Jonathan Goodman

John Bell, Marcus Day, Jonathan Goodman et al.

First-principles Markov Chain Monte Carlo sampling is used to investigate uncertainty quantification and uncertainty propagation in parameters describing hydrogen kinetics. Specifically, we sample the posterior distribution of thirty-one parameters focusing on the H2O2 and HO2 reactions resulting from conditioning on ninety-one experiments. Established literature values are used for the remaining parameters in the mechanism. The samples are computed using an affine invariant sampler starting with broad, noninformative priors. Autocorrelation analysis shows that O(1M) samples are sufficient to obtain a reasonable sampling of the posterior. The resulting distribution identifies strong positive and negative correlations and several non-Gaussian characteristics. Using samples drawn from the posterior, we investigate the impact of parameter uncertainty on the prediction of two more complex flames: a 2D premixed flame kernel and the ignition of a hydrogen jet issuing into a heated chamber. The former represents a combustion regime similar to the target experiments used to calibrate the mechanism and the latter represents a different combustion regime. For the premixed flame, the net amount of product after a given time interval has a standard deviation of less than 2% whereas the standard deviation of the ignition time for the jet is more than 10%. The samples used for these studies are posted online. These results indicate the degree to which parameters consistent with the target experiments constrain predicted behavior in different combustion regimes. This process provides a framework for both identifying reactions for further study from candidate mechanisms as well as combining uncertainty quantification and propagation to, ultimately, tie uncertainty in laboratory flame experiments to uncertainty in end-use numerical predictions of more complicated scenarios.

Emilio Zappa, Miranda Holmes-Cerfon, Jonathan Goodman

We describe and analyze some Monte Carlo methods for manifolds in Euclidean space defined by equality and inequality constraints. First, we give an MCMC sampler for probability distributions defined by un-normalized densities on such manifolds. The sampler uses a specific orthogonal projection to the surface that requires only information about the tangent space to the manifold, obtainable from first derivatives of the constraint functions, hence avoiding the need for curvature information or second derivatives. Second, we use the sampler to develop a multi-stage algorithm to compute integrals over such manifolds. We provide single-run error estimates that avoid the need for multiple independent runs. Computational experiments on various test problems show that the algorithms and error estimates work in practice. The method is applied to compute the entropies of different sticky hard sphere systems. These predict the temperature or interaction energy at which loops of hard sticky spheres become preferable to chains.

Leen Alawieh, Jonathan Goodman, John B. Bell

A new algorithm is developed to tackle the issue of sampling non-Gaussian model parameter posterior probability distributions that arise from solutions to Bayesian inverse problems. The algorithm aims to mitigate some of the hurdles faced by traditional Markov Chain Monte Carlo (MCMC) samplers, through constructing proposal probability densities that are both, easy to sample and that provide a better approximation to the target density than a simple Gaussian proposal distribution would. To achieve that, a Gaussian proposal distribution is augmented with a Gaussian Process (GP) surface that helps capture non-linearities in the log-likelihood function. In order to train the GP surface, an iterative approach is adopted for the optimal selection of points in parameter space. Optimality is sought by maximizing the information gain of the GP surface using a minimum number of forward model simulation runs. The accuracy of the GP-augmented surface approximation is assessed in two ways. The first consists of comparing predictions obtained from the approximate surface with those obtained through running the actual simulation model at hold-out points in parameter space. The second consists of a measure based on the relative variance of sample weights obtained from sampling the approximate posterior probability distribution of the model parameters. The efficacy of this new algorithm is tested on inferring reaction rate parameters in a 3-node and 6-node network toy problems, which imitate idealized reaction networks in combustion applications.

Jonathan Goodman, Kevin K. Lin, Matthias Morzfeld

Implicit samplers are algorithms for producing independent, weighted samples from multi-variate probability distributions. These are often applied in Bayesian data assimilation algorithms. We use Laplace asymptotic expansions to analyze two implicit samplers in the small noise regime. Our analysis suggests a symmetrization of the algo- rithms that leads to improved (implicit) sampling schemes at a rel- atively small additional cost. Computational experiments confirm the theory and show that symmetrization is effective for small noise sampling problems.