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4 THE ROLE OF THE MATHEMATICAL SCIENCES IN THE GRAND CHALLENGES The ultimate driving force behind the HPCC initiative is the promise of making so-called grand challenge problems computationally tractable. These are problems of significant societal and intellectual importance whose solution demands computing power far in excess of that available today. However, even with all the computing power to be available with successful completion of the HPCC program, the grand challenges cannot be conquered without significant mathematical sciences involvement. Such involvement should generally be through multidisciplinary teams or informal collaborations, since the scientific background required to contribute to these problems is formidable for one person, and often multifaceted. This section examines the interaction of the mathematical sciences with some of the grand challenges in recent years and describes some promising mathematical directions for the near future. Turbulence Research The quantitative statistical properties of fluid flows remain largely unknown. 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. From this work have come stochastic models of turbulence in which the original nonlinear chaotic dynamical system is modeled by more tractable linear random dynamical systems. These provide normal or gaussian approximations to the original unknown nongaussian statistical properties. More recent work on nonlinear mapping schemes is attempting to recapture the nh~erv~1 nongaussian properties of turbulence. ~cat ~rib In practical engineering applications, however, turbulence is neither isotropic nor homogeneous, and the best hope has been to use numerical simulation of the three-dimensional time-dependent flows of interest. The range of scales of motion involved far exceeds that resolvable on any present or future computer, but fortunately the important eddy transport properties of turbulent flows are determined by the larger, resolvable scales of motion. There remains, though, the problem of accounting for the effects of unresolvable motions on these larger eddies. Such turbulent subgrid-scale models have for many years been used to estimate the mean damping effects of associated 15

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eddy viscosity. Recently, the random forcing effect of unresolvable eddies on the large eddies has been approximated [Mason and Thompson, 19913. The form of such forcing terms has been determined by earlier work in the mathematical theory of turbulence. As the goals of the HPCC program are reached and the range of resolvable scales is increased, the behavior of subgrid-scale fluctuations is expected to become more purely probabilistic, and thus more amenable to stochastic mocieling. Further research on a mathematical theory of the interaction between subgrid-scale modeling and the properties of finite approximations to the Navier-Stokes equations is needed in order to derive full benefit from the HPCC program. Further study is also required of their long-time dynamical properties, such as the persistence of patterns and the occurrence of periodic and chaotic behaviors. 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. Because this reduction of an infinite-dimensional problem to a finite-dimensional one is exact and error-free, it holds the promise that approximation schemes based on the inertial manifold paradigm can lead to greater computational efficiency and better modeling of complicated systems, such as global weather phenomena or cardiac networks. Nonlinear Galerkin [e.g., Marion and Temam, 1989, 1990], or approximate inertial manifold (ATM) methods, are based on this development and show good computational promise, but more work is needed. Turbulence--and computational fluid dynamics in general--is of major importance to the chemical industry, among others. Chemical processing often involves multiphase chemically reacting flows, which lead to stiff time-dependent mathematical problems with moving boundaries. Such problems are beyond the capability of current computational science, without even attempting to include turbulent effects. Due to the size of the chemical industry, mathematical methods that could simulate, and hence help to optimize, these processes would have enormous economic impact. Computational Biology Proteins are the fundamental molecules of all living organisms. They are essentially linear chains of subunits called amino acids. There are 20 amino acids, and thus we can think of a protein as a word constituted from this 20-letter alphabet; this word is referred to as the protein's primary structure. The primary structure is encoded in the DNA of an organism by the genetic code: each amino acid is represented by a triplet of DNA base 16

