### New evolution equations to the joint response excitation PDF for stochastic modeling: Theory and numerical methods – NSF

We are developing new theory and corresponding numerical algorithms for addressing fundamental open questions in stochastic modeling of physical and biological systems, e.g., the curse-of-dimensionality, the lack of regularity and the long-time integration of stochastic systems. Such problems arise in applications involving processes with small relative correlation length or large number of random parameters, and for time-dependent nonlinear systems subject to uncertainty. We base our work on the time-evolution of the joint probability density function (PDF) between the system’s response and the stochastic excitation. To this end, by using functional integral methods we determine new types of linear deterministic partial differential equations satisfied by the joint response-excitation PDF associated with the stochastic solution of nonlinear stochastic ordinary and partial differential equations. So far we have developed theory for nonlinear and for quasilinear first-order stochastic PDEs subject to random boundary conditions, random initial conditions or random forcing terms. In preliminary work, we have verified the correctness and accuracy of the new evolution equations by comparing the solutions obtained from the joint PDF approach with the solutions obtained directly from the nonlinear stochastic equations using probabilistic collocation methods. For higher-order equations, such the stochastic wave equation or the Oberbeck-Boussinesq thermal convection equations, we propose to develop a new PDF method based on differential constraints for the PDF of the solution. This general methodology allows us to determine a new hierarchy of unclosed equations for the PDF of the solution and some of its derivatives. The new hierarchy of equations is expected to be formally equivalent to the Lundgren-Monin-Novikov and the BBGKY hierarchies. We plan to investigate the theoretical and numerical effectiveness of this new approach for high-dimensional random systems, such as random flows subject to high-dimensional random boundary or initial conditions in bounded domains.

The proposed work will have significant and broad impact as it will set new 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’s CAVE as an opportunity to educate students about simulation, predictability, and other issues of computational science and applied mathematics.