the very highest performance and (2) a new generation of local refinement methods and codes for atmospheric, oceanic, and land modeling. . . . The propagation of uncertainty through a coupled model is particularly problematic, because nonlinear interactions can amplify the forced response of a system. In addition, it is often the case that we are interested in bounding the uncertainties in predictions of extreme, and hence rare, events, requiring a rather different set of statistical tools than those to study means and variances of large ensembles. New systematic theories about multiscale, multiphysics couplings are needed to quantify relationships better. This will be important as atmospheric modeling results are coupled with economic and impact models. Building a better understanding of coupling and the quantification of uncertainties through coupled systems is necessary groundwork for supporting the decisions that will be made based on modeling results.5
ILLUSTRATIVE DEMANDS FROM ENGINEERING
The 2008 National Academy of Engineering report Grand Challenges for Engineering6 identified 14 major challenges. Following is a list of 11 of those challenges that depend on corresponding advances in the mathematical sciences, along with thoughts about the particular mathematical science research that will be needed.
• Make solar energy economical. This will require multiscale modeling of heterogeneous materials and better algorithms for modeling quantum-scale behaviors, and the mathematical sciences will contribute to both.
• Provide energy from fusion. This will require better methods for simulating multiscale, complex behavior, including turbulent flows, a topic challenging both mathematical scientists and domain scientists and engineers.
• Develop carbon sequestration methods. This will require better models of porous media and methods for modeling very large-scale heterogenous and multiphysics systems.
• Advance health informatics. Requires statistical research to enable more precise and tailored inferences from increasing amounts of data.
• Engineer better medicines. Requires tools for bioinformatics and simulation tools for modeling molecular interactions and cellular machinery.
• Reverse-engineer the brain. Requires tools for network analysis, models of cognition and learning, and signal analysis.
5 Ibid. p. 58.