My general research interests are in applied mathematics,
scientific computing and numerical analysis. My primary research interests
are in the numerical solution of a variety of partial differential equations. Specifically, I work in the areas of computational electromagnetics and computational magnetohydrodynamics which involve the numerical discretization of Maxwell's equations and the magnetic induction equation. I am also working on several problems in mathematical biology,
specifically involving population dynamics, epidemiology and spatial ecology. In all these fields, I am interested in multiscale aspects that arise due to a variety of mechanisms operating at varying spatial and temporal scales. In addition, I am interested in how uncertainty propagates through these systems and studying techniques for quantifying such uncertainty.
Current Research Projects
- Mathematical Epidemiology
- Multiscale Vectored Plant Diseases
Collaborators: Linda J. S. Allen (Texas Tech U.), Alison Power (Cornell), Lou Gross (UTK),
Mike Jeger (Imperial College, UK), Nick Cunnife (Cambridge, UK), Zhilan Feng (Purdue), Cherie Briggs (UCSD), Karen Garett (KSU), Frank Hilker (Germany/Institute of Applied Systems, Osnabruck Univ.), Fred Hamelin (Agrocampus Ouest,France)
Postdocs Carrie Manore (Tulane/LANL), Megan Rua (NIMBioS)
Funding National Institute of Mathematical and Biological Synthesis (NIMBioS), Working Group 2015-2017, UT Knoxville, TN.
This project involves current problems related to multiscale aspects of the spatial and temporal transmission and the evolution of vectored plant viruses. The goals are to derive novel mathematical, statistical, and computational methods that incorporate multiple hosts and multiple pathogens operating at varying spatial and temporal scales to bring insight into the effects of climate change and human activities on the emergence of new plant viruses.
- Stochastic Models and Optimal Control for Vectored Plant Disease Systems
Collaborators: Linda J. S. Allen (Texas Tech U.) and Suzanne Lenhart (UTK)
Past Funding National Institute of Mathematical and Biological Synthesis (NIMBioS), UT Knoxville, TN. Short Term Visit (V. A. Bokil and L. J. S. Allen)
This project involves construction and analysis of stochastic models based on continuous time Markov chains (CTMC) and stochastic differential equations (SDEs) for the spread of vectored plant diseases. Our case studies are the Barley/Cereal Yellow dwarf virus (B/CYDV) suite and the African Cassava Mosaic virus (ACMV). Optimal Control for controlling the spread of these viruses in crop systems is also an important part of this project.
Related Past Funding
- Co-PI, NIMBioS Investigative Workshop: Vectored Plant Diseases
(Co-PIs) Linda J. S. Allen (Texas Tech U.), E. T. Borer (U. Minnesota), and A. Power (Cornell)
- PI, NSF-AWM Mentoring Travel Award Stochastic Patch Models for the Spread of Disease in Heterogeneous Landscapes with Linda J. S. Allen, Texas Tech University
- Computational Magnetohydrodynamics
- Applying Computational Methods to Determine the Electric Current Densities in a
Magnetohydrodynamic Generator channel from External Magnetic Flux Density Measurements,
Collaborators: Nathan Gibson (OSU-Mathematics), Rigel Woodside (NETL)
Students Involved : Duncan McGregor (PhD expected June 2016)
Past Funding National Energy Technology Laboratory (NETL), Albany, OR
This project involves development of a computational approach to determine current densities in a magnetohydrodynamic (MHD) generator from magnetic field measurements. The approach may also be applicable to fuel cells or any other energy system where electric current paths are of interest. The physical basis for the anticipated approach is in the relation of magnetic fields to electric current as described in
Maxwell's equations. A model for the electric and magnetic fields inside of an MHD generator will be developed which incorporates fluid dynamics. The research activity will demonstrate the proof-of-concept by building and applying a numerical code to determine current density in a single section of the channel, starting with simple and known current paths. The method and code will be in full 3D, but will begin with the static case, and will be validated against model outputs from NETL’s MHD computational fluid dynamics (CFD) simulation development activity. The model will be discretized using mimetic finite differences and the corresponding inverse problem for the densities will be performed in a nonlinear least squares formulation. In order to account for measurement and model error, the inverse problem will be cast into a Bayesian framework. This allows credibility levels of detected arcs to be computed. Further, minimum sensor sensitivities and optimal placement for use in a deployed measurement system can be inferred from the framework.
