QUARTERLY OF APPLIED MATHEMATICS *09
APRIL, 1972
SPECIAL ISSUE: SYMPOSIUM ON
"THE FUTURE OF APPLIED MATHEMATICS"
MATHEMATICAL APPLICATIONS, COMPUTATION, AND COMPLEXITY
BY
HIRSH COHEN
IBM Thomas J. Watson Research Center, Yorklown Heights, New York
In this lecture I want to discuss some matters which are not themselves applied
mathematics but are about the profession and practice of applied mathematics. In the
trade, the measures of an article's quality are the sparsity of the prose, the swiftness of
the mathematics, and the sweet connection to the real world. This one will be rather
thick in prose and sparse in mathematics. I would have liked to have been able to
respond to Prof. Davis's invitation to be philosophical, but that has always seemed
to be a dangerous way to behave publicly. I offer these comments from the point of view
of an industrial mathematician and as one of the Old Browns.
When I arrived at Brown nearly twenty-five years ago, the world looked a little
different to a young applied mathematician than it does to a middle-aged applied mathe-
matician now. It was clear that what was to be learned at Brown was continuum mechan-
ics and the complex function and differential equation theory that was a part of it.
Fluid mechanics, elasticity, plasticity—we were beginning to grapple with those problems
that von Karman had said we were to in his 1940 Gibbs lecture [1]. We were really
not yet "tooled up" for nonlinearity, as he urged in the first issue of the Quarterly of
Applied Mathematics [2], but there was a realization of its importance. There was an
atmosphere of the beginning of something at Brown in the late 1940s and, in fact, it was a
beginning—a feeling that there were many, many problems to be solved in aerodynamics,
flutter and aero-elasticity, viscous flow and boundary layers, plastic design, vibration
problems. These were obviously good problems to solve because someone, somewhere
needed the solutions, or they were hard, or they were the next problem in their field, or
just because they were there.
We were (and, of course, I'm aware of that benevolent distortion that occurs in
looking backward this way) an optimistic crew who had a great deal of confidence
in the results, techniques, and styles of thought of several generations of European
applied mathematicians. One can observe that for American science, applied mathe-
matics was a rather new field then, not very populated or popular but with an apparent
good potential for growth. Furthermore, when scientists thought of the uses of mathe-
matics, they were used to thinking of the applications of classical analysis to the problems
of the physical sciences.
In a corner of the house at 27 Brown Street there was a large room full of girls and
machines—-our computers. It was a small but busy and necessary enterprise. Other
important locations were the softball field and the house on Benevolent Street where
the Friday afternoon post-colloquium sessions were held.
Before nostalgia overcomes, what happened to all of that? Where are we?
What I think has happened is that the needs for and uses of mathematics have
110 HIRSH COHEN
opened up very quickly and very dramatically during these twenty-five years. Science,
in general, received a large amount of support; mathematics, in general, participated;
and applied mathematics did too. Technological enterprise in government and industry
bloomed, and so did the number of people involved in the applications of mathematics.
A great deal of this had to do with the mathematics of continuum mechanics and
therefore led to grappling with the nonlinear problems that von Karman had pointed
towards. But there were other mathematical activities coming into being as well. The
new methods of discrete optimization, linear programming for example, were rapidly
developed. A number of other applied combinatorial fields came into vogue—-game
theory, graph theory, queueing theory, with applications in various military, business,
economic, and political operations. Control theory, in both its continuous and discrete
form, was evolved mostly originally in response to aerospace needs but then applied
in many industrial and finally even social processes. Numerical analysis, an old science,
advanced practically and theoretically. Operations research and systems analysis, for
better or worse, began to invade previously nonmathematized fields. As the number
of people working with these techniques increased, so did the areas of application for
them.
The largest effect on applied mathematics, however, came with the development of
computers. In the late 40s, our computing center at Brown typically handled formulas
that were the result of considerable analysis and manipulation. Gradually, as computers
became faster, could handle more data in storage, and gained more input-output facilities,
more problems were formulated so as to be done numerically at an earlier stage. We have
learned now that the amount of analysis we do on a problem has to be carefully balanced
against the ability to compute. There are classes of problems, for example in mathe-
matical programming, in which the size makes it clear from the beginning that computer
solution is necessary. In some modern physically-based problems that are formulated
in terms of differential equations, the approximate methods of applied mathematics,
in some cases, cannot make any headway at all. This may be because of the size, again,
but is more likely because of strong nonlinearities or even linear interactions that are
complex.
There are still other kinds of problems in which one is unsure of the formulation.
The model itself may be in doubt. With the computer, we are now able to experiment
with formulations, look at numerical solutions, and try new formulations in a selective
fashion. There are interesting and useful "black box" problems in engineering, biology,
ecology, and elsewhere, of this kind.