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lo pairs. Proteins fold up into complicated three-dimensional structures to form the geometrical shapes, called secondary and tertiary structures, needed to carry out their function. It has become a standard laboratory procedure to determine the DNA sequence . ~ encoding tor a protein and translate it, by applying the genetic code, into an amino acid sequence. However, it is quite difficult to determine the three-dimensional structure of a protein given its primary structure, even though all the information required is there. (Mathematical advances from areas as disparate as optimization methods and knot theory have helped with this problem, though.) This difficulty has led to a large discrepancy in the size of the sequence databases: there are approximately 2S,000 proteins (S million amino acids) in the protein sequence database and about 41,000 DNA sequences (51 million nucleotides) in the GenBank DNA sequence database, but the structure database currently has only about 400 proteins. If the plan to sequence the entire human genome becomes a reality, the database of human protein sequences will increase some 1000-fold in the next decade, thus intensifying the need for an effective computational means of analyzing this information. Because the problem of determining the function and structure of a protein from its primary sequence is so difficult, biologists have looked for other approaches. One highly successful technique is to compare a newly sequenced protein with al] of the known proteins in the database. Similarities in the amino acid or nucleotide sequence are used to infer similarities in the structure or function of the related proteins [DoolittIe, 19863. This approach makes sense for two reasons. First, proteins with similar biochemistry are apt to fold and to act in very similar ways under the laws of physics. Also, proteins that have large similar regions are often evolutionarily closely related, which indicates that the proteins may perform related functions or catalyze reactions in very similar ways. These comparison searches have been quite successful in uncovering relationships between apparently unrelated proteins. Several years ago it was discovered that some cancer-causing genes have a close similarity to genes that encode for proteins such as human growth factors tDoolittle et ai., 1983i, which suggests a possible mechanism by which these cancer-causing genes produce uncontrolled cell growth. The methodology just described has raised a number of interesting mathematical questions that impact the use of high-performance architectures in the area of computational biology. The main search algorithm that is used for finding similar regions in proteins is the dynamic programming algorithm first described in the 1950s [Bellman, 1957] and rediscovered by biologists for use in this problem in the 1970s [Needleman and Wunsch, 1970; Smith and Waterman, 19814. 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] and implementations of dynamic programming algorithms for parallel architectures [CouIson et ai., 1987; Edmiston and Wagner, 1987; Lander et ai., 17

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1988] and special-purpose hardware [Lipton and L~opresti, 1985; Hunkapiller et al., 1990]. Work on improving search techniques for the databases will have to continue as their size increases. Moreover the improved searching will create large databases of similarities that will have to be analyzed. The problem of the statistical significance of a match or a similarity is understood only in the simplest of cases, and much work will continue to be needed in this area [Arratia and Waterman, 1985; Arratia et al., 1986; Waterman et al., 1987]. Also, there is work to be done in analyzing the effects of varying certain parameters of the scoring metrics in the commonly used versions of the dynamic programming algorithm. The ability to do many searches with vaporing metrics at high speed wii] help provide the data for such a study [Lander et al., 1989], and perhaps lead to a more general theory. Dynamics of Cardiac Models One of the early successes in the area of HPCC has been the use of the new massively parallel computers by physiologists to do numerical studies of models of major tissues in the heart [Winslow et at, 19913. 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. It turns out that each cell is an oscillator with a stable periodic orbit for a certain range of the physiological parameters. The present massively parallel computers are large enough so that an entire mammalian sinus node, for example, can be simulated on the machine. These models become large systems (of order 105) of coupled oscillators. The challenging mathematical issue is to understand the long-time dynamics of these large systems. Since large mammalian hearts have physiological features that differ significantly from those of small hearts, one cannot expect that the dynamics of small coupled systems of oscillators will, or even can, emulate the dynamics of large hearts. New methods, which combine analysis and numerical studies, are needed to study complicated behavior like atrial fibrillation or dysrhythmia. Global Change Understanding and predicting global climate change is a leading grand challenge problem that will require great increases in the availability and utilization of new computing resources. In this context, "high-performance" means highly parallel, and a central problem is to move climate modeling work onto the coming generations of highly parallel computers. It is clear, however, that computing power alone is not enough, and much needs to be done to improve the effectiveness with which available and forthcoming 18

