## Faculty Research Areas

Many phenomena in sciences and engineering are described by
partial differential equations (PDEs). Exact solutions of most
of real world problems are often very difficult or impossible to
obtain. Solutions to such problems can be approximated using
numerical methods. Numerical Methods for Solving PDEs is a vast
area which deals with numerical errors, computational stability,
parallel algorithms, efficient computation, and solution of
challenging multi-physics problems. My research interests are in
the following areas of Numerical PDEs: orthogonal spline
collocation method, iterative methods for solving large sparse
systems of equations, numerical solution of nonlinear elliptic
PDEs, numerical solution of non-self-adjoint or indefinite
problems, and interface problems.

Google
Citation Profile
I work on problems at the intersection of global analysis on
manifolds, Riemannian geometry and mathematical physics. The
main part of my research is devoted to the study of spectral
asymptotics of elliptic partial differential operators on
manifolds via heat kernel methods and to the application of
these methods to various problems of spectral geometry,
mathematical physics, quantum field theory, quantum gravity and
financial mathematics.

Google
Citation Profile
My research interests are in optimization and applications of
optimization techniques in inverse problems. In
optimization my research has involved interior point methods for
linear and semi-definite programming with applications in
integer programming. The work on parameter estimation and
inverse problems has involved optimization, statistics, and
Tikhonov regularization.

Google
Citation Profile
My research focuses mostly on diagnostic methods and their
applications to reliability and engineering models. These two
areas are crucial within the statistical research community, and
are also favorable at technical institutions. In the past few
years, I have focused on other fields of statistics; the most
attention has been given to cover theory and applications of
multivariate analysis, survival estimation, Bayesian estimation,
and reliability.

Google
Citation Profile
My research focuses on determining analytic solutions of
Boundary Value Problems. For example, the types of mixed BVP's
which arise in analyzing the stress field in the vicinity of a
crack, or in the study of flow past an obstruction. I am
currently working with the Electrical and Microsystems modeling
group at Sandia National Lab. I work on solving BVP's, referred
to as compact models. The development of compact models that can
simulate the excess carrier dynamics in undepleted semiconductor
regions are important for many applications, including photonic
and power devices.

Google
Citation Profile
Modeling complex multidimensional data; in particular, spatial
data. Precipitation modeling. Geostatistics. Bioinformatics.
Statistical consulting.

Google
Citation Profile
Currently, there exists a successful physical theory known as
the Standard
Model, which gives accurate predictions of physical phenomena.
Yet it
remains a fact that despite decades of research, there is no
mathematically precise definition for a quantum field.
Consequently, my
research is focused on the mathematical analysis of Schrodinger
operators
with physically important but mathematically difficult
potentials, the
calculus of distributions and generalized functions as they
pertain to
quantum field theory, and mathematically precise extensions of
the Feynman
integral from quantum mechanical systems to the realm of quantum
field
theory.

Google
Citation
Profile
My main area of interest is the topology of strange attractors,
particularly the topological types of the interlocked periodic
orbits. The template construction of Birman and Williams
provides a theoretical framework with which the periodic orbit
structure can be transferred to the study of knots and links on
branched two-manifolds supporting a semi-flow. Some of my recent
publications explored various suspensions of maps to flows, so
that the simpler dynamics of periodic and chaotic orbits in a
map can be mapped onto a flow generated by a smooth vector
field. Recently I have been working with large coupled chaotic
systems, investigating the synchronization of identical units.

Google
Citation Profile
My research interests are Mathematical Biology and Mathematical
Modeling. Recently I have been increasingly working with
modeling of Fluids. In the past few years I have had students
working on modeling of glaciers, size-structured populations,
age-distribution of groundwater, resonances in the plasmosphere,
and other areas of modeling.

Google
Citation Profile
My research is centered at partial differential equations and
dynamical systems along with their applications. I am
particularly interested in the infinite-dimensional dynamical
systems generated by deterministic and stochastic PDEs defined
in bounded or unbounded domains. My current work has involved
random attractors, invariant manifolds, random periodic
solutions and random almost periodic solutions for equations
driven simultaneously by deterministic and stochastic forcing.

Google
Citation Profile