Research Interests

  • Stochastic modeling and uncertianty quantification
  • Numerical analysis and high-performance scientific computing
  • Data-driven and 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, 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, data-driven modeling, 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), Paris Perdikaris (now at U Penn).

Former Postdoctoral scholars: Jason M. Dominy

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

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

 

Appointments

  • 2019 - present, Associate Professor of Applied Mathematics, UC Santa Cruz.
  • 2015 - 2019, 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, physics, 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, e.g., DNS simulations of turbulent flows at reasonable resolution or atomistic systems with millions 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 ubiquitous in science, e.g., mathematical models of brain, large random networks of interacting agents/individuals, random heterogeneous materials, DNA and RNA folding, and mesoscopic descriptions of solids and fluids. In all these cases it is extremely difficult to derive a computable equation that accurately describes the collective behavior of the system, or the dynamics of a quantity of interest. 

Recent advances in data-driven modeling and artificial intelligence, however, open the possibility to discover equations from data, but at the same time raise deep philosophical questions regarding the appropriateness of the mathematics we are using to model such systems, and therefore the validity of our computations.  This fundamental problem requires a new vision, most likely a new type of mathematics, and perhaps a substantial re-thinking of the questions we are addressing.