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computing power is used. It is in this latter area that the mathematical and statistical sciences may play the largest role. The atmosphere is a reasonably well measured fluid, but the other major governors of climate, the ocean and some aspects of biogeochemical cycles, are not well measured. The thermodynamic and flow state of the global atmosphere is determined at least once per day by assimilating new observations into a prediction model, so that past information is included in the estimate, as are certain nonlinear dynamical constraints. There remain many fundamental statistical and mathematical problems associated with this process of finding the best statistical estimates on a nonlinear dynamical manifold in a state space of myriad dimensions. There is also much to be done yet in applying the mathematical methods of classical statistical physics to relate day-to-day weather fluctuations and long-term climate trends. 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. The time scales in the ocean vary from weeks in the surface mixed layer to millennia in the abyss, and the space scale of important eddies is much smaller in the ocean than in the atmosphere. These facts combine to make ocean models require much more computing power in spite of the simpler physics involved. The further addition of atmospheric chemistry to the climate models adds the dimensions of species space to the overall problem, and the range of reaction rates is so large that the associated differential equations are exceedingly stiff. The fundamental mathematical problem here is the development of a theory that could guide the lumping together of species and reactions to reduce the size of species space and to remove stiffness at a minimal sacrifice in accuracy. Both atmospheric and ocean models attempt to describe a wide range of scales of fluid motion. They are thus faced with the usual turbulence problem of being unable to resolve all scales with available computing power. Difficult as this problem is for turbulence modeling, it is far worse in climate models where important dynamical processes in the ocean are occurring at about the limit of resolution, and in the atmosphere many physical processes remain unresolved. In general the important transports in the atmosphere are carried by resolved eddies, so that representing the unresolved eddies by a subgrid-scale mode] has proved workable. The situation in the ocean is less clear, but progress on this fundamental problem would be important for efficient use of computers when the only alternative would be to increase spatial resolution. Climate modeling is beset by a complex interplay of uncertainty arising from (1) the imperfect match between the mathematical model and reality, (2) the approximate nature 19

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of numerical solutions, and (3) uncertainty in the input data. The merging of information gained from real observations, physical experiments, and numerical experiments creates statistical issues that must be understood. The use of prior distributions in developing methods for assessing accuracy looks very promising. The discussion so far has been of problems that arise in the more or less conventional approaches to climate research. There are also a number of untried new approaches with higher risk but potentially higher rewards. Two relate to how the continuous fields of relevance to climate are represented in models. Traditionally, they have been approximated by grid-point values or spectral amplitudes in a surface harmonic expansion. An alternate approach that has often been suggested is to use instead empirical orthogonal functions obtained by diagonalizing the spatial covariance matrix. These have the advantage of representing the variance in the system with the fewest variables, but the disadvantage of having more complicated dynamical laws [Rinne and Karhila, 19753. A newer possibility is the use of waveless for the representation. These combine the localness of grid-point representations and the scale properties of spectral representations. In recent years a wavelet transform theory has been developed in multidimensional Cartesian space complete with a fast transform [Daubechies, 1988~. For global climate models it would be useful to find a wavelet base for the surface of the sphere. - The wavelet approach has been found to have many interesting features for the representation and display of fields. It may be less satisfactory for the formulation of dynamical laws. , , ~ ~ , ~ ~ , ~ , ~ ~ 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. Another promising development is the recent discovery of global attractors for the Navier- Stokes equations on thin three-dimensional domains and the related fact that these attractors are well approximated by associated fluid flows on two-dimensional surfaces [Raugel and Sell, 1989]. This may be relevant to atmospheric modeling because, by resealing, the atmosphere can be mapped to a thin domain on the surface of a sphere. A new paradigm for the study of systems with many effective degrees of freedom suggests that they may be attracted to states of self-organized criticality in which small disturbances can have consequences on widely varying space and time scales (Bak et al., 1988~. The sandpile is taken as a prototype in which an added grain of sand may displace one other or create an avalanche down the whole pile. This is proving to be an interesting model of the ubiquitous 1/f noise found in so many physical and engineering applications. It is not immediately evident that the climate system should be in such a state, but if for some reason it were, then its low-frequency variability would be better understood. 20

