**Research Interests**

Determining the statistical properties of nonlinear dynamical systems is a problem of major interest in many areas of science and engineering. Even with recent theoretical and computational advancements, no broadly applicable technique has yet been developed for dealing with the challenging problems of high dimensionality, model uncertainty, lack of regularity, multi-scale features and random frequencies. My research activity has been recently focused on developing new theoretical and computational methods for uncertainty quantification and dimensional reduction in large scale stochastic dynamical systems. In particular, I have been working on the Mori-Zwanzig formulation, hierarchical tensor methods for the numerical solution to high-dimensional PDEs, and the numerical approximation of functional differential equations (e.g., Hopf characteristic functional equations).

**Current Ph.D Students:** Alec Dektor, Abram Rodgers

**Former M.S. Students: **Schuyler Krawczuk, Alec Dektor.

**Former PhD Students:** Catherine Brennan, Yuanran Zhu (now at UC Merced), Heyrim Cho (now at UC Riverside, co-advised with George Karniadakis), Paris Perdikaris (now at U Penn, co-advised with George Karniadakis).

**ICERM-2020: **http://icerm.brown.edu/programs/sp-s20/

**Numerical approximation of FDEs: ** http://www.soe.ucsc.edu/news/prof-venturi-addresses-long-standing-open-problem-computational-mathematics

**Appointm****ents**

**Research Vision**

Several multi-disciplinary areas bridging applied mathematics, engineering and computing sciences are currently in a situation that perhaps is unprecedented. On the one hand, we have enough computing power to simulate systems with billions of degrees of freedom, opening the possibility to perform deterministic DNS simulations of turbulent flows at reasonable resolutions or integrate atomistic systems with billions of molecules. On the other hand, there is an increasingly growing interest towards systems for which we do not know the governing equations or we might be able to determine them only locally and in an approximate form. Examples of such systems are stochastic models of brain or large random networks, heterogeneous materials, DNA and RNA folding, atomistic descriptions of fluids, solids and colloids. The computability of any realistic representation of such systems beyond micro/nano scale is out of reach of current scientific computing capabilities.

These observations raise deep philosophical questions regarding the appropriateness of the mathematics we are using to describe complex stochastic dynamical systems, and the validity of our computations. Local modeling and coarse graining are key elements for modern theoretical and computational approaches to large-scale stochastic systems. In this framework, we look for reduced-order equations for quantities of interest instead of attempting to determine the whole dynamics. The Boltzmann equation of classical statistical mechanics is a remarkable example of theoretical coarse graining, reducing a high-dimensional phase space (positions and momenta of N particles) to 6 phase variables. Such drastic dimension reduction can be effectively achieved, e.g., by using methods of statistical physics, or data-driven machine learning. Similar formulations can be applied to systems of stochastic ODEs or stochastic PDEs. In this setting, coarse graining can be seen as deriving reduced-order equations for phase space functions. An effective framework to compute the solution to such reduced-order equations, however, is still lacking, despite recent theoretical and computational advances. Modeling and understanding complex stochastic dynamical systems for which we dont have equations requires a new vision, most likely a new type of mathematics, and perhaps a substantial re-thinking of the questions we are addressing.