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Future Directions in Computational Mathematics,Algorithms, and Scientific Software. Report of thePanel.Society for Industrial and Applied Mathematics,Philadelphia, PA.National Science Foundation, Washington, D.C.85DMS-850348353p.Society for Industrial and Applied Mathematics, 1400Architects Building, 117 South 17th Street,Philadelphia, PA 19103-5052 (No charge while supplylasts).Viewpoints (120) -- Reports - General (140)

MF01 Plus Postage. PC Not Available from EDRS.Algorithms; Computers; *Computer Science; ComputerSoftware; *Federal Aid; Financial Support; HigherEducation; *Interdisciplinary Approach; Laboratories;*Mathematical Applications; Mathematical Models;Research Methodology

IDENTIFIERS *Computational Mathematics

ABSTRACTThe critical role of computers in scientific

advancement is described in this panel report. With the growing rangeand complexity of problems that must be solved and with demands ofnew generations of computers and computer architecture, theimportance of computational mathematics is increasing.Multidisciplinary teams are needed; these are found in most advancedand industrial laboratories, but rarely in universities. The existingeducational opportunities are not producing the required personnel tomeet substantial shortages. Therefore, the panel strongly recommendsincreased federal support for: (1) research in computationalmathematics, methods, algorithms, and software for scientificcomputing; (2) the development of interdisciplinary research teams;(3) the establishment and continued operation of a suitable researchinfrastructure for the teams; (4) graduate and post-doctoral studentsdirectly involved in the research of iome interdisciplinary team; and(5) young researchers and cross-disciplinary visitors. In the secon6section, research opportunities in a number of mathematical areas aredescribed. New modes of research are discussed next, followed bycomments on educational needs and a final section on fundingconsiderations. Appendices contain a list of related reports,information on laboratory facilities for scientific computing, andletters and position papers. (MNS)

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Report of the Panel on Future Directionsin Computational Mathematics,

Algorithms, and Scientific Software

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U.S. DEPARTMENT OF EDUCATIONOffic e or Educational Research and Improvement

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CENTER (ERIC)

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received hum the Person or organization

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"PERMISSION TO REPRODUCE THISMATERIAL IN MICROFICHE ONLYHAS BEEN GRANTED BY

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TO THE EDUCATIONAL RESOURCESINFORMATION CENTER (ERIC)."

FutureDirections in

ComputationalMathematics,Algorithms, &

Scientifk SoftwareWerner C Rheinboldt, Chairman

2

MEMBERS OF THE PANEL ON FUTUREDIRECTIONS IN COMPUTATIONAL MATHEMATICS,ALGORITHMS, AND SCIENTIFIC SOFTWARE

Werner C. Rheinboldt, Chairman

Stanley Eisenstat

Dennis Gannon

James G. Glimm

Gene H. Golub

Peter J. Huber

Mitchell Luskin

Andrew Majda

Thomas Manteuffel

Anil Nerode

Marcel F. Neuts

Andrew Odlyzko

University of Pittsburgh

Yale University

Purdue University

New York University

Stanford University

Harvard University

University of Minnesota

Princeton University

Los Alamos National Laboratory

Cornell University

University of Delaware

AT&T Bell Laboratories

REPORT OF THE PANEL ON FUTURE DIRECTIONS INCOMPUTATIONAL MATHEMATICS, ALGORITHMS, AND

SCIENTIFIC SOFTWARE

Werner C. Rheinboldt, Chairman

Supported by grant DMS-8503483 from theNational Science Foundation

Any opinions, findings, conclusions, or recommendationsexpressed in this report are those of the panel and do notnecessarily reflect the views of the National Science Foundation.

Distributed by theSociety for Industrial and Applied Mathematics

Philadelphia1985

4

Distributed by the

Society for Industrial and Applied Mathematics

For additional copies write:

Society for Industrial and Applied Mathematics1400 Architects Building

117 South 17th StreetPhiladelphia, PA 19103

CONTENTS

Executive Summary 5

1. Overview 8

2. Research Opportunities 13

3. New Modes of Research 25

4. Educational Needs 28

5. Funding Considerations 32

Appendixes 37

Appendix A: List of Related Reports

Appendix B: Laboratory Facilities for Scientific Computing

Appendix C: Letters and Position Papers

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FXECUTIVE SUMMARY

The use of modern computers in scientific and engineer-ing research and development over the last three decades hasled to the inescapable conclusion that a third branch ofscientific methodology has been created. It is now widelyacknowledged that, along with the traditional experimentaland theoretical methodologies, advanced work in all areas ofscience and technology has come to rely critically on thecomputational approach. Accordingly, for the advancement ofthe scientific and technological base of our nation, it isessential to maintain the U.S. leadership advantage inscientific computing.

In recent years, increasing concern has been voiced byvarious scientific and engineering groups and organizationsaboutihe competitive posture of the U.S. scientific comput-ing enterprise. In particular, the Panel on Large ScaleComputing in Science and Engineering El], chaired by Profes-sor P. Lax, identified the critical needs for supercomputersin science and engineering and recommended increased supportfor all related aspects of large-scale computing. During thepast year several funding agencies have responded to some of

, the recummendations of this report and of the subsequentNational Science Foundation (NSF) study [2]. These responsesinclude the Fast Algorithm initiative of the Air ForceOffice of Scientific Research, the establishment of theScientific Computing Program in the Office of the Directorof Energy Research at the Department of Energy, and the for-mation of the NSF Office of Advanced Scientific Computing.

Both of the above mentioned reports noted the need forincreased support of computational modelling and mathemat-ics. In a detailed report, entitled "Computational Modellingand Mathematics Applied to the Physical Sciences" ED, theNational Research Council (NRC) Committee on Applications ofMathematics forcafully detailed the integral and criticalrole of computational mathematics in the overall computa-tional process.

The NRC report illustrated and explained the inter-dependence of the various stages of the computational model-ing process. The committee observed that, at its best, com-putational mathematics involves the design and analysis ofmathematical models and algorithms which are fullyintegrated with considerations about their implementation asscientific software in computational models. This is partic-ularly important in the large scale scientific computingenvironment where new computer architectures are now being

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introduced. The NRC committee further recommended increasedsupport for the entire computational mathematics enterprise.

In the fall of 1984 the Panel on Future Directions in

Computational Mathematics, Algorithms, and Software wasformed to review perceived needs and to identify possiblecourses of future action. The panel's deliberations wereguided by the following realizations:

Computational mathematics is a high-leverage element ofour nation's scientific and technological effort. Itsimportance in scientific computing is increasing withthe growing range and complexity of problems that haveto be solved and with the demands of n..r generations ofcomputers and new computer architectur.

Modern research in computational mathematics increas-ingly depends on a multidisciplinary approach tran-scending the usual academic disciplines. Effective mul-tidisciplinary teams typically consist of severalscientists, engineers, and technicians who togethercover the relevant scientific and engineering discip-lines, applied mathematics, numerical analysis, statis-tics, probability theory, computer science and softwareengineering. Today, such research and engineering teamsare found in the most advanced national and industriallaboratories, but are rarely located in universities.

Although the field has developed vigorously during the

past decades, there are serious reasons for concernabout the future. There are presently substantial shor-tages of personnel on all levels and the existing edu-cational opportunities are not producing the requiredpersonnel. At the same time, low funding levels andlack of stability in the support for facilities arehindering the research effort at academic institutions.

In order to overcome these problems and to provide forthe needed strengthening of computational mathematicsresearch in this country, the panel strongly recommends the

following:

1. The appropriate federal agencies should strengthentheir support for research in computational mathemat-ics, methods, algorithms, and software for scientificcomputing.

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2. The agencies should support the development ofinterdisciplinary research teams in computationalmathematics and scientific computing.

3. Such support should permit the establishment andcontinued operation of a suitable research infra-structure for the teams, including computer hardwareand software and the associated support personnel.Moreover, the support should be of sufficient durationto insure continuity and stability of the researcheffort in the academic environment.

4. The agencies should increase the support for gradu-ate students and post-doctorals directly involved inthe research of some interdisciplinary team in computa-tional mathematics and scientific computing.

5. The agencies should increase the support for youngresearchers and cross-disciplinary visitors who cancontribute to the research work of one of the interdis-ciplinary teams.

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L. OVERVIEW

Scientific computation has become a vital component andeven a basic mode for research and development in scienceand engineering. The impact of computers upon science andtechnology has already been extremely profound. A principalreason for this certainly is attributable to the fact thatthe computer has increased and will increase enormously therange of solvable problems. The rapid progress which we seetoday is a strong sign that this impact of computers may beexpected to continue for some time to come.

