Technology for a Changing World

Some of the areas of excellence of our department are:

- (S) Bayesian nonparametric methods: One important area of Bayesian methods is nonparametrics, which involves placing probability distributions on functions (the statistics of the 21st century) rather than on scalars or vectors (the statistics of the 18th through 20th centuries). Professor David Draper works on this and other topics, including Bayesian hierarchical modeling, stochastic optimization, Markov chain Monte Carlo methods, model uncertainty, quality assessment in health and education, risk assessment and applications in the social and environmental sciences. Professor Athanasios Kottas works on Bayesian nonparametric modeling and inference, mixture models, probability order constraints, quantile regression, spatial statistics, and survival analysis. Associate Professor Abel Rodríguez also works on Bayesian nonparametric modeling and inference, financial econometric models and statistical models for genomic and protemic data.

- (S) Computationally intensive Bayesian inference, prediction, and decision-making: Modern methods of Bayesian statistics employ Markov chain Monte Carlo (MCMC) techniques to draw inferences and make predictions and decisions. These methods are highly computationally intensive, and are crucial to the continued success of the Bayesian approach in applied problem-solving. Professor Herbie Lee works on computational methods, inverse problems, computer models, spatial inverse problems, machine learning, model selection and model averaging. Professor Raquel Prado works on Bayesian analysis of nonstationary time series, multivariate time series, biomedical signal processing, wavelets, statistical models for genomic data. Professor David Draper is also involved in this area. As an example of our work in this field, in 2003 we hosted an International Workshop on Bayesian Data Analysis. This workshop brought together approximately 160 researchers from 15 countries on 5 continents for 26 invited talks and 75 posters.

- (AM, S) Envirometrics: Professor Emeritus Marc Mangel studies population biology of disease and quantitative fishery science. Professor Bruno Sanso works on Bayesian predictive modeling of environmental variables in space and time, statistical inference from climate model output, Bayesian spatial modeling, modeling of changes in atmospheric variables and geostatistical applications.

- (AM) Control theory: Professor Qi Gong works on computational optimal control, trajectory optimization with aerospace applications, optimal control of uncertain systems, motion planning of autonomous multi-agent systems, and state/output feedback control of nonlinear systems.

- (AM) Fluid dynamics: Faculty in AMS are specialized in the study of astrophysical and geophysical fluid dynamics. Professor Pascale Garaud works on mathematical modeling of natural flows, numerical solutions of differential equations, planetary formation, and internal dynamics of stars with applications to astrophysics and geophysics. Professor Nic Brummell works on fluid dynamics, compressible convection, magnetohydrodynamics, turbulence, dynamos and other highly nonlinear systems; numerical methods, simulations and supercomputing. Nic Brummell and Pascale Garaud are both members of TASC (Theoretical Astrophysics at Santa Cruz). Prof. Daniele Venturi works on stochastic CFD (hp-spectral element methods with MPI) on complex geometries. This includes direct and large eddy stochastic simulations of isothermal and non-isothermal flows using multi-element polynomial chaos, probabilistic collocation, and sparse grids. Prof. Dongwook Lee’s research includes computational fluid dynamics governed by compressible nonlinear PDEs of hydrodynamics and magnetohydrodynamics. Such physical fluid phenomena arise in many different applications of science and engineering including space astrophysics, laboratory astrophysics (or high-energy-density physics), radiation hydrodynamics, reactive flows, geophysical flows, and aerodynamics.

- (AM) Mathematical biology: Professor Emeritus Marc Mangel works on mathematical modeling of biological phenomena, especially the evolutionary ecology of growth, aging, and longevity; quantitative issues in fishery management; mathematical and computational aspects of disease). Professor Hongyun Wang works on modeling of protein motors, with applications to nanotechnology, theoretical biophysics, energy transduction mechanism of protein motors; thermodynamics of small systems; partial differential equations; statistical physics; classical analysis and numerical analysis.

- (AM) High-performance computing: Prof. Nic Brummell works on astrophysically-related numerical models especially solar interior magnetohydrodynamics. His research involves heavy use of distributed supercomputing resources and powerful local visualisation workstations in order to understand numerical solutions of idealised models of solar and planetary fluid dynamics problems, in particular those involving convection and turbulence. Prof. Dongwook Lee’s interests focus on developing stable and efficient numerical methods for nonlinear fluid dynamics in order to simulate multi-physical phenomena using time-dependent, high-order accurate mathematical algorithms, especially on large-scale parallel computing architectures. Prof. Daniele Venturi works on spectral element methods for uncertainty quantification in the context of MPI coding, including stochastic CFD on complex geometries by using multi-element polynomial chaos or probabilistic collocation, stochastic thermal convection, etc. He also is interested in numerical methods for high-dimensional PDEs using Adaptive Discontinuous Galerkin methods for probability density function equations.

- (AM) Stochastic modeling and uncertainty quantification: Modeling stochastic nonlinear systems and determining their statistical properties is of major interest across many disciplines ranging from fundamental physics to engineering. Professor Venturi's research activity focuses on developing theoretical and computational methods for uncertainty quantification (UQ) in stochastic nonlinear systems, e.g., systems of stochastic ODEs and stochastic PDEs, where initial conditions, geometry, boundary conditions, forces or physical parameters are set to be random: this includes stochastic modeling of high-dimensional systems by using kinetic theory and methods of irreversible statistical mechanics; stochastic fluid dynamics; Mori-Zwanzig approach to dimensional reduction and uncertainty quantification; reduced-order modeling techniques; and design of engineering systems under uncertainty. Professor Wang's research on stochastic modeling focuses on two types of applications: i) single molecule studies in which kinetic models are compared based on their distinct and measurable stochastic features, and in which optimal experiments are designed for estimating model parameters; ii) search for randomly located or stochastically moving targets.