Part I

Introduction to Applied Mathematics

I.1 What Is Applied Mathematics?

Nicholas J. Higham

1 The Big Picture

Applied mathematics is a large subject that interfaces with many other fields. Trying to define it is problematic, as noted by William Prager and Richard Courant, who set up two of the first centers of applied mathematics in the United States in the first half of the twentieth century, at Brown University and New York University, respectively. They explained that:

Precisely to define applied mathematics is next to impossible. It cannot be done in terms of subject matter: the borderline between theory and application is highly subjective and shifts with time. Nor can it be done in terms of motivation: to study a mathematical problem for its own sake is surely not the exclusive privilege of pure mathematicians. Perhaps the best I can do within the framework of this talk is to describe applied mathematics as the bridge connecting pure mathematics with science and technology.

Prager (1972)

Applied mathematics is not a definable scientific field but a human attitude. The attitude of the applied scientist is directed towards finding clear cut answers which can stand the test of empirical observation. To obtain the answers to theoretically often insuperably difficult problems, he must be willing to make compromises regarding rigorous mathematical completeness; he must supplement theoretical reasoning by numerical work, plausibility considerations and so on.

Courant (1965)

Garrett Birkhoff offered the following view in 1977, with reference to the mathematician and physicist Lord Rayleigh (John William Strutt, 1842–1919):

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Essentially, mathematics becomes "applied" when it is used to solve real-world problems "neither seeking nor avoiding mathematical difficulties" (Rayleigh).

Rather than define what applied mathematics is, one can describe the methods used in it. Peter Lax stated of these methods, in 1989, that:

Some of them are organic parts of pure mathematics: rigorous proofs of precisely stated theorems. But for the greatest part the applied mathematician must rely on other weapons: special solutions, asymptotic description, simplified equations, experimentation both in the laboratory and on the computer.

Here, instead of attempting to give our own definition of applied mathematics we describe the various facets of the subject, as organized around solving a problem. The main steps are described in figure 1. Let us go through each of these steps in turn.

Modeling a problem. Modeling is about taking a physical problem and developing equations — differential, difference, integral, or algebraic — that capture the essential features of the problem and so can be used to obtain qualitative or quantitative understanding of its behavior. Here, "physical problem" might refer to a vibrating string, the spread of an infectious disease, or the influence of people participating in a social network. Modeling is necessarily imperfect and requires simplifying assumptions. One needs to retain enough aspects of the system being studied that the model reproduces the most important behavior but not so many that the model is too hard to analyze. Different types of models might be feasible (continuous, discrete, stochastic), and for a given type there can be many possibilities. Not all applied mathematicians carry out modeling; in fact, most join the process at the next step.

Analyzing the mathematical problem. The equations formulated in the previous step are now analyzed and, ideally, solved. In practice, an explicit, easily evaluated solution usually cannot be obtained, so approximations may have to be made, e.g., by discretizing a differential equation, producing a reduced problem. The techniques necessary for the analysis of the equations or reduced problem may not exist, so this step may involve developing appropriate new techniques. If analytic or perturbation methods have been used then the process may jump from here directly to validation of the model.

Developing algorithms. It may be possible to solve the reduced problem using an existing algorithm — a sequence of steps that can be followed mechanically without the need for ingenuity. Even if a suitable algorithm exists it may not be fast or accurate enough, may not exploit available structure or other problem features, or may not fully exploit the architecture of the computer on which it is to be run. It is therefore often necessary to develop new or improved algorithms.

Writing software. In order to use algorithms on a computer it is necessary to implement them in software. Writing reliable, efficient software is not easy, and depending on the computer environment being targeted it can be a highly specialized task. The necessary software may already be available, perhaps in a package or program library. If it is not, software is ideally developed and documented to a high standard and made available to others. In many cases the software stage consists simply of writing short programs, scripts, or notebooks that carry out the necessary computations and summarize the results, perhaps graphically.

Computational experiments. The software is now run on problem instances and solutions obtained. The computations could be numeric or symbolic, or a mixture of the two.

Validation of the model. The final step is to take the results from the experiments (or from the analysis, if the previous three steps were not needed), interpret them (which may be a nontrivial task), and see if they agree with the observed behavior of the original sys- tem. If the agreement is not sufficiently good then the model can be modified and the loop through the steps repeated. The validation step may be impossible, as the system in question may not yet have been built (e.g., a bridge or a building).

Other important tasks for some problems, which are not explicitly shown in our outline, are to calibrate parameters in a model, to quantify the uncertainty in these parameters, and to analyze the effect of that uncertainty on the solution of the problem. These steps fall under the heading of UNCERTAINTY QUANTIFICATION [II.34].

Once all the steps have been successfully completed the mathematical model can be used to make predictions, compare competing hypotheses, and so on. A key aim is that the mathematical analysis gives new insights into the physical problem, even though the mathematical model may be a simplification of it.

A particular applied mathematician is most likely to work on just some of the steps; indeed, except for relatively simple problems it is rare for one person to have the skills to carry out the whole process from modeling to computer solution and validation.

In some cases the original problem may have been communicated by a scientist in a different field. A significant effort can be required to understand what the mathematical...