A decade ago this nation was pre-eminent in all aspectsof electronics, computers, and computational technology.This leadership position is being challenged from varioussides. Some expressions of serious concern have already beenpresented in several reports. In particular, the report ofthe Lax panel Cl] (see Appendix A) observed that "U.S.leadership in super-computing is crucial for the advancementof science and technology, and therefore, for economic andnational security", but that "under current conditions thereis little likelihood that the U.S. will lead in the develop-ment and application of this new generation of machines".

Computational mathematics plays a central and perhaps.critical role in this connection. The computational solutionof today's highly complex problems of engineering and sci-ence involves questions ranging from the design of suitablecomputer architectures and the study of algorithms to themodeling of physical, chemical, biological, and engineeringprocesses by means of mathematical equations. These issuesare tied together by mathematical theory, which seeks a fullunderstanding of the nonlinear phenomena contained in themodels, and by numerical mathematics which conceives, ana-lyses, and tests the required computational methods. In acomplementary and equally essential role, mathematics is thescience of algebraic and logical manipulation which, inturn, represents the basis of such widely differing opera-tions as the control of complex decisions through artificialintelligence, the design and encoding of large data sets, orthe exact solution of scientific problems of an algebraicnature. The span of this wide range of activities forms thesubject that has become known as computational mathematics.

Today's computer technology is dominated, on the onehand, by the introduction of micro-computers and theirwidespread acceptance, and, on the other hand, by thedevelopment of supercomputers with architectures thatrepresent a major departure from traditional sequentialmachines. These supercomputers and other machines advancingthe state of the art (e.g. in graphics or symbol manipula-tion), rather than the micros, are essential in areas of

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frontier science. In fact, it has been frequently observedthat the most challenging problems always require moreresources than can be provided by the fastest available com-puter. The reasons that these problems are under-computed isthat each advance in computing-power or algorithm designopens up new possibilities for further research. Just as theincreased resolution of the optical and electron microscoperesulted in the discovery of whole new realms of science, sosuccessive generations of computing hardware and softwarehave led and will lead to fundamental advances in scienceand technology.

Here, more than ever, computational mathematics is ofcrucial importance. The effective utilization of super-computers and, in particular, of the planned highly parallelarchitectures, is far from being understood. There is acritical need for developing new -- and re-examining oldalgorithms which will take full advantage of all the powerof these computers. New advances in numerical mathematics,algorithm design, and computer languages, will be essentialfor learning to exploit the new facilities. It is expectedthat new research contributions of the community of computa-tional mathematicians and computer scientists to these fun-damental issues will be at least as important in thedevelopment of new generations of machines as the newhardware developments themselves.

These observations certainly support the conclusionthat computational mathematics is a high-leverage element ofour nation's scientific and technological effort. The fieldhas developed vigorously during the past few decades. Butthe current vitality and level of achievement notwithstand-ing, there are reasons to be concerned about the future.There exists at present a considerable shortage of personnelon all levels in the area of computational mathematics.Moreover, the educational opportunities in the field arelimited and do not produce the required people, especiallythe much needed young researchers. At the same time, thecomputational mathematics research effort in academic insti-tutions is seriously hindered by low funding levels, lack ofstability in the support for facilities and support person-nel, and limited access to up-to-date computer equipment.These problems have already been noted by several previousreports. For instance, the NRC report £3] recommended thefollowing:

1. Increased research support for computational model-ing and applied mathematics.

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2. Increased support for computing facilities dedicatedto computational modeling and applied mathematics.

3. Increased support for education and manpowerdevelopment in computational and applied mathematics.

In part, some of these problems reflect those of themathematics community as a whole. In fact, as the Davidreport [4] documents, "(s)ince the late 1960s, support formathematical sciences research in the United States hasdeclined substantially in constant dollars, and has come tobe markedly out of balance with support for related scien-tific and technological efforts". Moreover, "(o)ver thehistorical period 1968-82 .. the annual number of U.S.citizens obtaining doctorates in the mathematical sciencesfrom U.S. institutions has been cut in half, from over 1,000to fewer than 500". "Overall demand for Ph.D.'s exceeds sup-ply".

Computational mathematics has fared somewhat betterwith respect to federal research support than mathematics asa whole. But nevertheless this support lagged far behind thegrowth of the field. Moreover, with respect to manpowerresources, the field clearly is in worse shape than mostother parts of mathematics. The David report observes inthis connection: "Particularly worrisome are the scarcity ofsenior personnel in this area and the extremely small numberof young researchers and graduate students". "A major effortin this area is needed to attract, educate, and support gra-duate students, postdoctorals, and young researchers, and toprovide the computational equipment essential for the properconduct of this research". "Significant additional resourcesfor the mathematics of computation may be needed in theyears ahead".

In order to meet the immediate need for making super-computers available to academic scientists and engineers,the National Science Foundation in April 1984 announced theformation of the Office of Advanced Scientific Computing.The office was formed to: (1) increase the access toadvanced computing resources for the scientific andengineering research community, (2) promote cooperation andsharing of computational resources among all members of thecommunity, and (3) develop an integrated research communitywith respect to computing resources. This is certainly amost important step toward overcoming some of the criticalproblems in scientific computing in the country.

But, in line with our earlier observations, any suchprogram of expanded access to supercomputers can only be a

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first step. Its benefits to the country will remain severelylimited unless there is the vigorous, accompanying programof academic research and education on computationalmathematics, software, and algorithms necessary for theeffective and efficient use of such large computer systems.

The panel does not find it sufficient to consider sim-ply an increase in the funding level for support of the workof individual researchers in computational mathematics. Thesolution of today's computing research problems in scienceand technology requires an interdisciplinary approachinvolving contributions from various areas of science,engineering, applied and numerical mathematics, statistics,and computer science. Moreover, such work cannot proceedwithout appropriate computer hardware and software facili-ties and the associated support personnel. Indeed it willnot be very effective unless there is a certain continuityand stability in this infra-structure which supports theresearch.

This calls for an increased emghasis on the support ofteams which have a common intellectual focus but are suffi-ciently interdisciplinary to span a significant portion ofthe broad range from an application area through applied andnumerical mathematics to scientific software and computerscience. In order to assure the required stability and con-tinuity of the research effort, such support has to havereasonably long duration. It also should include funds forthe initial acquisition of the necessary hardware andsoftware and for the subsequent operation and maintenance ofthese facilities. In this connection it is also importantto foster more cooperation and coordination betweenresearchers who often work on the same or similar problems.There is a great need for an extensive electronic networklinking groups in mathematics, computer science, and relatedareas. Such a network would lead to greater standardizationof software and less duplication of effort.

The proposed emphasis on group efforts does have theadded benefit of optimal utilization of the limited numberof active researchers in computational mathematics. More-over, such groups provide an excellent training ground forgraduate students and post-doctoral fellow.ci. This would helpin alleviating some of the manpower problems, although itwould not remove the need for a more concerted program ofstrengthening the educational programs in the mathematicalsciences in general and in computational mathematics in par-ticular. A broad plan for educational support in mathematicshas been presented in the David-report. In computationalmathematics there appears to be some added urgency forincreasing the level of support of graduate students andpost-doctorals, and to study effective modes of

13

undergraduate education which capture the experimental andlaboratory components of the field.

On the basis of its findings the panel strongly recom-mends the following actions:

1. The appropriate federal agencies should strengthentheir support for research in computational, methods,algorithms, and software for scientific computing.

2. The agencies should support the development ofinterdisciplinary research teams in computationalmathematics and scientific computing.

3. Such support should permit the establishment andcontinued operation of a suitable research infra-structure for the teams, im:luding computer hardwareand software and the associated support personnel.Moreover, the support should be of sufficient durationto ensure continuity and stability of the researcheffort in the academic environment.

4. The agencies should increase the support for gradu-ate students and post-doctorals directly involved inthe research of some interdisciplinary.team in computa-tional mathematics and scientific computing.

4. The agencies should increase the support for youngresearchers and cross-disciplinary visitors who cancontribute to the research work of one of the inter-disciplinary teams.

2. RESEARCH OPPORTUNITIES

2.1 INTRODUCTION: In this section we discuss severalresearch areas and directions in computational mathematicsand examine how results in this field have a high leverageeffect on broad areas of science and technology. The exam-ples chosen are illustrative only and are not intended to becomplete. Moreover, all the reports listed in Appendix Apresent such examples and there is certainly no need forduplication. Instead we endeavored to stress topics thatwere not covered very extensively in the cited reports.

