The Applied Mathematics Research program in the Office of Advanced Scientific Computing Research (ASCR) has a long history and focuses on mathematical research and software that impact the future of high-performance computing.

Applied Mathematics translates the physical world into algorithms – mathematical procedures – that computers can calculate and solve much faster and attack bigger and more complex problems than humans alone.  Without the mathematics behind them, such things as weather forecasting models and digital television would be impossible.  Applied Mathematics enables scientists to describe and predict phenomena like motion and gravity in mathematical terms.

The algorithms Applied Mathematics Research develops power high-fidelity simulation and analysis of physical, chemical and biological processes, describing them in discrete terms computers can calculate.

The program supports research on vital areas important to creating and improving algorithms:

• Numerical methods for solving ordinary and partial differential equations, especially numerical methods for computational fluid dynamics.  PDEs solve problems involving unknown relationships between several variables, enabling simulations of things like fluid flow, wave propagation and other phenomena.
• Computational meshes for complex geometrical configurations, which seek to translate domains of mathematical values into discrete points to simulate continuous processes like combustion.
• Numerical methods for solving large systems of linear and nonlinear equations.
• Optimization, which seeks to minimize or maximize mathematical functions and can be used to find the most efficient solutions to engineering problems or to discover physical properties and biological configurations.
• Multiscale computing, which connects varying scales in the same problem, such as relating processes and properties at the tiniest scales of time and space to those at the largest scales.
• Multiphysics computations, which simulate physical processes of different kinds, such as a chemical reaction at its boundary with a material.
• Math software and libraries – modular codes that can be incorporated in programs from diverse science areas, allowing developers to quickly build software that makes difficult calculations efficiently and rapidly.

In the years ahead, Applied Mathematics Research will focus on algorithms that take advantage of computers capable of up to 1 quadrillion operations per second (1 petaflop).  These computers will allow scientists to consider research never thought possible, such as predictive simulation of the physical properties of novel materials.

Applied Mathematics Research also is devoting more resources to “uncertainty quantification,” which gives a mathematical characterization of the confidence levels for calculations.  Algorithms are also being developed to help researchers sift through the mountains of data generated by increasingly detailed simulations and more powerful experiments.

The program also looks to the future by supporting fellowships and other programs to train new computational scientists.

The Applied Mathematics Research program at DOE has broadly influenced our ability to use computers for scientific discovery.  Work done 50 years ago, for example, provided the foundations for modern computational fluid dynamics (CFD), which studies the movement of gases and liquids. New research and continuous refinement of techniques has made modern CFD essential to climate modeling, nuclear reactor design, subsurface flow modeling, and many other applications of critical importance to our nation.

Applied Mathematics Research projects often take years before they directly contribute to new and better simulations and software.  Nonetheless, the fundamental advances these projects provide are essential to maintaining America’s global competitive edge.