# A Recent Report: Applied Math at the US Dept. of Energy

A few reports with some OR & Sustainability content were issued recently. **Applied Mathematics at the US Department of Energy: Past, Present and a View to the Future** was written by “an Independent Panel from the Applied Mathematics Research Community” chaired by David L. Brown of Lawrence Livermore National Laboratory.

**Complex Systems** emerge as a major theme in the report, as the title of Chapter 2 (of 3) suggests: *Advancing Mathematics for Modeling, Simulation, Analysis and Understanding of Complex Systems*. That chapter has a good defintion of a complex system through a listing of the typical properties of one. I tend to think of a colony of ants; lots of small, relatively simple components (the ants) whose combined actions and interactions lead to behavior that would be difficult to predict based on examination of the individual ants. More relevant to the subject at hand and among the examples in the report are complex engineered systems, such as:

the electric power grid, where models may involve inequality and other types of constraints, severe nonlinearities and discontinuities, a mixture of continuous and integer variables, a large number of variables, a huge range of scales, and non-unique solutions that may make it difficult to characterize the most physically reasonable result.

Some sections within the report with OR-relevance include:

- Modeling Stochastic Effects in Complex Systems
- Networks, Systems and Systems of Systems
- Sensitivity Analysis
- Uncertainty Quantification and Mitigation
- Using Complex Systems to Inform Policy-Making (with subsections Risk Analysis and Optimization)

Early on, the report states “The DOE Applied Mathematics program has also had significant impact on the field of mathematical optimization,” and goes on to mention foundational work in nonlinear optimization done at Argonne (unconstrained) and Stanford (constrained).

Later, the section on optimization discusses a number of challenges such as problems of mixed variable types or with non-smooth objective, multilevel and multi-objective optimization problems, stochastic optimization, and many others. Here is an excerpt regarding the first of those:

DOE applications increasingly result in nonlinear optimization problems with a mixture of variable types. For example, the design of fossil energy power generation systems involves both continuous quantities such as length and width, and integer variables such as the number of processing units. Categorical variables (those that lack a natural ordering, such as alternative energy technologies) are also common in policy-based applications, and occur, for example, in determining the optimal locations for placing sensors or designing nanomaterials.

The 3rd and final chapter is titled *Driving Innovation and Discovery in Applied Mathematics Through Effective Program Leadership*. It includes recommendations such as encouraging researchers to take risks and the use of multidisciplinary teams.

The energy/environment topics, from climate modeling to carbon sequestration to nuclear waste storage, mostly all have *something* to do with sustainability. But sustainability may not mean quite the same thing to the fossil fuel industry as it does to the renewable energy industry.

Independent of the discussion of energy and the environment, the report provides an interesting read on the state of the art in applied mathematics and the challenges that lie ahead for it particularly as applied to complex systems.

It is available through the US Department of Energy ~~SIAM (the Society for Industrial & Applied Mathematics)~~ at ~~this page (PDF).~~