Finally, when one knows only that there is a problem and does not even have a
mathematical model in terms of a set of equations, there is a collection of techniques
known as simulation. These often have to do with particle motions or their counterparts
(automobile movements, war personnel, or vehicles), and often models arise from such
simulations.
All of these uses of the computer in the old and new realms of applied mathematics
mean to me a gradual usurpation of some of the functions of the applied mathematician.
There are some formulations, some analyses, and some solutions that are just more
easily and more quickly accomplished with computation.
At this point, let me draw two observations from the description of change over
twenty-five years. First, the application of mathematics to physical problems, and
especially to problems in continuum mechanics, has continued to be an active and
MATHEMATICAL APPLICATIONS 111
fruitful preoccupation for applied mathematicians. However, although it is undoubtedly-
growing, it has become a small fraction of the total uses of mathematics already and
will probably become a smaller one. Second, the computer has usurped much of the
work of the applied mathematician.
Both of these observations point to a change in what applied mathematicians do and,
especially, in how we will train applied mathematicians for the future.
It seems clear to me that our reaction should be:
1. To broaden the training toward new areas of application. In effect, to recognize
that the application of mathematics to- the physical sciences now only represents a small
fraction of the uses of mathematics.
2. To understand that computation itself should be studied as a major branch of
applied mathematics; not only numerical analysis but a much larger view of the whole
computational process.
1. New applications. The fact is that if one looks around a bit, it is easy to find
mathematical formulations in almost every kind of endeavor. I am not speaking of
data analysis although, in my mind, that topic represents one end of the spectrum of
activities in mathematical formulation. There is, as I have pointed out elsewhere [3],
a long history of mathematics in physiology. There are uses in other areas of biology
and medicine. There is the famous marriage theory [4] in sociology, work in linguistics [5],
economics, many areas of commerce and business, many topics covered by operations
research and systems analysis-like formulations. There are, however, no successes like
those of the physical sciences. Is it because mathematics and physics have grown up
together? Is it because this was the part of the world men had most pressing need to
understand and, therefore, gave the most effort toward? Or is it because on our space
and time scale, these questions of physics are easier to put into mathematical terms
and to solve? I suppose the fact that we have gotten by with linear descriptions of
physical phenomena for such a long time may be one fact favoring the last view.
Whatever the reasons, in those areas where mathematics has had the greatest suc-
cesses, one also finds the largest incursion of computing. There were elegantly appropriate
uses and developments of eigenvalue methods that took place analytically in the early
part of this century to find their place in quantum mechanics and many kinds of vibra-
tion and spectral theory. One now finds massive calculations undertaken for the solution
of Schroedinger's equation or for many areas of continuum mechanics. In fact, calcula-
tions in these fields are now regularly part of design practices in aerodynamics, structures,
nuclear energy, etc. In meteorology, oceanography, solid geophysics, hydrology, per-
turbation theories or other approximations show so little of the complex phenomena
that people in these fields have turned to full- and large-scale computation.
If such is the case, that the classical realm of the applied mathematician has gone
over very much toward computation, then, those who desire to continue using the
beautiful approximation techniques of applied mathematics will have to look for new
application areas. Similarly, for those whose creativity lies in formulation I am saying
that I believe that all of those powerful techniques of approximation—-regular and
singular perturbations, integral approximations, comparison and bounding methods,
etc.—must now be taken to more fertile fields.
How to move? Surely, it's not quite as simple as just picking up the bag of tricks
and carrying it to the new subjects.
What I think is required is literally a mathematical immersion into new application
112 HIRSH COHEN
areas to the same depths that were reached in the physical sciences. Applied mathe-
maticians have come to know a great deal about fluid dynamics. From this have come
precise, detailed explanations of very complicated phenomena. We observe how this
in-depth mathematical knowledge is now being carried forward into oceanography,
astrophysics, meteorology and other parts of the physical world that depend very strongly
on fluid flows. On the other hand, I do not feel the same kind of impact is being made
in applying fluid dynamical mathematical techniques to problems of blood flow or
desalinization—-yet. It will require more serious penetration into these areas by those
who wish to make their mathematical contributions really meaningful.
For mathematics to become more effective in the social sciences and in social enter-
prises, I believe the same commitment is required. Instead of working on the fringes of
fields, we shall have to learn as much about them as the social science specialists in each
case. For example, colleagues of mine have prepared some tools, mostly computational
schemes, for dealing with some aspects of urban housing [6]. To understand where this
particular calculation (which has certainly shown itself to be useful to some of the
parties interested in low-cost housing) lies in the whole spectrum of problems requires
learning something of the sociological as well as the financial aspects of housing and
probably a good deal of the structural and architectural factors.
