Applied Mathematics Research

Research in the Applied Mathematics Department focuses on mathematical modeling, including dynamical systems, mathematical biology, fluid dynamics, control theory and uncertainty quantification, with applications to astrophysics, biology, engineering and geophysics.

Research Areas of Excellence

  • 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. Professor Halder works on dynamics and control of systems with uncertainties with application focus on large scale cyberphysical systems such as the unmanned aerial systems traffic management and the smart grid.
  • Fluid dynamics: Faculty in AM 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.
  • 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. Professor Gomez uses mathematical models to study single cell dynamics and collective behaviors, as well as, direct design of genetic networks that elicit specific dynamical phenomena. 
  • 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.
  • 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.