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

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The PI proposes to develop new effectrive methods for solving stochastic partial differential equations (SPDEs). In particular, the PI will address two outstanding issues in polynomial chaos (PC) methods for modeling uncertainty in computer 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 or large number of independent random parameters. The two approaches are complementary to each other as problems with very small correlation length can be effectively modeled by white noise processes.The new ideas are the use of ANOVA decomposition and the introduction of proper weighted Wiener chaosspaces and stochastic convolution products. ANOVA provides a 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 much lower than the nominal dimensionality. In preliminary work, the PI has demonstrated the effectiveness of the new approach in approximating efficiently problems with more than 500 dimensions.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 new experiments are designed and the type of questions that can be addressed, while the interaction between simulation and experiment will become more meaningful and more dynamic.The PI plans to incorporate these new ideas in engineering and applied mathematics courses at Brown. Sponsored graduate and undergraduate students will be involved in this research and will interact withall 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. He also plans 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 the PI plans to expand this activity nationwide.