If I'm on the right track, the implication may well be that training in applied mathe-
matics for some must include serious excursions into social and political areas. This will,
of course, be hard to come by in mathematics or applied mathematics departments and,
as has so often occurred in applied mathematics, we will undoubtedly see the work
beginning within the application fields themselves.
Even so, to encourage other areas to mathematize themselves we need to teach
mathematics in a fashion that is sympathetic to their needs. There are already some
beginnings of this. For example, as part of a biomathematics development at the Cornell
Medical College and Sloan-Kettering Cancer Research Institute in New York, courses
were given to graduate students in the medical sciences (Ph.D.'s in biochemistry,
physiology, pharmacology, etc.). These took the form of problems from various parts
of biology and medicine in which one could make very explicit gains by using differential
equations, probability methods, matrix and eigenvalue methods, and so on. The gains
were important to show; my own experience has been that medical students show very
little sympathy for nonlinear partial differential equation solutions for nerve pulses
when they can literally see the pulses in oscilloscopes.
Other courses of this type have begun in medical schools and universities. They
will play a useful role. They won't replace the immersion I spoke of, however. I can see
no other way than for young people in applied mathematics to learn as much as they
can about the rest of the world.
There is another aspect of change in applied mathematics that I would like to men-
tion. I would like to claim that we have so far mostly dealt with relatively simple systems.
When we turn to sciences other than the physical sciences, we are confronted with greater
complexity in structure, relationships, a larger number of variables, and often, it seems, a
higher degree of interdependence and hence nonlinearity. I can't prove this. It may only
seem to be the case because we have not yet perceived how to simplify and structure other
areas as we have done in the physical ones.
It is clear that we do not even have very good measures of the difficulty of such
structuring. With the mathematics of the physical sciences, there are far-reaching phys-
MATHEMATICAL APPLICATIONS 113
ical principles firmly tied to concepts of energy, entropy, work, etc., which enable us to
look for optima, measure efficiency in the physical processes themselves and, in turn, in
the methods of solving the problems. This is not the case in other sciences or in other
nonscientific enterprises. We may want to create human good, avoid delay, raise profit,
distribute goods more uniformly, increase democratic participation, make life easier,
or survive. All valid objectives in their own context, but how do we examine the diffi-
culties in solving problems? How close do we get to solutions? How efficient or effective
are our solutions in these areas and how efficient are our methods of obtaining the
solutions?
Let me take some examples from neurophysiology. As I have described elsewhere
[3, 7], there is a good theory for how nerve pulses travel along nerve fiber membranes.
This nonlinear system of partial differential equations uses, directly, physical relations
and, empirically, chemical kinetic data and hypotheses. Now the familiar form of these
Hodgkin-Huxley equations applies to nerve membrane that is continuous and con-
tinuously active along its length. In our own bodies and in most of the nerve systems of
vertebrates, nerve membrane is interrupted along the length of the nerve fiber. In fact,
the active energy input for the maintenance of the pulse propagation occurs only at the
interruptions which are called nodes of Ranvier. The voltage across the membrane
obeys a passive diffusion equation in between the nodes. The nonlinear system becomes
an ordinary differential equation set at each node. The nodes may be 1^ in length and
the regions between them at least 2 mm, so that the nodes can be looked upon as points
along the line. Again, one would like to be able to calculate the nerve pulse propagation
and, especially, how it is affected by changes in inter-node length, chemical environment,
membrane deterioration, etc.
The equation that governs the voltage across the membrane, v(x, t), between the
nodes is
C(dv/dt) + (v/r,) = (a/2ri){d2v/dx2), x, < x < xi + 1 , (1)
where the x{ are the node locations, C = membrane capacitance (fd/cm2), r, = cross
membrane specific resistance (ohm cm2), r, = specific resistance of material inside the
fiber (ohm cm) and a = radius of nerve fiber.
At the nodes x = x, the voltage v = v%(x,) is given by
Cn{dvjdt) + Im(v, m, n, h) = I{ ,
r 3, - [exp (V{ — VNa) — lj , _ 4/ — \ i t
Im = rriihig^Vi —exp (v.) — l ^ - vK) + IL,
It = ( lim (dv/dx) — lim (dv/dx))(l/rn),
x—*xi + 0 x—*xi~°
I0 = (l/r„)(r„Ie(t) + 2 lim (dv/dx)),
z—+0
dm/dt = (1 /Tm(Vi))(m — m„(y,)),
dh/dt = (l/TK(v,))(h - ha(Vi)),
dn/dt = (1/r„(yt))(n - n0„(t\)),
where Cn is node capacity, is the transmembrane current, 70 accounts for the stimulus /„