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Materials Science Understanding the macroscopic properties of materials--such as fiber-reinforced composites, metal alloys and shape-memory materials, polymers, and ceramics--from their microscopic structure is of great technological interest. The broad interaction between the mathematical sciences and materials science was recently surveyed [National Research Council, 1991]. The correct modeling of the macroscopic or bulk properties of complex materials is an extremely difficult problem that will benefit substantially from the accomplishments of the HPCC program. Many of the most promising materials on the horizon, especially for structural purposes, are composites, wherein the microstructural arrangement of constituents is critical. At present, we are unable to predict the properties of composites based on their microstructure, but mathematical scientists have begun to make progress toward this. 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. Many mathematical descriptions--including self-avoiding random walks, lattice models, Wiener functional integrals, and renormalization group methods--have been helpful in characterizing polymers. Established methods of statistical mechanics have helped in the prediction of some solid-phase polymeric properties--for instance, in predicting the elastic constants for glassy polypropylene. In the area of polymer processing, many serious and important mathematical problems are raised by attempts to model the non-Newtonian flows involved. Semiconductor Modeling Of all the grand challenge problems, semiconductor modeling is most closely linked with the success of the main HPCC hardware initiative. Future generations of semiconductors will involve the use of more exotic materials and processing techniques, as well as decreased feature size and expanded capabilities. This in turn demands more sophisticated physical models and simulation tools. For example, below one-half micron, the physical simplifications underlying the standard drift-diffusion model, which is the basis for most current device simulators, begin to lose validity. As devices become more densely packed, possible unintended interactions between nearby devices must be studied, which will require three-dimensional calculations. At the same time, more efficient iterative techniques must be developed to handle the 21

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larger sets of nonlinear equations that result from more complex models, and careful numerical analysis must be performed to ensure stable and accurate discretizations. Fast prototyping of these more demanding simulations through statistical modeling may become preferred to numerical experimentation, as the former is more adaptable to achieving robust engineering design. Quantum mechanics (QM) is essential to the study of semiconductors as well as many other materials. The present methods for accurate QM simulations have large scaling laws (o(N3) to o(N7)), which restrict their application to small- to moderate-sized systems of atoms. 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. The inverse problem of determining the structure and properties of the earth's interior from the elapsed travel time of sound waves is highly nonlinear. The solution of such problems involves repeatedly solving the forward problem--which can routinely require an hour of time on a Cray Y/MP for each run--with different assumptions about the unknown properties. The inverse problem is highly ill-posed and must deal with weak and noisy data. Mathematicians, as part of multidisciplinary teams, have made significant progress toward understanding and quantifying the nonuniqueness of the problem. They have also made advances in the stabilization and solution of the resultant least-squares formulation of the problem iSantosa and Symes, 19893. Mathematicians have also developed boundary and interface conditions necessary for discretizing complex geometries [Sochacki et al., 1991] and have also begun developing solution programs for parallel computers. Due to the ilI-conditioning of the inverse problem, a good initial guess is required for the iterative process. The increasing use of three-dimensional visualizations of inputs and outputs can give enormous intuition for understanding various seismic signatures. This adds to other depositional information to suggest a good initial guess for the true structure. Efficient three-dimensional visualization calls for data compression algorithms; the latest and most promising are based on wavelet technology. Geophysical techniques and inverse problems at different length scales are also required to characterize a reservoir once one is located. These methods must be coupled with complicated models of the multiphase flow processes in heterogeneous media to obtain reservoir properties through history matching. Control and optimization techniques have allowed great strides in automating the inverse problem, but they have not yet been 22

