People
Misha Kilmer William Walker Professor of Mathematics
Contact Info:
Tufts University
Department of Mathematics
503 Boston Avenue
BromfieldPearson
Room 103
Medford, MA 02155
Email @tufts.edu:
misha.kilmer
Phone: 6176272005
Personal site
Expertise:
Iterative Methods, Numerical Linear Algebra, Numerical Analysis, Scientific
Computing, Image and Signal Processing
Research:
Numerical Linear Algebra (NLA) is a field that straddles the line
between mathematics and computer science in that the goal is to use
and develop mathematical tools and theory to produce algorithms that
solve largescale, realworld problems as accurately and efficiently
as possible on a computer. These days, almost every incoming Tufts
student has (unknowingly) been exposed to Numerical Linear Algebra:
web searches using the Google search engine are possible in large
part because of algorithms that numerical linear algebraists have
designed to compute socalled dominant eigenvectors. The fact that
NLA is so useful in so many realworld problems, often in
nonobvious ways, appeals to the practical side of my nature. I am
fascinated by the fact that much of the mathematics, the algorithms,
and theory is relatively new, having been born since the dawn of the
computer era and/or inspired directly by a particular recent science
or engineering application. The mathematics and the algorithms must
continue to evolve as technology and society's needs change, hence
the challenge to those of us working in NLA. Because NLA is
inherently interdisciplinary in nature, working in this field
provides me a handson opportunity to learn about research within
other disciplines (chemistry, biology, psychology, economics,
biomedical engineering, etc.), and to work with researchers in
those disciplines to help advance the stateoftheart in those
disciplines as well as my own.
Numerical Linear Algebra is considered a branch of applied
mathematics to the extent that the resulting algorithms are applied
to solve problems in science and engineering. Applications that have
been a focal point of my research include detection of buried
landmines from electromagnetic scattering data, image deblurring,
and medical image reconstruction. These applications are considered
"inverse problems." To solve an inverse problem, noisy, measured
data is used to reconstruct discrete samples of an unknown input
function. For example, one might take electromagnetic measurements
on the surface of the earth with the goal of locating buried
ordinance, underground contaminant plumes, and the like. The
reconstruction process is based on a mathematical model of the
relation between the input and the data. In nonlinear inverse
problems, such as breast tissue imaging with diffuse optical
tomographic data, many "forward problems" (i.e., linear systems)
must be solved in order to reconstruct the appropriate images.
Furthermore, the solutions to underlying inverse problems must be
made insensitive to the noise in the data. My research, therefore,
is twofold: to build computationally efficient and stable forward
solvers and to formulate solution methods for inverse problems in
which noise is filtered during the reconstruction phase. Clearly,
the potential impact of such research is great: to "see," in
realistic time, the world (or other medium) from the inside out.
