My
research interests lie broadly in the
theory, computation, and applications of
inverse problems. Besides this, I am also
actively engaged in set-valued
optimization, quasi-variational
inequalities, optimal control, and
uncertainty quantification. In the
following, I briefly discuss some of my
main research contributions in these
areas.

See.

Elliptic Inverse Problems

Soft
tissue cancers in the interior of the
human body reign among the deadliest forms
of the disease, making up a majority of
the 7.6 million estimated cancer deaths in
2008 [Cancer (Fact Sheet No. 297), WHO,
2013]. The effectiveness of most
treatments hinges upon early detection but
the process of finding tumors inside the
body remains a difficult one. It is known
that the stiffness of soft tissue can vary
widely based on its molecular makeup and
differing macroscopic/microscopic
structure and that such changes in
stiffness correlate with changes in tissue
health. Palpation allows doctors to feel
directly for changes in tissue stiffness
and can detect such stiff lesions but the
practice is subjective and is usually
limited to finding exceptionally hard
nodules near the skin's surface.
Ultrasound can also be used to diagnose
tumors further within the body as well as
quantify their stiffness, but even hard
growths can lack the necessary acoustical
properties for effective detection.
Elasticity imaging inverse problem extends
the practice of palpation, making use of
the varying elastic properties of healthy
and diseased tissue to identify likely
tumors. A relatively small external
quasistatic compression force is applied
to the tissue and then the tissue's axial
displacement field is measured either
directly or indirectly through the
comparison of an undeformed and deformed
image. A tumor can be identified by
solving the inverse problem of determining
the tissue's underlying elastic properties
from this measurement. Although elasticity
imaging inverse problem is a well
established approach commonly used in a
clinical setting, the underlying
mathematical formulation is in terms of a
non-convex optimization framework and
hence it cannot ensure that a global
solution can be found. It has been a long
standing problem whether it is possible to
remedy this drawback.

From a mathematical stand
point this inverse problem seeks the
elasticity parameter Mu from a
measurement of the displacement vector
under the assumption that the parameter
Lambda is very large. This is due to the
fact that in most of the existing
literature on elasticity imaging inverse
problem, the human body is modelled as
an incompressible elastic object.
Although this assumption simplifies the
identification process as there is only
one parameter to identify, it
significantly complicates the
computational process as the classical
finite element methods become quite
ineffective due to the so-called locking
effect. One of the few techniques to
handle this problem is by resorting to
mixed finite element formulation. I have
developed various optimization based
approaches for this inverse problems.

Set Optimization:
Optimality Conditions and Applications

During my
stay in Germany I worked on a research
project sponsored by German Research
Foundation (DFG), devoted to a detailed
study of optimization problems involving
set-valued maps. In this field, one major
difficulty is of differentiating the
set-valued maps. (The study of
optimization methods for
non-differentiable functions also fall in
this category.) We established usefulness
of the notion of epiderivative of
set-valued mapping. This notion is a
cornerstone of set optimization and was
coined by Johannes Jahn, who was the main
investigator of the project. The basic
idea is to use certain tangent cones to
define the epiderivatives and use them to
state optimality conditions in set
optimization. Set optimization subsumes
nonsmooth optimization which, due to many
real-world applications, was one of the
most active branch of optimization before
the birth of set-optimization. Currently,
I am working on general higher-order
optimality conditions in set optimization,
and on applications of set optimization in
robotics and optimal control.

Ill-posed Variational
Inequalities

The theory of Variational and
quasi-variational inequalities is now an
independent branch of applied
mathematics and has far reaching
applications. My current interest in
this field is in developing
regularization techniques for ill-posed
quasi-variational inequalities. I have
published papers both of theoretical
nature and papers dealing with iterative
schemes.