DANIELE VENTURI 

Assistant Professor of Applied Mathematics



Appointments
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, multiscale features and random frequencies. My research activity has been recently focused on developing new theoretical and computational methods (including the development of parallel scalable methods for highperformance computing) for stochastic analysis in large scale dynamical systems where geometry, initial conditions, boundary conditions, forces or physical parameters are set to be random. In particular, I have been working on the NakajimaZwanzigMori formalism for probability density function equations, stochastic domain decomposition methods, numerical techniques for highdimensional PDEs, and numerical approximation of functional differential equations (e.g., equations involving Hopf characteristic functionals). Research Vision Several multidisciplinary 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 ubiquitous in nature, e.g., stochastic models of brain through complex random networks, heterogeneous random materials, DNA and RNA folding, atomistic descriptions of fluids, solids and colloids. The computability of any realistic representation of these systems beyond micro/nano scale is far beyond the capabilities of scientific computing for many years from now (in an optimistic view). 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 largescale stochastic systems. In this framework, we look for reducedorder 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 highdimensional 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 NakajimaZwanzigMori projection operator framework. A similar formulation can be applied to more general systems of stochastic ODEs, e.g., to the semidiscrete form of stochastic PDEs. In this case the phase variables are not simply positions and momenta of particles but rather Fourier coefficients of the series expansion of the solution to the SPDE. In this generalized framework, coarse graining can be seen as deriving reducedorder equations for suitable phase space functions (quantities of interest), e.g., the Fourier series representation of the solution to the SPDE. An effective framework to compute the solution to such reducedorder 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 rethinking of the questions we are addressing. Short Biographical Sketch I received the combined B.S. and Sc.M. degree in Mechanical Engineering at the University of Bologna in 2002. Then I joined the Department of Energy, Nuclear and Environmental Engineering (now Department of Industrial Engineering) at the University of Bologna where I received my Ph.D. degree in Applied Physics (thermofluid dynamics) in 2006. I was a Postdoctoral Research Associate at the same Department from 2006 to 2010. In those years, I performed research in theoretical, computational and experimental fluid dynamics, and taught courses in heat transfer, fluid mechanics and thermodynamics. Since 2010 I have been appointed as Assistant Professor of Applied Mathematics (Research) at Brown University and thought courses in scientific computing and numerical approximation of PDEs. My research interests embrace a wide range of topics. In particular, I have been working on reducedorder stochastic modeling, biorthogonal methods for random processes and fields, differential geometry, nonlinear functional analysis, field theoretic methods and principles of least action, probability density function methods, stochastic PDEs, WickMalliavin aprooximation, multifidelity modeling, data assimilation, stochastic domain decomposition methods, and functional differential equations. While at University of Bologna, I was also involved in a more applied research conducted in collaboration with various companies. In particular I developed projects on heat pipes, experimental fluid dynamics (3DDPIV), granular flows, solar tracking, and vacuum processes.
