Many mathematical and statistical problems demand innovative
ways of integrating numerous small pieces of information
into a larger picture. NSF supports research in a variety
of these areas, including:
 Inverse Problems. In the aftermath
of an earthquake, scientists are confronted with a plethora
of data from numerous seismic recorders scattered around
the affected region. The challenge is to combine those
measurements to answer key questions: Where was the quake
centered and how strong was it?
Mathematicians can help find the answers by using insights
from problems that work in the opposite direction. For example,
geologists often use explosive charges or mechanical thumpers
to apply a shock at specific locations and record the results
with arrays of remote sensors. The combined instrument readings
are then used to make deductions about rock structures far
below ground. The earthquake problem runs in reverse. That
is, the researchers take advantage of many years of accumulated
data on geological structure and then derive the location
and strength of a shock that could pass through those known
rock structures to produce the reported measurements. Because
the mathematics required for those conclusions results from
reasoning in a somewhat backward fashionmoving from
effects toward likely causesproblems of this sort
are often called "inverse."
 Image Creation. Medical imaging issues often take
the form of inverse problems. A celebrated example is the
CAT (computerassisted tomography) scan, which collates a
large number of lowdose Xray images into a crosssectional
image that doesn’t exist outside the computer. The
relevant mathematical technique was developed years before
it was needed in radiography. But the computations needed,
although theoretically straightforward, were not feasible
numerically until digital computers were widely available.
Modern mathematical methods allow radiologists to obtain
fast, accurate results with less patient exposure to Xrays
than ever before.
A dot or pixel in an Xray image is largely an estimate
of the total density of matter along the straight line through
which the Xrays travel from their source to that point.
Other technologies for medical imaging, such as PET and MRI,
measure different aspects of the body, leading to other kinds
of assembly problems that are an active subject of mathematical
research.
 Statistics. This field has been dramatically affected
by the power of modern computers, which permit ideas that were
at best theoretical 50 or 100 years ago to become part of the
routine toolkit. One such innovation combines a classic idea
from probability theory with a mid20 th century mathematical
technique to produce systems that can determine, for example,
which of a number of models is the best fit for a new data
set, or how various models fit with each other.
Managing and Modeling Complexity [Next]
