## What is Applied Mathematics and Computational Science?

**A SIAM Brief on the Applied Math & Computational Science Fields**

Applied mathematics is the branch of mathematics that is concerned with developing mathematical methods and applying them to science, engineering, industry, and society. It includes mathematical topics such as partial and ordinary differential equations, linear algebra, numerical analysis, operations research, discrete mathematics, optimization, control, and probability. Applied mathematics uses math-modeling techniques to solve real-world problems.

Computational science is an emerging discipline focused on integrating applied mathematics, computer science, engineering, and the sciences to create a multidisciplinary field utilizing computational techniques and simulations to produce problem-solving techniques and methodologies. Computational science has become a third partner, together with theory and experimentation, in advancing scientific knowledge and practice.

Applied mathematics and computational science are utilized in almost every discipline of science, engineering, industry, and technology. Industry relies on applied mathematics and computational science for the design and manufacture of aircraft, automobiles, textiles, computers, communication systems, prescription drugs, and more. Work with applied mathematics often leads to the development of new mathematical models, theories, and applications that contribute to diverse areas of science.

Applied mathematicians and computational scientists often hold jobs with titles such as actuary, statistician, scientific programmer, systems engineer, analyst, research associate, and technical consultant. Applied mathematicians and computational scientists work for federal and state governments, financial services, scientific research and development services, and management, scientific, and technical consulting services. Software publishers, insurance companies and aerospace, pharmaceutical, and other manufacturing companies also employ applied mathematicians and computational scientists. Many work in academia, teaching the next generation and developing innovations through their own research.

Some examples of the use of applied mathematics and computational science follow.

- Simulation and prototype testing are used in manufacturing design and evaluation. For example, automotive companies are using computer-aided design to test for performance, safety and ergonomics. In doing so, they dramatically lower the cost of constructing and testing prototypes.
- Computational simulations in aircraft design have been used to analyze the lift and drag of airfoil designs since the early days of computing. Advanced computation and simulation are now essential tools in the design and manufacture of an aircraft.
- A major advance in computing power will enable scientists to incorporate knowledge about interactions between the oceans, the atmosphere and living ecosystems, such as swamps, forests, grasslands and the tundra, into the models used to predict long-term change. Climate modeling at the global, regional and local levels can reduce uncertainties regarding long-term climate change, provide input for the formulation of energy and environmental policy, and abate the impact of violent storms.*
- Accurate simulation of combustion systems offers the promise of developing the understanding needed to improve efficiency and reduce emissions as mandated by U.S. public policy. Achieving predictive simulation of combustion processes will require terascale computing and an unprecedented level of integration among disciplines including physics, chemistry, engineering, mathematics, and computer science.*
- Meeting the needs of nuclear stockpile stewardship and management for the near future requires high-performance computing far beyond our current level of performance. The ability to estimate and manage uncertainty in models and computations is critical for this application and increasingly important for many others.*
- Applied mathematics and computational science is also useful in finance to design trading strategy, assist in asset allocation, and assess risk. Many large and successful hedge fund companies have successfully employed mathematics to do quantitative portfolio management and trading.

*From Graduate Education for Computational Science and Engineering Report,
which can be found at, www.siam.org/students/resources/report.php.

*10/5/08*