Research Narrative
I have devoted the greater part of my research career to the three “i”s: inverse
problems, integral equations, and ill-posed problems. Most classical problems in the
mathematical sciences are direct problems. Direct problems typically involve
determining exterior effects of internal causes or predicting future forms of present
conditions. In a sense one can say that in direct problems the orientation is usually
outward-directed or prospective, and direct problems are almost always analyzed using
the mathematical methodology designed to model temporal evolution or spatial
distribution, that is, differential equations. A canonical example of a direct problem is the
determination of future temperature distributions in a body given its present temperature
distribution (“initial condition”) and details of its interaction with its environment
(“boundary conditions”). But many important problems in current science and technology
are inverse problems. In these problems the point of view is typically introspective or
past-directed. Inverse problems often involve determination of interior structure from
exterior measurements (think: medical imaging, such as Computed Tomography,
Magnetic Resonance Imaging, and Ultrasound), or “post-diction” (think: assessment of
pre-historical topography or paleoclimatology). The inverse heat conduction problem,
which requires the reconstruction of past temperature distributions of a body given its
present temperature distribution, is an iconic example of an inverse problem. It should
come as no surprise, at least to those who can recall a bit from their first calculus course
(hint: differentiation and integration are inverse processes), that since direct problems are
usually modeled by differential equations, many inverse problems are modeled by
integral equations. Solving inverse problems modeled by integral equations generally
requires determining fine features of an unknown function given gross information
obtained through a process of smoothing or integration (as occurs in all physical
measurement processes). What makes these inverse problems interesting - and difficult is that they are often ill-posed: an inverse problem might not have a mathematical
solution (at least not in the conventional sense); it could have many (even infinitely
many) solutions; and – most important for practicing scientists – solutions of inverse
problems can be unstable. Instability means that tiny changes in the parameters or data of
an inverse problem can result in solutions that are wildly out of control. Taming such
mathematical instability is a major technical challenge.
I have researched approximation methods for inverse and ill-posed problems
modeled by integral equations (or more generally, operator equations) for more than three
decades and have made several contributions to these areas which have been termed
“fundamental.” The major methodology for dealing with instabilities in ill-posed inverse
problems, known as regularization, was developed by the legendary Russian
mathematician Andreĭ Nikolaevich Tikhonov and his research groups at Moscow State
University and the Soviet Academy of Sciences beginning in the 1960s. In 1984 I
published the first English language monograph (The Theory of Tikhonov Regularization
for Fredholm Equations of the First Kind, Pitman Advanced Publishing Program,
London, 1984) devoted exclusively to regularization theory and containing a number of
original results of my own, including the first precise elucidation of certain convergence
properties of Tikhonov’s method for linear inverse problems. This book has been
described in the journal literature as “very influential: it has … provided a firm basis for
much subsequent work on regularization methods.” The book “Inverse and Ill-posed
Problems” which I co-edited in the eighties with Heinz W. Engl (currently Rector of the
University of Vienna) is considered an influential document in the field. Two of my
widely cited books on inverse problems published in the nineties have helped shape and
popularize the field of inverse and ill-posed problems. Inverse Problems in the
Mathematical Sciences (Wiesbaden, 1993; Japanese translation, Tokyo, 1996) has been
described as “a masterful work” in Zentralblatt für Mathematik and “unique in the
mathematical literature” in SIAM Review. My book Inverse Problems (Washington, D.C.,
1999; Japanese translation, Tokyo, 2002; Mandarin Chinese translation, Beijing, 2006) is
termed “a remarkable contribution … in the field of inverse problems” and “a genuine
treasure in the literature of inverse problems” in published reviews. My latest research
monograph on stabilization theory for ill-posed problems (Stable Approximate
Evaluation of Unbounded Operators) was published by Springer-Verlag in 2007. This
monograph provides a general framework, from the perspective of inverse problems, of
the basic procedure of evaluation of unbounded operators such as those that appear
frequently in inverse problems in mathematical physics.
My research on approximation methods for inverse and ill-posed problems has
been presented in invited lectures on every continent of the globe except Antarctica. My
work has been supported by the National Science Foundation, the Air Force Office of
Scientific Research, the NATO Division of Scientific Affairs, and the Science and
Engineering Research Council (U.K.) and I have held visiting research appointments in
England, Germany, Switzerland, and Australia.
The year 2010 brought a couple of notable honors in recognition of my
contributions to the three “i”s. : The Journal of Integral Equations and Applications, an
international research quarterly, dedicated its Summer and Fall issues in my honor. I was
also one of ten mathematicians nation-wide elected Fellows of the American Association
for the Advancement of Science (AAAS) in 2010. Founded in 1848, the AAAS is the
world’s largest scientific society and includes 262 affiliated societies and academies
serving 10 million individuals worldwide. Since 1874 the AAAS has bestowed, upon
election by peers and approval by the AAAS Council, the designation of Fellow on select
members whose “efforts on behalf of the advancement of science or its applications are
scientifically or socially distinguished.” I was elected in recognition of my “distinguished
contributions of the application of mathematics to science, particularly in the areas of
inverse or ill-posed problems, approximation theory, and mathematical modeling.”