In the last two decades many developments in nurericalanalysis have had a large impact on scientific computations.As examples we mention here only the fast Fourier t-ansform,the evolution of the finite element methods, variablestep/order ordinary differential equation solvers, adaptivemesh-refinements, spline approximations, spectral methods,fast matrix algorithms including, for instance, multigridtechniques, and effective optimization methods. But thereare a host of problems waiting to be understood. Perhaps thegreatest opportunity provided by the modern computationalapproach is that it has opened the wide realm of stronglynonlinear phenomena to systematic, relatively accurate, andefficient modeling, improving the chance that importantphenomena can be isolated and analyzed. This represents oneof the great ciallenges to numerical anEL2ysis today.

2.2 NONLINEAR ELLIPTIC EQUATIONS: The solution of coupledsystems of nonlinear elliptic partial differential equationsis important in many applications. These systems arise insteady state problems and as steady state subsystems orimplicit solution steps within a dynamical problem. Manyefficient discretization and solution methods have beendeveloped over the past thirty years by computationalmathematicians.

The solution of coupled nonlinear partial differentialequations plays an especially important role in semiconduc-tor simulation. It is now possible to analyze many differentdesigns without building a prototype. The numerical problemsencountered here, however, require computing equipment ofthe most modern type. Undoubtedly, more powerful computerswill stimulate the use of more complex models, especiallyfor three-dimensional problems.

The selection of suitable discretization procedures is

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influenced by various considerations. These include thegoals of the analysis of the physical problem, knownmathematical properties of that problem and of the solutionalgorithm, hardware considerations, such as parallelity,data management requirements and other computer science con-siderations, and, last but not least, restrictions on compu-tation time and expense. For any choice of discretizationthere are questions about the accuracy and stability of themethod, the efficient solvability of the resulting equa-tions, the robustness of the method with respect to theinput data, and, for nonlinear problems, the separation oflegitimate approximate solutions from spurious approximatesolutions, which do not correspond to any exact solution.Much important work on the various discretization methodsremains to be done, especially when it comes to adaptiveforms of these methods.

2.3 LINEAR AND NON-LINEAR EQUATIONS: The mentioneddiscretizations typically give rise to systems of linearequations. In particular, large sparse systems with tens orhundreds of thousands equations are relatively common, andthe iterative solution of nonlinear problems usuallyinvolves such a linear system at each step. Much researchhas gone into the development of efficient methods for thesolution of systems of linear equations. This includesdirect solution techniques, such as the various forms ofmatrix decompositions, banded and frontal elimination tech-niques, nested dissection, and different approaches tosparse elimination, such %- the minimal degree algorithn,etc. We also refer to itera . processes, such as the suc-cesive overrelaxation metho.4., the alternating directionmethod, the conjugate gradient method, as well as the hybridmethods such as block and multi-grid methods. The study ofefficient algorithms for solving large systems of linearequations remains a very active field of research. In par-ticular, interest centers on fast methods for non-symmetricsystems, the ievelopment of methods for computers with dif-ferent ardhitectures, especially parallel machines, and thedevelopment of methods which are effective for problemsinvolving singularities. Moreover, partitioning of largeproblems has just begun to be studied and represents an areaof considerable challenge.

There are various classes of solution algorithms whichare of similar importance. These include, for example,methods for solving linear eigenvalue problems, and optimi-zation methods for linear and nonlinear problems. Forinstance, large scale linear least squares problems comefrom a variety of sources. A particularly challenging set ofsuch problems arises in geodetic computations. Here the data

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often are obtained from remote sensing devices and verylarge data sets are acquired. Fortunately, the nature ofthese problems leads to matrices which are highly structuredand for which some of the modern iterative techniques areapplicable. Other classes of such problems cannot be handledeven with the most modern computing devices and in thesecases new algorithms are required which take account of thefine structure of the matrices.

The optimal power flow (OPF) problem is an example ofan important application whose solution has become practicalonly because of improved optimization methods. Broadlyspeaking, the goal is to optimize the distribution ofelectrical power over a network; the constraints are non-linear, and the number of variables in any realistic case isvery large. The OPF problem occurs in many areas of powerengineering and power engineers have been working on it forat least 20 years. The methods used in the 1960's -- whenthe computer solution of the OPF problem was first attempted-- were slow and unreliable. Because of inadequacies in themethods, it was widely considered that large OPF problemswere essentially impossible to solve. For example, theobjective function was believed to be almost "flat", withmany local optima, simply because older methods usuallyfailed to make any significant progress after the first fewiterations. Within the last two years, however, sequentialquadratic programming methods have been applied with greatsuccess to general OPF problems of a size previously con-sidered intractable. In fact, very large OPF problems cannow be solved routinely. This breakthrough received consid-erable attention in the power industry after the resultswere reported at a national power engineering conference inFebruary 1984.

Despite such success stories, improved computationalpower is still necessary if certain problems are to besolved effectively. For example, stochastic linear programshave been the subject of intense study since World War II.The "natural" approach to such problems would be to explorethe largest possible number of alternatives at each step ofthe solution process. However, the inherent complexity ofthis approach has meant that less satisfactory methods basedon averaging had to be used instead. With the advent of newcomputer architectures involving substantial parallelism, itappears to be likely that, for the first time, stochasticproblems can be treated in their "natural" most desirableway.

In the case of nonlinear equilibrium problems we usu-ally encounter a number of parameters. Then the solution setforms a manifold in the combined state and parameter space,and for a deeper understanding of the system it is necessary

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to analyze the shape of this manifold. Here continuationmethods are to be used. Moreover, questions about the sta-bility of the solutions typically involve the determinationof the location and character of the singular points of thissolution set. Another important but as yet incompletelyexplored question concerns the comparison of the solutionmanifold of the original equations with the manifold definedby a discretized form of these equations.

The study of singular points on the mentioned solutionmanifolds constitutes an aspect of bifurcation theory -- thestatic case. The dynamic case is of equal importance inmodern computational mathematics. In fact, it has beenremarked that one of the most engaging problems in nonlineardynamics is that of understanding how simple deterministicequations can yield apparently random solutions. Both staticand dynamic bifurcation problems involve challengingmathematical and computational questions and representactive areas of research.

2.4 NONLINEAR HYPERBOLIC EQUATIONS: Nonlinear hyperbolicequations are equally important for a number of applicationsincluding, for instance, high speed gas dynamics. Theseproblems are often characterized by the simultaneous pres-ence in their solution of significantly different time andlength scales. The solutions to these models will haveregions of strongly localized behavior, such as shocks,steep fronts, or other near discontinuities. In recentyears, a series of improved methods for these problems havebeen introduced by computational mathematicians, culminatingin currently very refined agreement between computation andexperiment for complex shock wave diffraction patterns. Werefer here to upwind methods, flux limiters, higher orderGodunov methods and methods based on solutions of Riemannproblems. Areas of research important in coming yearsinclude the extension of such highly accurate solution tech-niques to more difficult problems, including those arisingin reactive chemistry and combustion, and especially also toproblems in three space dimensions.

2.5 ADAPTIVE MESH METHODS: In these problems, as well as inthe earlier mentioned elliptic boundary value problems,adaptive mesh construction has come to be of critical impor-tance. Such techniques allow extra computational effort tobe concentrated in regions where the solutions are singularor nearly so. The design of efficient adaptive mesh refine-ment methods is an active research area. An importantrelated research topic is the development and application of

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a posteriori error estimates. Such estimates can be used toassess the errors of the computed results and to form thebasis of adaptive procedures. The requirement of designingerror estimators with appropriate properties for realisticclasses of problems, different solution methods, and suit-able error norms certainly represents a demanding researchtask in computational mathematics.

Often local mesh refinement leads to nested levels ofmeshes which can be solved effectively on parallel computersas independent tasks. This makes this topic also very impor-tant for use in connection with modern computer architec-tures. For situations in which the localized behavior isknown to be approximated well by very sharp fronts (e.g.shocks or flames), front tracking methods often have signi-ficant advantages. Such methods, as well as related vortexmethods, insert analytic information concerning solutionsingularities into the numerical algorithm.

2.6 ILL-POSED PROBLEMS: The notion of a well-posed problemis due to Hadamard: a solution must exist, be unique, anddepend continuously on the data. The term "data" can have avariety of meanings; in a differential equation it mayinclude, for instance, the boundary or initial values, theforcing terms, or even the coefficients of the equation.Since data cannot be known or measured with arbitrary preci-sion, it was thought for a long time that real physicalphenomena had to be modeled by mathematically well-posedproblems. This attitude has changed considerably in recentyears, and it is now recognized that many important appliedproblems require the solution of less well-posed problems.

One important class of such applications falls, looselyspeaking, under the heading of tomography. In a narrowsense, tomography is the problem of reconstructing the inte-rior of an object by passing radiation through it andrecording the resulting intensity over a range of direc-tions. Tomography has also come to include applicationswhere the source of radiation is inside the object or whenradiation is reflected by features inside the object.

