(79d) Comparison of Sampling Strategies for Kriging-Based Reduced Order Models of Nanoparticle Dynamics

The task of identifying an empirical model is difficult for nonlinear dynamic systems, especially when the sampled data is limited, expensive or noisy. Since an empirical model is based on sampled data, the choice of samples taken plays an important role in the prediction and accuracy of the resulting model. This is particularly true in nonlinear systems when the functional form of the empirical model is not known a priori. Here we consider the effect that sampling strategies have in a regression-based reduced order model to describe a nonlinear system. Here we apply our approach to nanoparticle dynamics.

One of the the most challenging problems in nanoparticle synthesis is the control over particle size and distribution, while sustaining a high yield of the process [1]. Platinum nanoparticles are needed for catalysts in fuel cells and for drug delivery, and a monodisperse size distribution is required for these applications. This nanoscale process can be simulated using a kinetic Monte-Carlo (kMC) method, based on a stochastic model that represents the sequence of chemical reactions to synthesize platinum nanoparticles on carbon nanotubes.

Because of the high computational demands of the kMC simulations, an approximated model is needed for engineering tasks like process optimization and control. A new methodology to create approximate models for multivariate stochastic dynamic simulations is employed here, using simulated stochastic data sampled from the kMC simulations. The methodology is based on kriging [2], a statistical technique?coming from geostatistics?that interpolates the value of a random field at an unobserved location, using observations of its value at nearby locations. We combine kriging with model reduction techniques to create a reduced state for the kMC simulation data.

We compare sampling strategies for building and refining an approximated model for stochastic dynamic simulations. Specifically, we compare a sequential design of computer experiments, guided by the mean squared prediction error of the kriging model, to a uniform sampling strategy. This work describes the impact of this additional sampling in the performance of the approximated model, for the local and global prediction of the state variables, and contrasts the results with the original kMC simulation. Optimized selection of new sampled data from the kMC simulation can improve the performance of the approximated model, while minimizing the number of sample points and thus the computational time for building the approximated model.

References

[1] Y. Zhang and C. Erkey, ?Preparation of supported metallic nanoparticles using supercritical fluids: A review,? Journal of Supercritical Fluids, vol. 38, pp. 252?267, 2006.

[2] N. Cressie, Statistics for Spatial Data. Wiley Interscience, 3 ed., 1993.