Funding NSF – DMS-0915077

Overcoming the Bottlenecks in Polynomial Chaos: Algorithms and Applications to Systems Biology and Fluid Mechanics

Intellectual Merit: We propose to develop new theory and algorithms for addressing two outstanding issues in polynomial chaos (PC) methods for modeling uncertainty in simulations of physical and biological systems. The first one is related to treating effectively many stochastic dimensions while the second one is related to modeling accurately white noise. Such problems arise in applications with small relative correlation length A/L or large number of independent random parameters. The two approaches are complementary to each other as problems with very small A/L can be effectively modeled by white noise processes. The new ideas are the use of ANOVA decomposition and the introduction of proper weighted Wiener chaos spaces and Wick products. Specifically, we will combine multi-element polynomial chaos (MEPCM), which can effectively deal with discontinuities in parametric space, with hierarchical functional decomposition that exploits the effective dimensionality of the system. This type of dimension-wise decomposition can effectively break the curse of dimensionality in certain approximation problems in which the effective dimensionality is significantly lower than the nominal dimensionality. The introduction of weighted Wiener spaces in polynomial chaos allows representation of stochastic solutions with infinite variance not possible with the existing PC theory. In preliminary work we have demonstrated the effectiveness of MEPCM-ANOVA in approximating efficiently problems with more than 500 dimensions. We also present convergence theory and numerical results for white noise processes demonstrating exponential decay of the error in the newly defined norms. We propose to develop error estimates for both approaches and extend them to time-dependent stochastic PDEs. We also plan to apply these methods to sensitivity analysis in cellular signaling networks with hundreds of uncertain parameters, and in modeling external random disturbances in turbulent flow past a cylinder.

Broad Impact and Outreach: The proposed work will have significant and broad impact as it will set rigorous foundations in uncertainty quantification, data assimilation and sensitivity analysis for many physical and biological systems. For example, in computational fluid dynamics, it will establish a robust and efficient framework to endow simulations with a composite error bar that goes beyond numerical accuracy and includes uncertainties in operating conditions, the physical parameters, and the domain. The proposed work is transformative as it will make stochastic simulations the standard rather than the exception. It will also affect fundamentally the way we design new experiments and the type of questions that we can address, while the interaction between simulation and experiment will become more meaningful and more dynamic.

The new knowledge will contribute towards understanding noisy dynamical systems, stochastic PDEs, data assimilation, and parametric uncertainty. We plan to incorporate these new ideas in engineering and applied mathematics courses we teach at Brown. Sponsored graduate and undergraduate students will be involved in this research and will interact with all senior personnel that includes several international visitors.

The PI will work closely with undergraduate students who are involved with outreach activities through two very effective organizations at Brown that target women in science and engineering and also middle school students. We also plan outreach activities for inner-city high schools by developing along with the teachers computer-based interactive math learning strategies. Preliminary results working with the MET school have been very encouraging, and we plan to expand this activity nationwide. We will use immersive flow visualizations at Brown University's CAVE as an opportunity to educate students about simulation, predictability, and other issues of computational science and applied mathematics.

Publications