Medical Imaging

We focus on mathematical modeling, analysis, reconstruction, and visualization in various imaging modalities. Often the problem arising in imaging is the typical inverse problem, which is highly ill-posed. To overcome such difficulties, a number of mathematical tools are required: Partial differential equations, harmonic analysis, numerical analysis, and scientific computing. Additionally, several image processing techniques are applied to the quality of reconstruction and diagnosis. Research includes:

  • Electrical Impedance Tomography (EIT)
  • Magnetic Resonance Electrical Impedance Tomography (MREIT)
  • Magnetic Resonance Imaging (MRI)
  • Computed Tomography (CT)
  • Ultrasound Systems

Computational & Theoretical Fluid Dynamics

Faculty and students will focus on computational turbulence to study the fundamentals of turbulent flows, to develop a high-fidelity model for complex flows, and to pursue various applications in mechanical and environmental engineering. Research will include:

  • Understanding fundamental turbulence
  • Particle-laden turbulent flow
  • Numerical algorithm for complex flows
  • Turbulence modeling
  • Applications in environmental fluid dynamics

Numerical Analysis & Scientific Computing

This team focuses on developing novel computational algorithms and related mathematical tools to address fundamental scientific and engineering problems. It covers mixed finite element methods, finite volume methods, adaptive procedure, multiscale computations and hybrid discontinuous Galerkin methods for elliptic and parabolic equations and their applications to Navier-Stokes equations. In interdisciplinary teams, research will be conducted in numerical methods and mathematical modeling, including:

  • Flows in porous media
  • Numerical weather prediction
  • Mathematical biology
  • Medical imaging and CFD

PDE Theory & Analysis

We focus on mathematical theory for partial differential equations arising from various physical and biological problems. Understanding of nonlinear phenomena is one of the most important motivations for developing various mathematical tools for analyzing differential equations and their discretization. Research will include:

  • Regularity theory for nonlinear elliptic PDEs
  • Elliptic and parabolic boundary value problems
  • Optimal transportation and related topics
  • Mathematical theory for fluid flows
  • Geometric maps and properties of solutions
  • Applications to inverse problems