Research activities

I conduct scientific modeling by mathematical and numerical analysis with research focused on high-performance methods for solving partial differential equations (PDEs) in the study of wave propagation for natural phenomena. Much of my work is motivated by the fundamental desire to construct methodologies that faithfully preserve the underlying physics of computational models, and seeks to identify interdisciplinary problems in science and engineering that can provide mutual validation of both numerical simulation and experiment. Current interests are centered on
    • frequency-domain boundary element methods: fast multipole methods with applications to acoustic scattering, anisotropic mesh generation (Figure 1)
    • time-domain Fourier continuation methods: linear elasticity with applications to NDT (Figure 2) and seismology/geophysics (Figure 3), arterial flow with applications to cardiac physiology (Figure 4)
    • high-performance computing for both shared and distributed memory parallel clusters
mesh
Figure 1. Iterative mesh refinement for geometric singularities.
phasescatter
Figure 2. Phase field for scattering by a thru-hole [1,2].
timeresponse
Figure 3. FC-simulated seismogram of a classical earthquake scenario [5].
boundarymodel
Figure 4. Boundary model for truncated vasculature [3].

International publications & patents

Talks

  • On anisotropic mesh adaptation for BEM with the fast multipole method:
    • Contributed talk at the Paris-London BEM Workshop 2017, University College London, UK (06/2017)
    • Contributed talk at the 5th BEM on the Saar, Universität des Saarlandes, Germany (05/2017)
    • Contributed talk at the 13th International Conference on Mathematical and Numerical Aspects of Waves, University of Minnesota, USA (05/2017)
  • On Fourier continuation for linear elastodynamics:
    • Contributed talk at the 13th International Conference on Mathematical and Numerical Aspects of Waves, University of Minnesota, USA (05/2017)
    • Invited seminar in Seismology, Institut de Physique du Globe de Paris, Université Sorbonne Paris Cité, France (11/2015)
  • On other topics:
    • Invited lecture on delay differential equations with clinical application in respitory physiology, Dept. of Medical Engineering, California Institute of Technology (Caltech), USA (01/2015)
    • Contributed poster on optimized sensor location for advection-diffusion equations, NSF-VIGRE Symposium 2006, Rice University, USA (04/2006)
    • Invited joint talk on flux corrections for the Boltzmann equation, Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, USA (08/2004)

Teaching activities

California Institute of Technology (Caltech)

  • Lecturing, writing course materials, managing teaching assistants and their sections for:
    • Course: ACM 95/100a Complex Variables (2007-2013)
      Students: 3rd and 4th year undergraduate and 1st year graduate
      Topics: Analyticity, Laurent series, contour integration, residue calculus.
    • Course: ACM 95/100b Ordinary Differential Equations (2007-2013)
      Students: 3rd and 4th year undergraduate and 1st year graduate
      Topics: Initial value problems, boundary value problems, Sturm-Liouville theory, eigenfunction expansions, transform methods, Green’s functions.
    • Course: ACM 95/100c Partial Differential Equations (2007-2013)
      Students: 3rd and 4th year undergraduate and 1st year graduate
      Topics: Heat equation, separation of variables, Laplace equation, transform methods, wave equation, method of characteristics, Green’s functions.

Rice University and Baylor College of Medicine

  • Writing solutions, grading and delivering tutorials for
    • Course: CAAM 415 Theoretical Neuroscience (2005)
      Students: graduate and medical
      Topics: Theoretical methods used to mathematically model the properties of nerve cells and the processing of information by neuronal networks.
    • Course: CAAM 336 Differential Equations (2004-2005)
      Students: 3rd year undergraduate
      Topics: Classical solution techniques including Green's functions, Fourier series, finite element methods for initial and boundary value problems arising in diffusion and wave propagation phenomena.