Date: Apr 01, 2009 Author: William J Browning Source: INFORMS Roundtable (
click here to go to the source)
ROUNDTABLE PROFILE
Applied Mathematics Offers Wide Range of Services
Consulting firm develops and implements mathematical models for
everything from submarine warfare to agriculture.
Applied Mathematics, Inc., is a privately held technical consulting
firm with offices in Gales Ferry, Conn., near the U.S. Naval Submarine
Base New London. Also close by on the Thames River are the
Electric Boat Company, a builder of nuclear submarines; the U.S.
Coast Guard Academy and the Coast Guard Research and Development
Center; and Pfizer Central Research facilities.
Applied Mathematics develops and implements mathematical
models to help our clients better understand complex systems in order
to improve system performance. Since 1980, we have completed more
than 120 research and development projects for U.S. and foreign government
agencies and industry.Areas of applications include submarine
warfare, search and tracking, search and rescue, clinical informatics
and vineyard analytics. Algorithms developed by Applied
Mathematics are in use on U.S. and U.K. submarines. In addition, we
occasionally design, build and test prototype systems. Examples
include a portable launch system for analog torpedoes, a micro-fluidic actuation system for active flow control of an airfoil and an environmental sensor for use by a submarine.
Every Applied Mathematics staff member has a degree in mathematics, physics or computer science, with the majority having M.S. and Ph.D. degrees. Mindful of Isaac Newton's
counsel that "it is better to send mathematicians
to sea to collect data than to send data collected
at sea to mathematicians ashore," our
staff have participated as test directors and technical
observers in more than 100 operations
and tests conducted at sea on U.S. and U.K.
submarines and naval vessels and military aircraft
throughout the world. In addition, for
more than 30 years, our mathematicians and
physicists have been embedded at a submarine
squadron working closely with submarine officers
in developing and evaluating tactics.
In our technical reports for our clients we
include all the data and the assumptions made
in the analysis, and we describe the models
and algorithms in detail so that the client can obtain a second opinion
should he or she desire.We also regularly conduct pro bono projects;
our staff members volunteer their time and services to numerous
professional and non-profit organizations.
Following are brief descriptions of three application areas in
which we have worked for many years.
Submarine Tactics. The field of operations research began with
operational analysis of anti-submarine warfare during the Battle of
the Atlantic in World War II. A good reference for this fascinating
history is "Slide Rules and Submarines," by Montgomery Meigs, an
historian and Army general.
The role played by operations analysis in World War II was not
lost on the U.S. Navy. Throughout the long Cold War between Soviet
and U.S. submarines, Ph.D. analysts from Applied Mathematics
and other organizations were embedded in Navy fleet ASW and
submarine commands.
Some of the submarine tactics areas we have worked on include
submarine search tactics (e.g., choice of speed, track and depth), target
tracking using passive sonar, torpedo evasion tactics, submarine
evasion tactics, optimal allocation of sonar system signal processing
and torpedo and cruise missile tactics. Mathematical methods we
have used include simulation, game theory, control theory, optimization
and statistical estimation.
One important assignment for more than 20 years was the Submarine
Search Manual. This manual gives search speed, track and
depth guidance for a submarine when conducting a search for
another submarine.
Search is a basic activity of all organisms -- the search for food and
the search for shelter from predators. The science of how organisms
search for prey is called foraging by biologists. Three types of searches
are considered by biologists -- cruise, saltatory and ambush.
Energy demands -- e.g., lack of endurance and the "prey handling"
problem, e.g., what to do when you catch the prey, which is
often a matter of life and death for both parties -- dictate the type of
search that organisms conduct.
A cruise, or constant speed search, maximizes range at time of
detection and is used when the large sweep rates possible from a
saltatory search are either not available due to short detection ranges
or not necessary due to a small search area. Sharks and hawks use a
cruise search strategy. A saltatory, or sprint and drift, search is used
when the search area is large, detection ranges are long and the prey
is not uniformly distributed. The e coli bacteria and some fish use a
saltatory search.
