What are the Applied Mathematical and Computational Sciences?

The first step in a real-world problem is often the construction of a mathematical model, a description of the problem in mathematical terms. The model then is studied by using analytical or numerical methods to obtain exact or approximate solutions. Finally, the conclusions are interpreted in the language of the original problem in terms more familiar to a client or user. Often the model is changed to be more realistic or to include more features of the problem. Thus, the modeling process may involve false starts, modifications, and simplifications.

Mathematics enters primarily in the second stage: the solution of mathematically well-formulated problems and the development and analysis of the underlying theory. This stage may include analytical or numerical methods. The approach can range from specific algorithms and formal methods to abstract, general theories. It is often not clear which mathematical skills will be useful in the study of a new problem; thus, applied mathematicians need to be broadly trained so they will have a wide variety of mathematical tools available.

The mathematical scientist must not only be a competent mathematician but must be knowledgeable in the area to which mathematics is being applied. Thus, the applied mathematician must be concerned with the construction and interpretation of appropriate models. Students must communicate with scientists in that field in their language.

The art of formulating models requires that the modeler make choices about which factors to include and which to exclude. The goal is to produce a model that is realistic enough that it reflects the essential aspects of the phenomena being modeled, but simple enough that it can be treated mathematically.

Often the model is constructed to answer a specific question. Sometimes the modeler must either simplify the model so it can be analyzed or devise new mathematical methods that will permit an analysis of the model. Often a combination of analytical and numerical methods are used. The modeling process may involve a sequence of models of increasing complexity. Problems sometimes lead to new mathematical methods, and existing mathematical methods often lead to new insights into the problems. The successful applied mathematical scientist must be comfortable and confident in both mathematics and the field of application.

Applied mathematical and computational sciences is a name used to encompass the many analytical and numerical methods to solve classes of scientific problems. This name more accurately reflects the nature of modern "applied mathematics" since it explicitly includes the area of scientific computing, which includes many components of computational processes, such as numerical analysis, algorithms for machines with vector and parallel architectures, visualization, simulation, and computer-aided design.

Scientific computing already is being called a third science, complementing theoretical science and laboratory science. For example, in the design of automobiles and aircraft, many engineering issues are resolved through computer simulation rather than through costly prototypes, test models, and wind tunnel experiments. This trend toward design by computer simulation has been powered by computing hardware and software, computational methodologies and algorithms, and the availability and access to high-performance computing systems and infrastructure.