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SUMMARY OF THE AREAS WHERE NEW 1\lATHEMATICAL SCIENCES RESEARCH IS NEEDED In order to accomplish the HPCC program and to successfully attack the grand challenges, new mathematical sciences research will be needed, as described in Sections 3 and 4. Although it is very difficult and somewhat presumptuous to identify precisely the most promising and needed research areas, they certainly include the following: . Numerical algorithms, especially multigrid and domain decomposition methods and parallel algorithms, and adaptive mesh generation for partial differential equations, as they arise in modeling semiconductors, geophysics, turbulence, and elsewhere; . Homogenization methods, as they arise in the modeling of oil reservoirs and the atmosphere, in materials science, and in any situation where the range of relevant length scales exceeds our capability to resolve them by brute-force computations; . Dynamic graphics and other visualization methods for addressing high-dimensional data, and high-dimensional surface fitting for process control and product design; . Queueing theory and network flow algorithms to design efficient large-scale communication networks; . Efficient pattern matching (including dynamic programming) for problems in vision, molecular biology, and human-machine voice interactions; . Model validation and assessment of uncertainty based on data from numerical experiments combined with physical experiments or observations, for studying global change, materials science, and many other research areas; Development of user-friendly software for libraries; . Nonlinear wave propagations, both deterministic and random, in communications, geophysical explorations, ocean modeling, and stealth technology; . Numerical methods in nonlinear dynamical systems, as they arise in weather forecasting, climate modeling, and turbulence; and Graph theory, graph embeddings, and network algorithms. 27
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This list is by no means exhaustive: inverse problems in geophysics and medicine, large-scale optimization in protein folding and other areas, and complexity theory are just a few of the additional areas that are fundamental to either the HPCC program or the grand challenges. Effective development of the above research areas will also require continued support of the general areas of partial and ordinary differential equations, statistics, computational geometry, control and optimization, and numerical analysis, to the extent that these areas actively interface with the topics itemized above. It is vital that, as the overall research effort in high-performance computing and communications is stepped up, relevant areas of the mathematical sciences also receive commensurate attention and support. 28