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Research
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My research is in the area of applied
mathematics with a special emphasis on analysis and development of new numerical methods. I am
interested in developing new finite element methods to approximate solutions of partial differential
equations defined on arbitrary surfaces. In recent years, I got interested in biological and medical
applications of partial differential equations and started several collaborations in these areas.
My current work related to these applications is on spatial epidemic models, avian influenza,
inverse problems arising in optical tomography and PET scan imaging. My research can be summarized under
the following topics;
- Finite Element Method on Arbitrary Surfaces
- Mathematical Biology
- Inverse Problems
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Finite Element Method on Arbitrary Surfaces
- Pattern formation on evolving surfaces
My current research in the area of finite element method focuses on developing new methods for reaction diffusion
systems defined on evolving surfaces. Typical examples of biological applications of reaction-diffusion systems are:
pattern formation in hydra, animal coat markings, butterfly wing pigmentation patterns, and shell pigmentation
patterns. Detailed studies of these models have been carried out on spatial domains of one dimension and two dimensions
with simple geometries. In nearly all these cases, the domain was considered fixed in size, however recent experimental
evidence shows that domain growth may play a crucial role in pattern formation. My research focuses on approximating solutions
of reaction diffusion systems that model pattern formation on evolving surfaces.
- Radially projected finite elements
Finite element method applied to domains like spheres, ellipsoids, cylinders, is a challenging problem
due to the geometry of such domains. Partial differential equations defined on such domains have many
applications in climate modeling, weather prediction and in many engineering problems. We developed and
analyzed a new method to approximate solution of such pdes. Finite element discretization of spheres (or ellipsoids, cylinders, and torus)
are obtained by the radial projection. Thus finite elements constructed using radial projection are called radially projected finite elements.
Following figures are the examples of meshes generated by radial projection. A uniform and non-uniform grid on the sphere, a grid on a
cylindrical shell, and a grid on ellipsoidal shell.
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Mathematical Biology
- Two strain SIS model with diffusion
My interest in mathematical biology includes spatially heterogeneous
deterministic epidemic models. Epidemiological parameters such as transmission and recovery rates
are clearly spatially dependent. However, it is surprising to see that in most epidemic models these
rates are assumed to be constants. We let transmission and recovery rates be spatially dependent to better
understand the effect of spatial heterogeneity on the persistence and extinction of a disease. Furthermore,
we study the effects of spatial heterogeneity on the competition of pathogen strains. The main question that
we address is whether the presence of spatial structure would allow the two strains to coexist, as the
corresponding spatially homogenous model leads to competitive exclusion. Following figure shows a simulation of
coexistence of the strains using the finite element method. This numerical simulation supports our theoretical result
that spatial heterogeneity is essential for coexistence of the strains.
- Avian influenza
My current research in the area of mathematical biology focuses on comparing avian
influenza models by their fit to the human avian influenza cases. In particular, we are interested in multi-species
mathematical models of avian influenza. The purpose of my current research is to understand
the reasons of periodicity (or seasonality) in the H5N1 avian influenza cases. The following figure is one of the
models we have predicting the future H5N1 cases.
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Inverse Problems
- Optical tomographic imaging of brain
Optical Tomography (OT) is the technique of using light in the near-infrared
range (wavelength from 700 to 1200 nm) for imaging specific parts of the body to obtain information about
tissue abnormalities, such as breast or brain tumors. Optical tomographic imaging is performed by obtaining a
spatially resolved map of the optical properties within the part of the body under investigation. In the following figures,
the exact values of diffusion and absorption coefficients are shown on the left. These coefficients determine the location
of a brain tumor. On the right, diffusion and absorption coefficients are estimated using an iteratively
regularized Gauss-Newton method and finite element method (on a disc domain) for the forward problem.
- Groundwater hydrology problem
Groundwater hydrology problem, in general form, is the movement of
groundwater in an aquifier and is modeled by a diffusion equation. It is of practical importance to
find the diffusivity of the sediment in order to explore the internal structure of the aquifer, and
this is an inverse problem.
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