An ambush, or barrier search, is used when the track of a target
can be predicted. Scorpions and snakes use an ambush search.
Each species has evolved to search in only one way except the
nuclear submarine. A nuclear submarine with its modern longrange
sensors and its unlimited source of energy is unique in that it
can and does conduct each of these distinct types of searches for its
prey -- ships and submarines.
Ocean Surface Current Estimation.A core mission of the Coast
Guard is search and rescue (SAR) of people in distress at sea. The
Coast Guard conducts more than 50,000 SAR cases annually.
Mathematically, a search consists of a probability distribution in
space and time of the search object, the detection range of the sensor(s)
and constraints on search effort, either number of searchers,
time or both. The problem is to optimize some measure of effectiveness,
such as probability of detecting the person while still alive
subject to constraints on search effort. Because SAR detection
ranges are usually very short and the ocean is so large, it is important
to obtain as good an estimate as possible on where the search
object may be drifting in the ocean due to the effects of wind and
surface current.
One method used in a large area open ocean search is to drop
self-locating data marker buoys (SLMDB) from an aircraft at the
last known position of the search object. These buoys then periodically
report their GPS position via satellite.
When measurements are scarce, expensive or difficult to obtain, it
is important to maximize the information obtained from the measurements.
The more useful the mathematical concept, the more
names it has. Kriging, geostatistics, objective analysis, weighted least
squares and optimal interpolation all refer to the same mathematical
concept that is used in this and many other applications.
For the Coast Guard, we developed time-dependent kriging
algorithms to blend the reported GPS positions of the buoys to
obtain a surface current velocity vector field and associated uncertainty.
The velocity field was then used with the Fokker-Planck
equation to propagate in time the area of uncertainty for possible
target locations. This model resulted in a significant improvement
in the probability distribution of the search
object compared to the procedures that had
been in use.
Grape Crop Estimation. More than
23,000 farms grow grapes in the United States
with close to a million grape-bearing acres
and a total crop value in excess of $3.5 billion
in 2005. About 90 percent of the grape crop
in the United States is grown in California.
However, grape production in the U.S. is less
than 5 percent of the world production.
There are almost 5,000 bonded wineries in
the U.S., located in all 50 states. The retail
value of wine produced in the U.S. was $24
billion in 2005.
The inability to accurately forecast grape
crop yield costs the wine and grape industries
hundreds of million of dollars a year. In
recent years, 30 percent to 50 percent errors
in yield estimates have occurred.
Accurate and timely forecasts of wine
grape crop yields are required for the efficient operation of a modern
winery. Yield estimates are used throughout the winery to support
a wide range of activities including scheduling labor in the
vineyard; procuring cooperage, labor and equipment for crush; and
developing marketing strategies. Yield estimates are needed during
the dormant season -- late fall -- and during the growing season -- at
bloom, mid-season and at harvest.
The increased use of mechanical harvesters in California,
currently more than 60 percent of the acreage, has also
increased the demand for more accurate yield estimates in
order to better manage harvest.
Yield components from the current growing season such as
cluster count, cluster size, berry set and berry weight are used
to estimate yield using statistical methods. A commonly used
method is cluster counting. Cluster counts sampled statistically
prior to bloom are multiplied by an estimate of cluster
weight at harvest to predict yield per vine.
We have been working with growers and research viticulturists
in California to improve the sampling and kriging
algorithms used to estimate crop yield.
Using GPS-equipped mechanical harvesters, technicians have
recently conducted precision harvesting experiments on California
vineyards. The vineyard is sampled using a portable Near Infrared
Spectrometer to prepare a quality map of the vineyard prior to harvest.
The mechanical harvester then automatically segregates the
fruit into separate gondolas. We are currently analyzing data from
these experiments.
In summary, the ease with which massive amounts of data can
be automatically recorded today -- and the availability of powerful
and inexpensive computer processing -- offers the promise for significant
advances and widespread use of mathematical modeling in
areas as diverse as agriculture and submarine warfare.
