A not-so delightful aroma entered through the car vents as I drove past scores of dairy farms along interstate 5. I couldn't help but wonder to what extent environmental scientists study the effects on the ozone of methane gas produce by California Central Valley cows alone. And what about the dairy farmers who must own enough milk-producing cows to stay in business? But what, aside from some temporary olfactory discomfort do any of these concerns have to do with a mathematician?

As the story goes, the USDA once wanted to make cows that would produce more milk.

So, they assembled a team of the foremost biologists and recombinant DNA technicians, gave them unlimited funding, and asked them to build a better cow. Two years a minor bacterial epidemic later, they came back with the "new, improved cow" having a milk production improvement of 2% over the original.

Disappointed, the USDA went to the greatest chemists around. The chemists worked for six months, and managed a 5% improvement in milk output.

Finally, in desperation, the USDA turned to the foremost mathematician of his time. Upon hearing the problem, he told them to come back in the morning at which time he would present his results. And indeed, in the morning, he handed them a piece of paper with the computations for the new, 300% improved milk cow.

The plans began:

A Proof of the Attainability of Increased Milk Output from Bovines.

Consider a spherical cow, completely filled with milk.... ¹

I don't blame the mathematician for failing to consider the impracticality of his solution. Most of the problems I solved throughout my undergraduate mathematics courses either had solutions that I could write down in closed-form or did not exist at all. With an exception or two, most of these problems were not based on any real-world system, and the few that were had been so simplified as to be useless. To supplement my undergraduate training, I entered a doctoral program in computational and applied mathematics that extends classical mathematics training to include computational science and engineering. Using the tools of computational and applied mathematics, which includes optimization, numerical analysis, and linear algebra, mathematicians like me could develop truly useful models and thus avoid problem-solving that calls for mathematically perfect cows. Let's review some of those tools now.

Optimization and Operations Research
Some mathematical problems involve a few to a million variables that can give rise to multiple solutions. Optimization entails finding a best (or optimal) solution amongst all solutions satisfying the given conditions. Optimization problems consist of a cost function to minimize or maximize, a set of variables affecting the cost function value, and a set of constraints specifying the allowable range of variable values. In milk production, we might want to maximize the milk revenue or minimize the cost of maintaining a bovine farm. The variables might include the number of milk-producing cows or the time spent milking each cow. A set of constraints might be the maximum cow capacity of the farm, the capital you're willing to spend to purchase any additional cows, the demand for milk, or the average cost to maintain a cow. In designing a bovine drug or hormone, we might maximize bovine lactation; the variables might be different amino acids; the constraints might be molecular bond lengths, angles, and rotations.

Numerical Analysis and Linear Algebra
Numerical analysis and linear algebra consist of the theoretical and computational investigation into the numerical algorithms solving a mathematical problem. These investigations determine, among a vast many things, stability to measure changes in the solution caused by small changes to the data, and rates of convergence to measure how fast the algorithm computes the solution - because, after all, we want to live to see the answer. Suppose we believe we modeled milk production fairly well, but we now need to change the time it takes to milk a cow by an additional 30 seconds. If our new numerical solution corresponded to losing the farm because of this small change, then either our model wasn't correct or our numerical algorithm was unstable - oops! Back to the numerical drawing board.

Interdisciplinary Study and Grand Challenge Problems
When I first started working on a real-world modeling problem (and, it wasn't about cows) in graduate school, I consulted with a mechanical engineer to ensure I was making the correct physical assumptions. Every so often I would use a mathematical term that resulted in both of us eyeing each other suspiciously followed by a prompt for each other's credentials. Are you sure you're a mathematician? Are you sure you're an engineer? This occurred for some time before we figured out that a mathematical term might have a different connotation in engineering and vice versa. Because of the importance of cross-discipline interaction, most computational and applied mathematics programs require courses in computer science, biology, chemistry, geology, physics, statistics, engineering, or economics ². The breadth of the computational and applied mathematician becomes important in the context of grand challenge problems. These problems are defined by the HPCC³ as "a fundamental problem in science and engineering with broad economic and scientific impact, whose solutions can be advanced by applying high performance computing techniques and resources." Grand challenge problems include weather prediction, analysis of fuel combustion, ocean modeling, and mapping the human genome. Because of the sheer magnitude of work required to solve a grand-challenge problem, it is broken down into smaller research tasks assigned to interdisciplinary teams. Those teams that are able to communicate with each other and constructively work together have a higher probability of succeeding.

The (Computational) Science Process of Modeling
Using the tools of computational and applied mathematics, together with our communication skills across disciplines, we are ready to tackle any scientific problem. Most scientific problems, across disciplines⁴, can be broken down into five steps^{5, 6}.

1. Identify an interesting real-world phenomenon.
2. Develop a conceptual model containing observations and measurable properties. Most of the uncertainty in the final model occurs at this phase either because you did not recognize the existence of some process or you did not completely describe it.
3. Develop a mathematical model using various constitutive laws of physics such as conservation of mass and energy together with the empirical laws describing the observations. The mathematical model usually corresponds to a well-defined mathematical problem that looks like a differential equation or an integral equation subject to some initial and/or boundary conditions that must be solved together over some spatial and/or temporal domain.
4. Develop a numerical model. You must choose or create a numerical algorithm that solves the mathematical problem and you must choose the programming language and the computer system for executing the model. Using known parameters or a perturbation of model parameters, validate the model by matching model results to experimental data. Return to step 2 or 3 to refine the model. Determine the stability and convergence rates of the numerical algorithm. Return to step 3 or 4 to refine the model.
5. The end product is called a simulator because it is used to simulate the true phenomenon. Use the simulator to predict system responses to system stresses.

Final Remarks
Just as it is important for the mathematician to be grounded in real-world problems, so it is important that you understand the tools, the modeling process, and the language used to solve real-world problems about which you hear or read. Now, how would you design a super cow?

Endnotes:
¹ A version of this fictional story can be referenced to Chet Murthy from Cornell University. Another version can also be found in the book Consider a Spherical Cow: A Course in Environmental Problem Solving. 1988, John Harte, Mill Valley, California, University Science Books, 283 p.
² These course requirements vary from program to program.
³ The High Performance Computing and Communication is a program of the National Coordination Office for Computing, Information, and Communications that in turn advises the Executive Office of the President as well as Congress on science and technology issues.
⁴ Whether we are aware of it or not, we use this five-step process when we search out research projects -- from choosing an interesting problem on which to work to determining which small part of it we're going to solve that isn't already solved.
⁵ These five steps are modified from the outline Anatomy of the Computational Science Process. Richard A. Tapia, Rice University. A detailed discussion can be found in Incorporating Uncertainty into Aquifer Management Models, Steven M. Gorelick, p101-111 in Subsurface Flow and Transport: A Stochastic Process. 1997, G. Dagan and S. P. Neuman eds., Cambridge University Press.
⁶ Missing from this five-step scientific process is a controversial sixth step involving evaluation and review of the final product for social and ethical implications.