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OCR for page 27
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.
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OCR for page 28
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.
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Representative terms from entire chapter:
differential equations