Probably the most widely known applications of tomogra-phy are in medicine. Computer assisted tomography (CAT scan)uses x-rays directed from a range of directions torecosntruct the density function in a thin slice of thebody. Recent advances in medical tomography include nuclearmagnetic resonance (NMR), where strong magnetic fields areused to make the hydrogen atoms resonate. By varying thefields and their direction the plane integrals of the den-sity of hydrogen can be measured and an "approximate" two or

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three dimensional hydrogen density can be reconstructed. Oneadvantage over the CAT scan is that the use of potentiallyharmful x-rays can be avoided.

Another important source of applications of tomographyis non-destructive evaluation (NDE). There is considerableneed in industry to evaluate the integrity and remainingreliable lifetime of components and structures, as, forexample, nuclear reactor vessels, bridge girders, or turbinedisks. Once again the component is subjected to penetratingradiation with the aim of deducing information about itsinternal state. Types of radiation used include ultrasound,x-rays, and neutrons. In other applications, electricalcurrents are induced in the material which produce fieldsthat may allow the determination of existing cracks.

The search for oil depends heavily upon the analysis ofseismic data. This is another example of the reconstructionof internal features of a body from monitored reflections ofradiation or energy flows. Another example occurs in thestudy of confined plasmas in reverse-pinch machines that arebeing studied as a possible source of fusion energy. Radia-tion is reflected off the plasma in an attempt to determinethe cross-sectional density function. In this example only ahandful of data are available. In an even more compellingexample, a suggested method of on-site monitoring of a lim-ited test ban treaty depends upon the ability to resolve aproblem of this type.

The idealized mathematical problem underlying all theseproblems is the reconstruction of a function f(x,y) of twovariables from its integral along lines. This is aninteresting example of a mathematical problem that was con-sidered and solved long before its applicability was seen.In fact, this problem, as well as its three-dimensional ver-sion, was solved by J. Radon in 1917 and later rediscoveredin various settings such as probability theory (recovering aprobability distribution from its marginal distributions)and astronomy (determining the velocity distribution ofstars from the distribution of radial velocities in variousdirections). Of course, much work was needed to adapt theRadon inversion formula to the incomplete information avail-able in practice. The computational solution of ill-posedproblems of the form arising in the general area of tomogra-phy is a very active research topic in computationalmathematics. Recent years have brought much progress in thisfield, but much work remains to be done. Because of theoverwhelming importance of the applications that lead tothese problems, this area of research may be expected toyield many important dividends.

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2.7 COMPUTATIONAL STATISTICS: As in mathematical physics andengineering, methodological breakthroughs in statistics tendto be a combination of many factors. For example, the majorbreakthrough in numerical spectrum analysis in the 1960'sinvolved the theory of stationary stochastic processes, thefast Fourier transform, statistical sampling theory, and,behind all that, driving examples from geophysics. Statisti-cal research is now poised for some major methodologicaladvances through access to improved computing resources ofvarious types.

An increase in computing power makes it feasible toapproach problems that are intractable in closed form. Thisincludes, for example, the establishment of properties ofstatistical procedures through Monte Carlo simulations (as,for instance, in Efron's bootstrap methods), and the solu-tion of problems in Bayesian statistics through the directcomputation of multi-dimensional integrals. On a more basiclevel, it now becomes possible to attack concrete statisti-cal tasks through direct numerical optimization. We notehere that a majority of statistical problems are naturallyformulated in terms of optimality principles (such as theclassical maximum likelihood estimates, optimal designs,robust alternatives to least squares, projection pursuitmethods, etc.), but that they rarely have closed form, oreven easily calculable solutions. Often, the function to beoptimized is not available in closed form and must be foundby integration and Monte Carlo simulation, or the optimiza-tion itself occurs inside a simulation loop. Frequently, notonly the location of the optimum is needed, but also contoursurfaces around the optimum (for confidence regions); thiscertainly compounds the computational problems. The researchchallenge here is to develop the methodology in a safe andefficient manner.

Large data sets represent a central problem for compu-tational statistics. Here the computational work is typi-cally storage bound rather than processor bound. The longrange research challenge is to identify basic buildingblocks (both in hardware and software) that are suitable formassively parallel data base operations, and to build toolsfor analyzing large, highly structured, inhomogeneous datasets. Research opportunities in this area are wide open; infact, none of the currently available data base managementsystems is designed to deal with the specific requirementsof scientific/statistical data.

Statistical data analysis requires software andhardware suitable for interactive, often graphical, improvi-sation. The computing requirements here are similar to thosein artificial intelligence work and include an integratedprogramming environment with instant, high resolution,

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easily improvisable, real-time graphics, and a mixture ofsymbolic and numerical manipulation capabilities. Theserequirements are beginning to be met by the single-usersuper-workstations now reaching the market (with a raw powerexceeding that of a VAX-ll/780). There are major challengesfor the computational statistician; we need to developbetter interactive languages for statistical data analysis,and we must design and implement effective methods that pro-vide the required intelligent machine assistance to theanalyst.

2.8 STOCHASTICS AND COMPUTATIONAL PROBABILITY: Stochasticprocesses and the need for probabilistic reasoning arise inthe solution of numerous problems. These range from queuingand inventory studies to the fault tree analysis of safetydesign factors, and from economic and financial studies tothe analysis of voting behavior. Modern applications of pro-bability theory involve problems of very high dimensionwhich present massive computational problems. The analysisof the related probabilistic models has called for newmethodological approaches. Among recent advances are theproduct-form solution used in the analysis of networks ofinteracting queues, and the matrix-analytic methods ofstructured Markov chains. The product-form solution hasstimulated extensive research on stochastic networks and iswidely used in the design of interconnected computer sys-tems. Structured Markov chains arise commonly in telecom-munication engineering in the design of buffers and in thestudy of operating protocols. The computation of variousmeasures of performance leads to the numerical solution ofvarious types of nonlinear matrix-integrals and matrix-equations.

As noted earlier, most probability models lead to veryhigh dimensional computational problems. Often already verysimple descriptions require the study of a higher-dimensional stochastic process. For example, a typicaleconometric model of a country or group of countries may becharacterized as a stochastic nonlinear system of hundredsor thousands of equations. They often involve some non-concavity in the system, and hence, already the computata-tional problems in analyzing the dynamic properties of thesystem are formidable. Moreover, the process of search forapproximate optimal control strategies for policy authori-ties, especially when more than one active authority isinvolved, introduces game theoretical aspects. The resultingsystem will tax most of the available computing resources.As this example already indicates, progress in this arearequires not only statistical methods of a nonclassicaltype, but also new approaches in nonlinear numerical

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analysis, and new techniques for data handling and computergraphics.

2.9 PAITERN RECOGNITION AND SYMBOLIC COMPUTATIONS: The areaof pattern recognition, including image processing, acousti-cal processing and speech recognition, requires integratedmathematical techniques from several areas including signalprocessing, mathematical logic and linguistics, numericalmathematics, Fourier transforms, and algebra. The objectiveis a classification of sequences of images as they mightarise in such widely differing applications as text readingor submarine detection. In the case of image reconstruc-tions, random disturbances may blur or obliterate atransmitted image. One recently proposed method for themathematical reconstruction of such images involves the useof stochastic dynamics based on Ising models. A dynamicalevolution to an image with a lower Ising model energyfilters out much of the noise while leaving the originalimage unchanged.

Numerical and non-numerical computing are oftenregarded as two different areas which use their ownsoftware, interfaces, and algorithms even when sharing thesame hardware environments. An important form of non-numerical computation involves the use of symbolic manipula-tion systems for carrying out algebraic operations on acomputer. These systems perform indefinite integration, fac-tor polynomials, simplify complex expressions, and operateon matrices. They have been applied very successfully inseveral situations, for instance in the computation oforbits of space objects, and in the calculation of Feynmanintegrals in physics. In each of these cases the requiredalgebraic computations were so extensive that only computerscould perform them sufficiently rapidly and accurately. Thedevelopment of computer algebra systems required the inven-tion of numerous new algorithms and the adaptation of vari-ous old ones. Some of the most notable ones are those forindefinite integration and polynomial factorization. Furtherwork is needed on algorithms that are more efficient thanpresently known ones and on algorithms than can deal withnew classes of problems. There is also need for newcomputer-algebra systems that are more portable than presentones, run on smaller machines, and can be interfaced moreeasily with other programs. Moreover, there is a strongneed for studying more closely the coupling of automaticreasoning algorithms with mathematical modeling and database techniques to produce effective control processes forcomplex systems and for effective modeling of complex sys-tems involving human and mechanical components.