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successfully applied to the forward problem. The importance and role of mathematics in these applications have been discussed in some detail in Glimm [1991~. Machine Vision Automation is a cornerstone of modern military and industrial competitiveness. An important impediment has been a traditional and growing gap between often spectacular improvements in computational speed and memory as well as other hardware advances, on the one hand, and a relative lack of progress in the development of algorithms and the associated software, on the other hand. Substantial progress in algorithm development often first requires appropriate tools from the mathematical sciences. Robotics, for instance, increasingly draws upon (and contributes to) 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. Image formation in both transmission tomography (CAT) and emission tomography (PET and SPECT) is based upon sophisticated mathematical and statistical formulae. Other examples of the interplay between machine vision and the mathematical sciences include the use of methods from partial differential equations and numerical analysis in the definition and computation of natural boundaries [Mumford and Shah, 1989], the computation of three-dimensional surfaces from shading information tHorn, 1986], and the restoration of blurred or noisy images [Lions et al., ]9913; the use of spatial stochastic processes, such as Markov random fields, for texture analysis [Cross and Jain, 1983], image restoration [Grenander, 1983; Besag, 1986], and reconstruction in emission tomography [Geman and McClure, 1987; Green, 1990~; and the use of regularization methods for various inverse problems, such as motion detection and stereo mapping [Poggio et al., 19854. Not surprisingly, it is the high-level machine vision tasks that have proven the most challenging, and it is here that mathematics is likely to play a most pivotal role. The issue of accommodating textures in automated industrial inspection was already discussed in Section 3. A second example is the problem of machine recognition of handwritten characters. Among many other applications, a robust solution would allow for fully automated processing of checks and credit card slips at great savings to banks and various other companies, such as utilities, that handle large volumes of checks. All that is needed is the recognition of the ten numerals, zero through nine, yet no company has yet been able to offer a commercially viable solution. Model-free approaches have largely failed. Explicit models are needed, especially for the infinite variety of stroke shapes and sequences that constitute the allowable presentations of a particular digit. This combination of structure and variability again suggests the use of stochastic processes. A 23

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promising example is the class of hidden Markov models, which have already been successfully applied in automatic speech recognition [Rabiner, 19893. These processes lend nemse~ves to a n~gn~y e~r~c~ent computational algorithm known as "dynamic programming," ~. . ^ ~- ~, , . 1 . ~ ~se ~- and are provably versatile: Every stochastic process can be arbitrarily well approximated by a hidden Markov model. 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. It is unreasonable to expect engineers, computer scientists, cognitive scientists, or others involved in the practice of machine vision to have a working knowledge of the various mathematical methods that may be brought to bear. Indeed, it is certainly true that existing mathematical tools will need to be extended and improved before they are suitable for applications. Thus, direct involvement of mathematical scientists would appear to be an essential ingredient in any effort to fill the growing gap between sophisticated computing hardware and available algorithms and software for machine vision. Human-Machine Communication Natural voice is the preferred means for information exchange by humans. As computers and communications systems become more complex, the incentive for ease of use will grow. Voice control and conversational interaction will be expected to bear more of the burden for easy, natural, and effective human-machine communication. Meeting this expectation will depend on expanded mathematical understanding. Several specific areas can be mentioned. Fluent conversational interaction with machines requires large recognition vocabularies and extensive computational models of language. Algorithms for the recognition of large vocabularies are not yet in hand. Mathematics research on higher-order Markov models is likely to contribute to this goal. Then, given accurate, reliable acoustic pattern recognition to produce word-string candidates, semantic and pragmatic interpretation must be accomplished. Present indications are that statistical models of language can contribute strongly to this capability, and again mathematical techniques for programming and manipulating probabilistic models of natural language are needed [Flanagan, 1991~. Very similar pattern recognition and semantic interpretation problems exist in a variety of fields as diverse as underwater acoustics, biology, medicine, and image compression. On the other side of the coin, our knowledge has not yet reached the point of being able to provide natural-sounding synthetic voices for computers. One reason is that the computation available cannot support the physical detail required, and the necessary mathematical understanding is not complete. For example, representation of the unvoiced sounds of speech requires accurate simulation of turbulence within a simulated larynx and 24

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oral cavity. The mathematical understanding and the computational techniques for this are not yet established. Detailed application of Navier-Stokes computations of turbulence will aid this understanding. Finally, natural communication with machines (as well as large-group conferencing) benefits from hands-free sound pickup with stereo spatial reality. Autodirective microphone arrays are now possible thanks to inexpensive high-quality microphones and single-chip signal processors that can be dedicated to individual sensors. 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. 25

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