Research Interests

  • Stochastic modeling and uncertianty quantification
  • Numerical analysis and high-performance scientific computing
  • Reduced-order modeling
  • Mori-Zwanzig formulation
  • Numerical approximation of functional differential equations
  • Theoretical and computational fluid dynamics

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).

PhD Students: Yuanran Zhu, Catherine B. Brennan

Master Students: Benjamin Gruey, Nathan Heller. 

Postdoc: Dr. Jason Dominy

Appointments

  • 2015 - present, Assistant Professor of Applied Mathematics, UC Santa Cruz.
  • 2010 - 2015, Research Assistant Professor of Applied Mathematics, Brown University.
  • 2006 - 2010, Postdoctoral Research Associate at the Department of Energy, Nuclear and Environmental Engineering, University of Bologna.

 

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, in the upcoming future, deterministic DNS simulations of turbulent flows at reasonable resolutions or integrate atomistic systems with billions and 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 (large random networks), heterogeneous random 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 stochastic dynamics, which is beyond current (and future) computational capabilities. 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 through a truncation of the BBGKY hierarchy arising from the Liouville equation or, equivalently, by using the Nakajima-Zwanzig-Mori projection operator framework. A similar formulation can be applied to more general systems of stochastic ODEs or stochastic PDEs. In this setting, coarse graining can be seen as deriving reduced-order equations for phase space functions (quantities of interest). An effective framework to compute the solution to such reduced-order equations, however, is still lacking, despite recent theoretical and computational advances. Thus, the problem of  high dimensions, lack of regularity and model uncertainty in large scale simulations of nonlinear stochastic dynamical systems is still unsolved, and most likely it requires a new vision, perhaps a new type of mathematics and certainly a substantial re-thinking of the questions we are addressing.