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2.10 COMPUTATIONAL METHODS IN PURE MATHEMATICS: Some of themost interesting applications of symbolic mathematics are inmathematics itself. Areas of both pure and applied mathemat-ics, including coding theory, cryptography, probabilitytheory, analysis, combinatorics, and number theory, have allgained from the availability of symbolic manipulation tools.These tools have been used to prove a number of resultsdirectly. Their main application, however, has been toobtain insight into behavior of various mathematicalobjects, which then led to conventional proofs. This rathernew, direct impact of computation upon pure mathematics isnot limited to symbolic manipulation and will no doubtexpand considerably in coming years. In doing so it may leadto the development of computational tools, methods, and con-cepts of broad applicability. For instance, the four colorproblem was reduced to a check of a very large but finitenumber of specific cases. This final step, too tedious forhumans, was performed on a computer. Computers have beenused in exploratory studies of the iterates of mappingswhich in turn led to new insight into the nature of chaosand turbulence. They also played a role in the classifica-tion of all finite simple groups. Finite groups certainlyplay a very important role in the study of many topics inapplied mathematics and science, such as discrete computa-tional structures, enumeration of combinatorial designs, thedefinition of stereo-isomers in organic chemistry, thecataloguing of crystallographic structures, etc.

The recent invention by Lovasz of a fast algorithm forfinding good bases for lattices has had a striking impact onseveral fields. The problem of finding the shortest non-zero vector in a high-dimensional lattice is considered tobe very hard, but has been of great interest because manyother problems in diverse areas can be reduced to it. TheLovasz algorithm does not in general find the shortest non-zero vector, but it does rapidly find a relatively shortone, and in many situations that is sufficient. The algo-rithms has been applied, for example, to the determinationof integer solutions of linear programs and to the factori-zation of polynomials. But perhaps its most important appli-cation is to cryptography. Among public key secrecy ssystems(that is, those that do not require the two communicatingusers to exchange secret keys beforehand) there was a largeclass for which the presumed security relied on the diffi-culty of solving the knapsack problem. Essentially all ofthese knapsack-based secrecy systems have been shown to bebreakable by application of the lattice-basis reductionalgorithm. As a result the only public key cryptosystemsthat still appear credible are those for which securitydepends on the difficulty of factoring large integers orcomputing logarithms in finite fields.

2.11 QUALITY SOFTWARE: Quality software is an indispensibletool in many modern science and engineering projects. Thecontinuing developments in micro-electronics will decreasethe cost of computing and increase the speed and memoryavailable. But these hardware improvements will allowpresent codes to be pushed only a limited amount beyondtheir current capabilities. Additional capabilities mustcome from better mathematical models and more efficientnumerical methods. Because computer architectures will cer-tainly change, there is a need to re-examine traditionalmethods and to develop new methods that use the inherentstructure of the new machines. But, at the same time, itwill be important to avoid locking the software into a par-ticular architecture. Software flexibility, modularity,reliability, efficiency, restricted data flow and documenta-tion are here fundamental requirements. There is certainly aneed for the development of high quality mathematicallibraries to support large-scale scientific computing. Thiswill require close cooperative work between applicationsprogrammers, numerical analysts, and the scientific comput-ing research community.

2.12 PARALLEL COMPUTING: Despite the fact that electroniccomponents are becoming faster, the limits of raw machinespeed are now clearly visible. Further gains will eventuallyhave to be made by the use of novel architectures and algo-rithms, and, in particular, by some form of parallelism.Many of the new designs now under consideration represent asharp break from previous computer architectures and intro-duce issues of scheduling, coordination, and communicationwhich previously did not arise. Thus it will be necessary torethink and reconstruct the vast number of familiar numeri-cal methods from the point of view of these new issues. Itappears that in many cases software, not hardware, will posethe most difficult problems and will account for the mostimpressive advances.

As an illustration we discuss here some issues relatingto parallel computing. Recent progress toward solving largeproblems, such as full body air flow simulations, has con-centrated on the use of local mesh refinements and adaptivemethods. Data structure design is evolving toward the pro-duction use of networks of overlapping grids some of whichare coarse and stationary while others are fine grids whichmay be moving to track a front. For the solution of ellipticproblems the algorithms that show the greatest promise aremultilevel methods such as hybrids of multigrid methods andmethods that are well suited to substructured grids, such asthe conjugate gradient method. From the point of view ofsoftware design there are two important features of these

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25"

techniques. First, the algorithm and data structure aredynamic. Second, while the global grid structure is verynon-uniform, the local structure is based on deformations ofuniform grids. These two facts have a large impact on theproblems of programming the next generation of computers. Inparticular, we face three fundamental research problems:

(I) Numerical algorithms: We must understand thedynamic structure and formulate the mathematical model interms of high level parallel operations. For example, thecomputation may be viewed and expressed as a system ofhigher order tasks which encapsulate the large scaledynamic/adaptive structure of the algorithm. Each higherorder operator can be composed of lower order paralleloperations on uniform fine structures such as subgrids of alarge network. The point is that no matter how the algorithmis formulated, we will need to employ much more parallelismthan we currently do in order to utilize the power of a mas-sively parallel machine.

(2) Algorithm analysis: A more accurate model of paral-lel algorithms is needed to describe the expected perfor-mance of the new .7,oftware. In particular, parallel algorithmanalysis must capture the cost of communication, processsynchronization, and initiation. An ideal model would bemachine independent but have parameters that are easily setto a specific architecture. Such a model also could be ofgreat value in the design of new architectures.

(3) The mapping problem: Given a parallel architectureand a parallel specification of an algorithm, find anoptimal mapping from one to the other. This problem takes onspecial significance in the case of massively parallel non-shared memory systems. In this case, the machine can beviewed as a graph where processors are nodes and the commun-ication network describes the edges between the nodes. Thealgoithm is a graph of communicating processes (or data flowgraph) which, in the case of adaptive methods, has a dynamicstructure.

Clearly, as this brief sketch already shows, researchin this important area of computational mathematics willrequire well integrated team efforts between algorithmdesigners, numerical analysts, and researchers in the appli-cations areas.

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3. NEW MODES OF RESEARCH

Traditionally, individual researchers working alone orin pairs have characterized the style of much of the workin the mathematical sciences. This situation is different incomputational mathematics where increasingly a multidisci-plinary team approach is required. There are several compel-ling reasons for this.

First and foremost, as noted already several times inthe previous section, problems in modern scientific comput-ing transcend the boundaries of a single discipline. In gen-eral, the computational approach has made science moreinterdisciplinary than ever before. There is a unity amongthe various steps of the overall modeling process from theformulation of a scientific or engineering problem to theconstruction of appropriate mathematical models, the designof suitable numerical methods, their computational implemen-tation, and, last but not least, the validation andinterpretation of the computed results. For most of today'scomplex scientific or technological computing problems ateam approach is required involving scientists, engineers,applied and numerical mathematicians, statisticians, andcomputer scientists.

Unlike theoretical mathematics, computational mathemat-ics, by its very own nature, has a strong experimental com-ponent. As a result, research work proceeds in part in alaboratory mode similar to that in the experimental sci-ences. The laboratory equipment required for modern scien-tific computing ranges from local workstations, micro- andmini-computers, to mainframe machines of various sizes andsuper-computers. This hardware is complemented by appropri-ate software systems and libraries. Moreover, the multi-disciplinary nature of the work requires a new level ofinterchange between researchers in widely diffurent fields.For this the communications capabilities of mode n computernetworks form an ideal mechanism to link scientists t.romwidely differing fields and to allow them to share ideas andinformation. As in the traditional laboratory sciences, sup-port personnel is required to ensure efficient usage of allfacilities and to avoid any unnecessary waste of researchtime. Under current fundin;; practices projects involvingsupport staff are certainly not very c,..amon in mathematics.

Clearly, the investment costs, as well as the longerduration of typical computational projects especiallywhen extensive software developmev is involved -- necessi-tate a certain continuity and stability of the entireresearch infra-structure and hence calls for a sharing ofthe facilities among several researchers. Once againresearch teams are best suited to sustain such an

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operation.

As has been noted before, computational mathematicscurrently faces a critical manpower problem. Any nationaleffort to strengthen the field has to take special accountof this fact and has to include steps toward making optimaluse of existing manpower resources. In the opinion of thepanel, the most advantageous approach will be to ensure theestablishment of a sufficient number of viable researchteams in computational mathematics. Such groups are not onlymost appropriate for research in the field, but they ensurethat the most effective use is made of the scarce talents ofestablished researchers, and they provide a means of bring-ing additional young researchers as quickly as possible intothe mainstream of the work.