- Compatible Discretizations for Efficient Uncertainty Quantification of Magneto-Hydro-Dynamic Models
Collaborators: Yingda Cheng (MSU), Fengyan Li (RPI)
Past Funding ICERM, Collborate@ICERM, June 6-10, 2016 .
Mathematisches Forschungsinstitut Oberwolfach (MFO), Germany, August 7-20, 2016.
Magnetohydrodynamic (MHD) models arise in many applications in astrophysics, electrical engineering and aerospace (such as MHD power generation), supernovas, and fast magnetic reconnection leading to solar flares and auroras, among others.
In this project, we propose to construct and analyze new numerical discretizations of the magnetic induction equation (MIE), a sub-model of MHD equations, with very general constitutive laws that correspond to ideal, resistive and Hall MHD models. In addition, we will quantify the effects of uncertainty in the velocity field and other parameters of the MHD system on the induced magnetic field, by modeling the velocity and parameters as random variables to obtain stochastic MHD models.
- Computational Electromagnetics
- Mimetic Methods for Maxwell's Equations
Collaborators: Vitaliy Gyrya (LANL), Konstantin Lipnikov (LANL)
Students Involved : Duncan McGregor (PhD expected June 2016)
Funding National Science Foundation, Los Alamos National Laboratory.
We study a novel strategy for minimizing the numerical dispersion error in edge
discretizations of Maxwell's equations on square and rectangular meshes
based on the mimetic finite difference (MFD) method. We call this strategy M-adaptation. We have recently constructed M-adapted methods that exhibit fourth order numerical dispersion for Maxwell's equations in non-dispersive dielectrics. Our current work involves the non-trivial extension of the M-adapted method to Maxwell's equations in dispersive media and metamaterials.
- Operator Splitting Methods for Maxwell's Equations
Students Involved : Puttha Sakkaplangkul (PhD expected June 2017)
Funding National Science Foundation.
Operator Splitting is a powerful technique for solving complicated multi-physics problems, in which the model for a given physical/biology system involving multiple mechanisms operating at varying spatio-temporal scales are replaced by a sequence of time discretized models each involving a single mechanism. We apply these methods to Maxwell's equations in non-dispersive as well as dispersive media to obtain efficient numerical discretizations.
Past Grant Funding
- PI, NSF-COMPUTATIONAL MATHEMATICS :
Time Domain Numerical Methods
for Electromagnetic Wave Propagation Problems in Complex Dispersive
Students Supported: Aubrey Leung (REU, 2010, BS Thesis 2011), Anna Kirk (MS Thesis 2011), Olivia Keefer (MS Thesis 2012), Duncan McGregor (MS 2013, PhD ongoing).
- Stochastic Numerical Methods for Interface Problems
- Co-PI, NSF-MATHEMATICAL BIOLOGY :
Residence and First Passage Time Functionals in Heterogeneous Ecological Dispersion
Edward Waymire (PI), Nathan Gibson (Co-PI), Enrique Thomann (Co-PI) and Brian Wood (Co-PI).
- Co-PI, NSF-MATHEMATICAL GEOSCIENCES, OPPORTUNITIES FOR RESEARCH CMG :
Mathematical and Experimental Analysis of Reactive
Transport in Discontinuous Porous Media.
Brian Wood (PI),
Enrique Thomann (Co-PI), Edward Waymire (Co-PI) and Dorthe Wildenschild
Mont St. Michel, Brittany, France, Oct 2016
Dr. Bokil's Departmental Homepage
Department of Mathematics Events
Department of Mathematics
Oregon State University