bayesian framework developed by Kennedy and O'Hagan ( J. Royal Stat. Soc. B, 36, 2001, 425). Within the Bayesian framework, model parameters become random variables. The associated probability distributions characterize parameter uncertainty. Bayes' theorem links a prior probability distribution for the parameters — which characterizes our knowledge of parameter uncertainty before any data is taken into account — to a posterior distribution representing our new assessment of parameter uncertainty in light of the experimental data.

Ab initio calculations are used to establish the prior distributions, which are designed to quantify the uncertainty present in these calculations due to the various approximations employed. Uncertainty in the model itself is quantified through a stochastic function called the model form discrepancy. The experimental data is therefore considered to result from a sum of terms: the physical model, the model form discrepancy, and a term representing experimental error.

The process leading to the posterior distribution for the model parameters also leads to a posterior form for the model discrepancy. The physical model and the posterior model discrepancy then combine to become a new, probabilistic model for the system. The predictions of this new model are precise in regions close to where experimental data exists, with increasing amounts of uncertainty as the system moves away from known experimental points. This provides for an "extrapolation penalty" that can be extremely useful in the context of multiscale modeling.

The utility of this new framework has been demonstrated in the context of equilibrium models of amine-based CO ₂ sorbents.

Bivariate marginal probability distributions for model parameters appearing in a model of a CO2 sorbent

Bayesian Methods in Multiscale Modeling

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