Generally, the emphasis should be on a concentration ofeffort. In other words, teams should have a common intel-lectual focus yet be interdisciplinary, spanning a signifi-cant portion -- and ideally all -- of the broad range froman application area through applied mathematics and numeri-cal analysis to mathematical software and computer science.This concentration of effort on a common research interestwill allow the group to take a problem all the way from theapplication to working software. This commonality and rangeof research interest is considerably more important than aparticular administrative structure, such as a center orinstitute. Of course, for vitality of such an organizationalunit it may be desirable to ensure a certain stability in aparticular academic setting, but it should not be the soledriving force.

In order to be viable, it appears that a group shouldconsist of at least three faculty level researchers, onepost-doctoral fellow, five to eight Ph.D. level graduatestudents and one full-time support person. In most cases itwill be highly desirable 1-,1 include additional juniorfaculty level researchers in buch teams. The support personwould do both systems and applications programming for thegroup, and generally ensure smooth operation of the localcomputing facilities. While some groups might survivewithout such a person, they would do so only at the expenseof researdhers performing these tasks which is a waste ofvaluable and scarce talent.

As noted before, each group will require some localcomputing facilities, but access to supercomputers can beprovided by suitable high-speed networks. The quality of thecomputing facilities has a decisive effect on the overallresearch effort. It is highly desirable that the group hasat least high-performance personal workstations, to supportthe research work of the individual members, to provide for

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efficient access to appropriate large computing facilities,and to allow for very-high bandwidth communication, such asneeded in graphical post-processing. If the group has tooperate more substantial computing facilities, then thereshould be at least two support people. But in all this, itis important to note that the requirements described hereare minimal rather than average. Some further discussion ofpossible laboratory facilities is presented in Appendix Bbelow.

In order to assure the required stability and con-tinuity of the research effort, funding should be providedon a multi-year basis, say, on the order of five years.Moreover, it is necessary to support both the initial pur-chase and the subsequent annual maintenance of the hardwareand software. The overall funding for such a group need notbe in the form of one large institutional grant, but maywell be in the form of several grants and contracts fromfrom different funding sources. Tn some cases, it may alsobe appropriate to establish (.:ooperation with industrialresearch groups and to secw:- some support from non-governmental sources.

This multidisciplinary team work in a laboratory set-ting represents a significant departure from the traditionalstyle of research in mathematics. It is also a departurefrom the standard funding mode of the federal agencies sup-porting computational mathematics. It will take specialefforts on the part of these agencies and the mathematicalcommunity to accept the changing situation and to help inthe development of this new mode of research in computa-tional mathematics. In line with this, it is of considerableimportance that the proposed new funding effort for theestablishment of such research teams is meant to complementrather than replace the traditional project support forindividual researchers in the field. The proposed teamapproach is intended to focus on different categories ofscientific computing than are feasible for individualresearchers and to strengthen the productivity and effec-tiveness of the researchers involved in such interdisci-plinary projects.

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4. EDUCATIONAL NEEDS

The Lax report CA (see Appendix A) recommended along-term National Program on Large-Scale Computing whichincluded as one of its goals the "(t)raining of personnel inscientific and engineering computing", and in [3] this wasre-iterated in the form of a recommendation for "(i)ncreasedsupport for education and manpower development in computa-tional and applied mathematics".

Basically, the educational needs addressed here arepart of a much broader problem concerning all of mathemati-cal education in the country. Throughout the past two yearsseveral pieces of legislation were introduced in theCongress in response to the report of the National Commis-sion on Excellence in Education entitled "A Nation at Risk".Then, in August 1984 the President signed into law the"Emergency Mathematics and Science Education and Jobs Act".

The David report EU addressed some of the educationalproblems in mathematics on the post-secondary level. ItsAindamental observation was that "(t)he field is not renew-ng itself".

There is little question that there are formidableproblems with our undergraduate and graduate educationalprograms in all areas of mathematics; that is, in puremathematics, computational and applied mathematics, statis-tics, operations research, etc. The quickly broadening needfor computational modeling in all areas of science,engineering, and business has produced an increasing demandfor more college-level mathematical education. This is evi-dent by the 33% increase of enrollments in the mathematicalsciences at four year institutions during the period 1975-80. But during the same period the number of facultymembers increased only by 8%. At the same time, mathematicsis failing to secure its share of qualified young people.When compared with 1970, the number of undergraduate degreesin the mathematical sciences has decreased by 40%, and thenumber of Ph.D.degrees has been dropping monotonicallysince 1970. Moreover, the percentage of foreign recipientsof the Ph.D. in mathematics has now increased to about 40%of the total. As the David report observes a "gap has beencreated between demand for faculty and supply of newPh.D's. It may well widen as retirements increase in the1990's".

The reasons for the difficulties in the mathematicalsciences are complex. A strongly contributing factor is cer-tainly to be found in the drastic decreases in federal fund-ing for mathematics during the past fifteen years as docu-mented in EU. Another factor is the earlier noted and

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widely discussed range of problems with mathematics educa-tion at the elementary and high school level. At the sametime, the typical undergraduate mathematics curricula arelargely inflexible and have not responded well to the chal-lenges introduced by the fact that today much of thestudent-interest centers in areas of applied mathematics,computational modeling and scientific computing. As aresult, talented young people often turn away disappointedlyfrom the field.

The problems are even worse in computational mathemat-ics. The rapid growth of the field has largely outstrippedthe. available educational opportunities. There is a profoundlack of senior faculty members in the field and a paucity ofgraduate students and young researchers. At the same time,there has been a growing demand for computational mathemati-cians in industry. In fact, during the last few years nearlyone-third of the new Ph.D.'s in the mathematical scienceshave taken positions in industry. At the same time studentsoften terminate their studies early to accept high payingpositions in the computing industry. Similarly, the bestyoung faculty members in computational mathematics are fre-quently lured away to industry not only by the attraction ofhigher salaries but also by the much better computingenvironments.

Significant additional manpower resources for computa-tional mathematics will be needed in the years ahead. Infact, the David report presents the following estimate:"Expectations are that a few hundred supercomputers foracedmic, industrial, or governmental use will be put inplace over the next decade. Each machine will requireapproximately 10 scientists with sophisticated knowledge ofapplied mathematics related to computation. Demand for suchnew scientists may run 500-800 per year. Even though numbersof these scientists will come from computer science, thephysical sciences, or engineering, the demand for new Ph.D.mathematical scientists in computing could easily reach 100per year in the future. Federal support of a subfield ofthis size could not be absorbed within the resources we haverecommended". The projections of the report appear to berather conservative. In fact, recent studies by the Insti-tute for Constructive Capitalism of the University of Texasat Austin indicate that by 1990 the projected number ofsupercomputer installations will be most likely around 200per year and that worldwide supercomputer sales may exceed1,500 per year by 1993.

These brief remarks already indicate that there is noeasy response to the recommendations in DA and [3] thatthere be an increase in the education and manpowerdevelopment in computational and applied mathematics. The

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David report recommended a broad "National Graduate andPostdoctoral Education Plan in the Mathematical Sciences".There is no need to repeat here thAl details. But we noteparticulary the strong emphais c additional support forgraduate students, post-doctorals, and young investigators.

The special situation of computational mathematicsrequires added attention to these recommendations. In thefield there exists a very urgent need for increased supportof graduate students to help in inducing more of them tocontinue their studies up to the Ph.D. degree. However, careneeds to be taken that such students are not assigned tostandard teaching duties but should participate as much aspossible in the work of multidisciplinary research teams incomputational mathematics.

In contrast to chemistry, physics, and biology, thereexists very little funding for post-doctoral positions inmathematics. In part this corresponds to the sociologicalfact that post-doctoral studies are not a traditionalrequirement in the core areas of mathematics. In computa-tional mathematics, however, there are growing indicationsthat this situation is changing. Typically a new Ph.D. hashad time to become somewhat familiar only with one particu-lar application area. As in the other experimental sciences,there is a need for a subsequent deeper involvement in someother application and problem area. Without such additionalpost-doctoral experience the young Ph.D.'s research islikely to remain severely restricted. At the same time, theproposed research teams represent an ideal training mechan-ism for such post-doctoral fellows. We recommend stronglythat the National Science Foundation increase its supportfor post-doctoral positions in computational mathematics.

In view of the severe shortage of computationalmathematics faculty, it appears to be desirable to considerretraining qualified persons from other areas of mathematicsor other fields. In part, this process can be aided by meansof longer visits of such people at places where there areestablished computational mathematics teams. Some considera-tion should be given to providing appropriate support forsuch visits, for example, in the form of competitive grantsto the potential visitors.

There appears to be a need for a closer study of suit-able modes of undergraduate education in computationalmathematics. In analogy with the experimental sciences,courses certainly require a meaningful laboratory component.This raises questions about suitable hardware and software,and even more critically about well designed textualmaterial. The funding agencies, and, in particular, theNational Science Foundation may be able to stimulate some

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developments in this area by means of some well targetedsupport. For instance, this may be in the form of some sup-port for interdisciplinary study groups to prepare materialfor novel courses in computational mathematics, or of chal-lenge grants to design new and innovative programs. But thisalso raises the question of funding of appropriate computa-tional facilities for educational purposes.

