Mathematical Foundations of High-Performance Computing and Communications (1991) / Chapter Skim
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4 THE ROLE OF THE MATHEMATICAL SCIENCES IN THE GRAND CHALLENGES
4 THE ROLE OF THE MATHEMATICAL SCIENCES IN THE GRAND CHALLENGES
Pages 15-26
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From page 15... ...
Yet the need to estimate mean turbulent transport is crucial to a wide range of computational fluid dynamics applications, from weather prediction to aircraft design to chemical processing. In recent decades, the mathematical theory of turbulence has dealt primarily with the properties of homogeneous isotropic turbulence and has turned to massive numerical simulations of fluid flows for experimental verification.
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From page 16... ...
In this connection, the new mathematical theory of approximation dynamics, which seeks to compare the long-time dynamical properties of a given equation with those of an approximate equation, is relevant [Plies and Sell, 19914. The recent discovery of inertial manifolds and related topics have enabled mathematicians to give a rigorous foundation to the observation that the formation of patterns, the occurrence of periodic and chaotic behavior in certain turbulent fluid flows, can be described exactly in terms of finite systems of ordinary differential equations.
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From page 17... ...
Current versions of this algorithm allow greater sensitivity in the searches and the use of different metrics based on biochemical properties of the amino acids and on evolutionary relationships between them. It is quite time consuming to do large numbers of comparisons on conventional computers, and so recent work has focused on developing faster serial algorithms for dynamic programming [Yao, 1982; Lipman and Pearson, 1985]
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From page 18... ...
These models consist of a network of single heart cells that obey certain rules on the transmission of electrical currents between neighboring cells. The behavior within each single cell is governed by a system of ordinary differential equations that describes the chemical concentrations and the electrical currents within that cell.
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From page 19... ...
In the validation of atmospheric climate models, in the design of new climate observing systems, and in the use of fluctuation dissipation relations to estimate the properties of the system related to its rapid response to external influences, determination of the fast space- and time-correlation properties of the atmosphere is crucial. Only within the past few years have global ocean models begun to be used to study the current systems coupling the various ocean basins.
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From page 20... ...
, , ~ ~ , ~ ~ , ~ , ~ ~ e, ~ e ~ ~ It is realistic to expect that the theory of nonlinear dynamics, and especially the promising developments expected as a result of the HPCC program, will offer opportunities for mathematical scientists to make significant contributions to global weather and climate predictions. Hardware and software capabilities are fast approaching a level that will allow serious investigations of the dynamics of the attractors arising in the very large systems of ordinary differential equations used to model atmospheric flows.
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From page 21... ...
Advances in the understanding of weak convergence, homogenization, Young measures, and numerical nonlinear analysis have been critical, while methods for averaging and computing the solutions to nonlinear systems of partial differential equations have contributed more generally. Mathematical challenges also abound in creating a theoretical description of polymers so as to relate bulk properties to the structures and interactions of the individual intertwined molecules.
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From page 22... ...
The development of new mathematical algorithms, probably utilizing massively parallel computing, would significantly increase the ability of researchers to study industrially important problems dependent on the quantum-mechanical nature of materials. Geophysical Modeling Propagation of elastic waves in the earth is of great interest in geophysics, with important applications to seismology, exploration and reservoir characterization for petroleum production, nuclear waste disposal, and remediation of contaminant leakage.
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From page 23... ...
modern control theory and other aspects of dynamical systems theory, while machine vision relies fundamentally on the theories of filtering, probability, statistics, and stochastic processes. For example, modern statistical tools have been used in the interpretation of satellite data for weather and crop yield prediction, pollution assessment, and geological applications.
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From page 24... ...
In these and other machine vision problems, the central issue is modeling, both for the articulation of regularities and for the accommodation of variability. The mathematical sciences offer some basic tools, mostly involving probability, statistics, and stochastic processes.
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From page 25... ...
Estimates for full spatial realism in sound capture and projection suggest the use of hundreds of microphones and a teraflop computer to achieve three-dimensional spatial selectivity [Flanagan et al., 19903.
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