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.
- Co-PI, NIMBioS Working Group: Multiscale Vectored Plant Viruses
(Co-PIs) Linda J. S. Allen (Texas Tech U.) and A. Power (Cornell), Multiscale aspects of vector transmission of plant viruses. Meeting dates : December 14-16, 2015; June 22-24, 2016; December 19-23, 2016
- 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.
- Computational Electromagnetics
- Compatible Discretizations for Maxwell Models in Nonlinear Optics
Collaborators: Yingda Cheng (MSU), Fengyan Li (RPI)
Current Funding NSF Computational Mathematics Grant #1720116
OP: Collaborative Research: Compatible Discretizations for Maxwell Models in Nonlinear Optics .
Past Funding ICERM, Collborate@ICERM, June 6-10, 2016 .
Mathematisches Forschungsinstitut Oberwolfach (MFO), Germany, August 7-20, 2016.
Nonlinear optics is the study of the behavior of light in nonlinear media. This field has developed into a significant branch of physics since the introduction of intense lasers with high peak powers. Compared with the huge amount of literature on simulations of Maxwell's equations in linear optical media, developing mathematically well-understood computational tools for space-time models in nonlinear optical media is relatively less tackled by the computational math community. Major advancement in this aspect can provide the scientific community reliable and accurate tools to simulate and to understand nonlinear optical phenomena, which hence can be better harnessed for practical applications.
The objective of the collaborative research program is to make significant advances in the understanding and simulations of Maxwell models in nonlinear optics with the aim of: (1) providing robust simulation tools for the nonlinear optics community, (2) developing novel mathematical and numerical techniques that are specifically tailored for different types of nonlinear models. The specific technical aspect includes the development of energy-stable time discretizations as well as two classes of spatial discretizations, discontinuous Galerkin methods and mimetic finite difference methods, for the propagation of electromagnetic waves in nonlinear (dispersive) optical media. Both macroscopic phenomenological and microscopic quantum descriptions will be considered for modeling the nonlinear material responses. Applications involving femtosecond soliton propagation, harmonic generation, self focusing, among others will be simulated and compared to existing time domain methods. This collaborative program is strengthened by a cohesive research plan that relies on the complementary expertise of each principal investigator. The educational components are integrated through the training of graduate students.
- Mimetic Methods for Maxwell's Equations in Dispersive Media
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 linear and nonlinear dispersive media and metamaterials.
- Operator Splitting Methods for Maxwell's Equations in Complex Media
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, electrically dispersive and magnetic media to obtain efficient numerical discretizations.
- PI, NSF-COMPUTATIONAL MATHEMATICS :
OP: Collaborative Research: Compatible Discretizations for Maxwell Models in Nonlinear Optics.
- 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
Dr. Bokil's Departmental Homepage
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Department of Mathematics
Oregon State University