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5. FUNDING CONSIDERATIONS

In paraphrasing the Bardon-Curtis report [2] (seeAppendix A) the Committee believes strongly that the generalrole of the Federal funding agencies supporting scientific

computing should be to maintain strong programs that aredesigned to provide:

support of basic and applied research in universitiesin computational mathematics and advanced computer con-cepts including software, algorithms, architecture,subsystems, and components;

support of basic and applied research in problems that7)quire computing across all areas of science and

ingineering;

training of mathematicians, scientists, and engineersin the concepts, design, and use of computing systems;

assured access for researchers and graduate students to

computing facilities.

During the past year several funding agencies respondedto some of the recommendations of the Lax report [l] and theBardon-Curtis report C23. We mention here only the Fast

Algorithm initiative at the Air Force Office of ScientificResearch, the establishment of the Scientific Computing Pro-gram in the Office of the Director of Energy Research at theDepartment of Energy, and the formation of the Office of

Advanced Scientific Computing at the National Science Foun-dation.

The Air Force initiative is based on the earlier notedobservation that new advances in numerical analysis, andalgorithm design are required to exploit the new computerarchitectures that are now being introduced. In line withthis, a moderate number of team-research efforts for the

development of "fast algorithms" for large scientific prob-lems are being funded.

The aims of the Scientific Computing Program at DOEagree closely with many of the ideas and recommendationsdescribed in Section 3. The program consists of the AppliedMathematical Sciences ResE:rch Program and the EnergyReseardh Advanced Computation Project. The Applied

Mathematical Sciences Program has initiated several majorinterdisciplinary projects at universities and laboratories

through its initiative in the development of Advanced

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Computing Concepts and the Energy Research Advanced Computa-tion Project in providing access to supercomputers at theNMFECC and the Supercomputer Computation Research Instituteat Florida State University via the nationwide MFE network.

The formation of the Office of Advanced Scientific Com-puting (OASC) by the National Science Foundation representsan important step toward providing much needed supercomputerservice for a broader community involved in academicresearch and science education. The OASC SupercomputingCenters Program is supporting access to advanced scientificcomputing resources and advanced prototype computers, aswell as projects in the area of software productivity andcomputational mathematics for supercomputers. In additionthe OASC Networking Program will facilitate the establish-ment of a national network and support of local access tothat network.

In connection with the OASC support for software pro-ductivity and computational mathematics it should be notedthat the emphasis is here not on basic research but on pro-jects that enhance the effectiveness of scientists andengineers in their use of supercomputers. But as pointedout in Section 1 above, there is a fundamental need forestablishing a companion program of support for research andeducation in computational mathematics itself. Evidentlythis is not part of the current program of OASC, althoughthere may well be reasons to consider modifications of thatprogram.

As summarized in Section 1 our fundamental recommenda-tion is to expand and strengthen the programs of the vari-ous funding agencies in support of research and education inthe computational mathematical sciences and scientific com-puting. In line with the discussions in the previous sec-tions, such expanded programs should have at least the fol-lowing components:

1. New funding programs specifically focused on theestablishment of a number of viable research teams incomputational mathematics and scientific computing andon ensuring the continuity of their work for severalyears.

2. Support programs for establishing and maintainingan appropriate research infra-structure for suchresearch teams consisting of suitable computer hardwareand software and the associated support personnel.

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3. A concentrated effort for overcoming some of themanpower shortages in computational mathematics. Thisshould include, in particular, support for graduatestudents, post-doctoral fellows, and young researchers.

In order to give some indication of the required funds,the following table presents an estimate of the yearly costof a team of five researchers: (All figures are in thousandsof dollars)

SALARIES5 investigators at an average of $45

two months summer support each

two months released timeper academic year each

2 full time support persons

1 post-doctoral fellow

FRINGE BENEFITS (at 25%)

GRADUATE STUDENTS8 students at $12 each for stipend

and tuition benefits

ASSOCIATED RESEARCH COSTSsecretarial costs, printing andreproduction, travel, etc (5 times

INDIRECT COSTS (50% of direct costs)

TOTAL PER YEAR

$7)

$50

50

60

25

185

46

96

35

$362

181

$543

There are currently about 15-20 small more or lessinformal research groups in computational mathematics andscientific computing around the country. Clearly they wouldconstitute natural nuclei for the proposed research teams.This would suggest that funds are needed for the establish-ment and continued support of about 3-4 such teams each yearfor a period of five years.

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As noted in the earlier sections, this support need notbe in the form of one grant. In particular, funding for thework of the senior investigators might already exist in theform of project grants. However, the currently availablesupport for post-doctoral fellows, support personnel, andadditional graduate students is certainly inadequate andneeds to be expanded. Moreover, most importantly, somemechanism has to be found to ensure some stability and con-tinuity in the support of such a team over a period ofseveral years. For this the current, relatively short termproject grants, e not well suited. A long-range approachcan sustant1j increase the productivity, and efficiencyof computational mathematics research in this country.

The above figures do not include computer costs. Herefunds are needed for the acquisition of computer hardwareand software, the continued operation and maintenance ofthese facilities, and access to national networks andsuper-computers.

The existing support program for computer equipmentappears to be in need of considerable expansion if it is tomeet the requirements of the proposed strengthening of com-putational mathematics research and the formation of theindicated research teams. Some cost-data for these labora-tory facilities are given in Appendix B. As summarizedthere the minimal equipment cost for the above research teamis of the order of $250,000 to 300,000.

The cost of operating and maintaining computing equip-ment is an increasingly difficult problem for most collegesand universities. In fact, frequently, institutions find iteasier to acquire some computing equipment through one-timefunds, gifts, or bequests, than to operate and maintain it.Such costs have been estimated at about 15-20% of equipmentcosts per year. Since these computing facilities are anessential part of the computational mathematics research,there is a need to establish some support for these ongoingcosts as part of the normal funding of this work.

The mentioned third component of support for computa-tional facilities, namely, the access to a national networkand suitable supercomputers is a part of the already esta-blished program of the NSF Office of Advanced ScientificComputing. But as noted earlier there is an equal need for abroadly based network which links groups in mathematics,computer science, and related areas, and which provides themuch needed closer cooperation and coordination betweenresearchers who often work on the same or similar problems.

As discussed in Section 4 above, there is no easy solu-tion to the required strengthening of the educational

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situation in mathematics, in general, and in computationaland applied mathematics, in particular. The proposed estab-lishment of a number of research teams in computationalmathematics will certainly have a positive effect in thisregard, at least on the graduate and post-doctoral level.As noted earlier, these teams can also provide the opportun-ity for faculty level visitors interested in switching tocomputational mathematics. All funding agencies should givestrong consideration to an expansion of their support forgraduate research assistants, post-doctoral fellows, younginvestigators, and faz1..Ity level visitors in the generalarea of the computational mathematical sciences. In addi-tion, through support of studies by professional societiesand special groups, there exist possibilities for mobilizingthe mathematical sciences research community to address thecritical problems of mathematical education in this computerage.

APPENDIX ALIST OF RELATED REPORTS

C13 REPORT OF THE PANEL ON LARGE-SCALE COMPUTING IN SCI-ENCE AND ENGINEERING, Sponsored by DOD and NSF incooperation with DOE and NASA, P.D.Lax, Chairman,National Science Foundation, Dec. 26, 1982

1E23 A NATIONAL COMPUTING ENVIRONMENT FOR ACADEMICRESEARCH, NSF Working Group on Computers forResearch, K.K.Curtis, Chairman, Report prepared underthe direction of M.Bardon, National Science Founda-tion, July 1983

[3] COMPUTATIONAL MODELING AND MATHEMATICS APPLIED TO THEPHYSICAL SCIENCES, National Academy ofSciences/National Research Council, Committee on theApplications of Mathematics, W.C.Rheinboldt, Chair-man, National Academy Press, 1984

1E43 RENEWING U.S.MATHEMATICS, CRITICAL RESOURCE FOR THEFUTURE, National Academy of Sciences/ NationalResearch Council, Ad. Hoc Committee on Resources forthe Mathematical Sciences, E.E. David, Chairman,National Academy Press, 1984

APPENDIX BLABORATORY FACILITIES FOR SCIENTIFIC COMPUTING

The Committee discussed various workstation networkconfigurations which would be suitable for scientific com-puting teams. On the basis of this discussion, one of thecommittee members, D. Gannon, compiled the following sum-mary.

