Publications

Bayesian statistics provides a foundation for inference from noisy and incomplete data, a natural mechanism for regularization in the form of prior information, and a quantitative assessment of uncertainty in the inferred results. Inverse problems - representing indirect estimation of model parameters, inputs, or structural components - can be fruitfully cast in this framework. Complex and computationally intensive forward models arising in physical applications, however, can render a Bayesian approach prohibitive. This difficulty is compounded by high-dimensional model spaces, as when the unknown is a spatiotemporal field. We present new algorithmic developments for Bayesian inference in this context, showing strong connections with the forward propagation of uncertainty. In particular, we introduce a stochastic spectral formulation that dramatically accelerates the Bayesian solution of inverse problems via rapid evaluation of a surrogate posterior. We also explore dimensionality reduction for the inference of spatiotemporal fields, using truncated spectral representations of Gaussian process priors. These new approaches are demonstrated on scalar transport problems arising in contaminant source inversion and in the inference of inhomogeneous material or transport properties. We also present a Bayesian framework for parameter estimation in stochastic models, where intrinsic stochasticity may be intermingled with observational noise. Evaluation of a likelihood function may not be analytically tractable in these cases, and thus several alternative Markov chain Monte Carlo (MCMC) schemes, operating on the product space of the observations and the parameters, are introduced.

This program was focused on the development of a fluorogenic chemosensor family that could tuned for reaction with electrophilic (e.g. chemical species, toxins) and nucleophilic (e.g. proteins and other biological molecules) species. Our chemosensor approach utilized the fluorescent properties of well-known berberine-type alkaloids. In situ chemosensor reaction with a target species transformed two out-of-plane, weakly conjugated, short-wavelength chromophores into one rigid, planar, conjugated, chromophore with strong long wavelength fluorescence (530-560 nm,) and large Stokes shift (100-180 nm). The chemosensor was activated with an isourea group which allowed for reaction with carboxylic acid moieties found in amino acids.

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The production of Ultra-cold molecules is a goal of many laboratories through out the world. Here we are pursuing a unique technique that utilizes the kinematics of atomic and molecular collisions to achieve the goal of producing substantial numbers of sub Kelvin molecules confined in a trap. Here a trap is defined as an apparatus that spatially localizes, in a known location in the laboratory, a sample of molecules whose temperature is below one degree absolute Kelvin. Further, the storage time for the molecules must be sufficient to measure and possibly further cool the molecules. We utilize a technique unique to Sandia to form cold molecules from near mass degenerate collisions between atoms and molecules. This report describes the progress we have made using this novel technique and the further progress towards trapping molecules we have cooled.

The Bayesian setting for inverse problems provides a rigorous foundation for inference from noisy data and uncertain forward models, a natural mechanism for incorporating prior information, and a quantitative assessment of uncertainty in the inferred results. Obtaining useful information from the posterior density - e.g., computing expectations via Markov Chain Monte Carlo (MCMC) - may be a computationally expensive undertaking, however. For complex and high-dimensional forward models, such as those that arise in inverting systems of PDEs, the cost of likelihood evaluations may render MCMC simulation prohibitive. We explore the use of polynomial chaos (PC) expansions for spectral representation of stochastic model parameters in the Bayesian context. The PC construction employs orthogonal polynomials in i.i.d. random variables as a basis for the space of square-integrable random variables. We use a Galerkin projection of the forward operator onto this basis to obtain a PC expansion for the outputs of the forward problem. Evaluation of integrals over the parameter space is recast as Monte Carlo sampling of the random variables underlying the PC expansion. We evaluate the utility of this technique on a transient diffusion problem arising in contaminant source inversion. The accuracy of posterior estimates is examined with respect to the order of the PC representation and the decomposition of the support of the prior. We contrast the computational cost of the new scheme with that of direct sampling. © 2005 American Institute of Physics.