Misha Kilmer

Misha Kilmer

William Walker Professor of Mathematics
Bromfield-Pearson, Room 103
503 Boston Avenue, Medford, MA


Iterative Methods, Numerical Linear Algebra, Numerical Analysis, Scientific Computing, Image and Signal Processing

Research Interests

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 large-scale, real-world 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 so-called dominant eigenvectors. The fact that NLA is so useful in so many real-world problems, often in non-obvious 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 hands-on 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 state-of-the-art 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 non-linear 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.