A basic laboratory suitable for three senior research-ers and five graduate research assistants should consist ofa network file server and gateway to a long haul network tosupercomputers, two high performance workstations and fourlow end workstations. These pieces of equipment should havethe following minimal characteristics:

1. The network (X.25) gateway and file server shouldhave at least 500 MBytes disk storage and tape backup.The cost range for such a file server ranges between$50,000 and $100,000 or more.

2. The two high end workstations should have at least4 MBytes memory each, and should include color grahpics(approx. 1024 by 1024 and 8 to 24 bit planes) as wellas hardware floating point accelarators. The cost rangeis about $40,000 to $80,000 each, but may easily bemuch higher than that.

3. The four low end workstations should have at least3 MBytes of memory each and a bit mapped black andwhite display. They need not incorporate floating pointhardware. The cost range is about $15,000 to $20,000for each station, but once again can go also muchhigher.

4. The software requirements will vary considerably,but FORTRAN and C compilers, UNIX, and network supportare most likely needed. For some networks the cost ofsuch software may be as high as $25,000 while in othercases only $1,000 to $3,000 may be required.

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du

The performance of the currently available syscems ofthis type varies as much as the cost. In fact some of thelower priced systems may need to be connected to a suitablesuper-mini computer to form an acceptable configuration. Thecost of the above laboratory after customary universitydiscounts is at least of the order of 0250,000 to$300,000, but may easily be double that amount. The yearlymaintenance for this equipment is roughly 15-20% of the pur-chase price per year.

A general overview of the computing facilities neededin support of a small research team in scientific computingwas prepared by the Computing Center of the University ofMichigan. In the remainder of this appendix we present averbatim copy of this thoughtful specification:

Computing facilities to support a small group ofresearchers in scientific computation should consist of acollection of workstations of varying and/or specializedcapabilities joined in a LAN with one or more especializedservers and network gateways. The workstation capabilitiesand specialized servers must be selected to meet the generalreqlsirements delineated below as well as any specialrequirements needed by the researchers for whom the systemis designed. Facilities custoized for specific researchareas should be anticipated. For example, workstations pro-viding unique computing capabilities and servers for spe-cialized hardware devices (high-quality printing, hardcopygraphics, or photographic output) should be anticipated.

The workstations can be characterized in terms of fivequantitative requirements: megapixel displays, megabytes ofRAM, MIPS of computing power, megabytes of storage, andmegabit transmission rates within the LAN. rhese five cri-teria provide the outline of a "5M" workstation and will beassigned different emphasis depending on the specificresearch activities under consideration. For example, a(1,4,2,1,1) configuration might be used for computationallyintensive activities, while a (8,4,1,4,1) might be moreappropriate for graphic intensive applications.

Although written in terms of hardware characteristics,the 5M criteria should be viewed in the context of the LAN.Thus, for example, reduced function workstations need nothave these hardware characteristics in isolation from theLAN provided that the equivalent facilities are available byusing system components within the LAN. Nevertheless, thetotal performance of the computing facility should reflectthat available from 5M workstations, i.e., reduced functionworkstations should not degrade total system performanceappreciably. In particular, the megabyte of RAM and MIPS

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processing rate and the megabyte of storage can be providedwithin the LAN and need not necessarily characterize eachworkstation, e.g., the LAll could include one or more multi-MIPS, multi-megabyte processors and a file server incor-porating a large disk system. The megapixel and megabittransmission rate requirements, however, are very worksta-tion specific.

The megapixel reci,);.vJnt means that the product of theveritical and horizontal ,-esolutions and the bits per pixelexceed one million. For example, 1024 x 1024 resolutionwith 8 bits of color provides 8 megapixels. In associationwith the megapixel criterion, the display should be bit-mapped, and the software should support multiple displaywindows and a 'mouse' for both interactive graphics and amenu/token driven environment.

Although 1.2ct an explicit requirement, virtual memory isalmost mandatory to meet the many demands that will beplaced on these systems. Because routine numerical applica-tions require 4 megabytes of memory, total system memorydemand will be /ery large. Real memory systems must becarefully considered if reduced function workstations areircluded. Further, if the local system is to be used forprogram development for production systems accessiblethrough the ne,work gateways, the memory capacity and per-formance of the local system must be a significant fractionof the production system so that reasonable testing can beperformed locally. This is particularly applicable tomemory capacity becaufze memory restrictions play a more sig-nificant role in &lgorithm and program design than the otherelements of the 5M criteria.

Use of secondary, or low-end, workstations for produc-tion computations should be anticipated, and their perfor-mance capabilities should not be significantly less than thehigh-end workstations. For scientific applications, thefloating-point performance available to a workstation shouldbe sufficient for the users to observe actual performance ofat least .1 MFLOPS, and the floating-point characteristicsshould be consistent with those of the IRPF floating-pointstandard. A floating-point coprocessor or other provisionfor hardware floating-point will probably be necessary toachieve these speeds.

The software system sLould support both menu/token andcommand driven environments, bit-mapped interactive graph-ics, multiple display windows, a language-independent pro-gram development system, both internal and external (usingthe network gateways) messaging, network support for themost commonly used network protocols (e.g., X.25 andTCP/IP), and file transfer and program portability with the

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systems routinely available through the network gateways.Program portability should be a primary consideration insystem and compiler selection.

The program development system should be languageindependent and provide full-function editing capabilities,a file/program management facility, a symbolic debuggingfacility, and a program performance analysis facility. Theprogram development system cannot be assessed in the limitedcontext of the workstation and its LAN. Although much ofthe program development will be for the workstation, muchwill be for remote systems.

The system should provide 'expert' compilers that sup-port program portability not only in terms of language syn-tax and semantics but also program performance. Thisrequirement is more stringent than simple conformance tospecific language standards. The compilers must encourageand guide program development in directions that will resultin acceptable performance both on the local system and vec-tor systems available through the network gateways. Byimplication, the workstation must provide sufficient perfor-mance and memory capacity for this type of use.

APPENDIX CLETTERS AND POSITION PAPERS

4 4

STANFORD UNIVERSITYSTANFORD, CALIFORNIA 94305

Gene H. GolubProfessor and Chairman Telephone:DEPARTMENT OF COMPUTER SCIENCE (415) 497-3124

September 10, 1984

Dr. John ConnollyOffice of Advanced Scientific ComputingNational Science FoundationWashington, D.C. 20550

DeEr Dr. Connolly:

Many thanks for making your excellent presentation at the SIAM meetingin Seattle. I am sure we all found it interesting to hear of NSF'sinitiative in supercomputing.

My colleagues and I, however, are quite concerned about certain aspectsof this project. In particular, we feel that it is still not completelyunderstood how to use supercomputers in their most effective manner. Inparticular, there is still a need for developing new (or possibly reexamining old) algorithms which will take advantage of all the power ofthese computers. It is important that the numerical analysis/scientificcomputing community be involved in this, since we have the greatestexpertise in devising and analyzing numerical algorithms which arerelevant to scientific computing.

In a similar vein, I think it also important that the computer sciencecommunity have some association with this project. We see increasinglythat simply having a supercomputer is not sufficient to accomplish largescale computation. A supercomputer environment can easily degenerateand the supercomputer would not be of great use if one does not haveexcellent means of communications and supporting systems software.

As you can see, I feel that supercomputers should involve a largerscientific cammunity than was initially envisioned. There is a lot ofexpertise now in developing numerical software and in developing computersystems. It think it is important these these communities be also takenin account in the supercamputer initiative.

Your4 incerely

W,4 1-1e-4H. Gol

,CC: Jack DongarraDr. John PolkingKent Curtis

Dr. Bruce BarnDr. M. Ciment

HIRSH G. COHENPRESIDENT

SOCIETY for INDUSTRIAL and APPLIED MATHEMATICS1400 ARCHITECTS BUILDING 117 SOUTH 17th STREET PHILADELPHIA. PA 19103-5052 (215) 564-2929

December 6, 1984

Dr. John C. PolkingDivision DirectorDivision of Mathematical SciencesNational Science FoundationWashington, D.C. 20550

Dear John,

REPLY TO:IBM RESEARCH CENTERPO BOX2IBYORKTOWN HEIGHTS. NY 10596

I don't know where you stand on shaping the mathematics ofcomputation initiative but I thought you might find theenclosed statement useful.

As will be obvious, I prepared it for use in our dealingswith DOD. It became a part of a report on 6.1 Researchsubmitted by Ivan Bennett's committee for the DOD-UniversityForum. Some of the comments and ideas may be useful to you.

I'm also sending a copy of a letter I sent to Ivan Bennettto reinforce the proposal.

I hope these are useful. I'm sending copies to WernerRheinboldt.

Sincerely yours,

Hirsh Cohen

/emk

Enclosures

cc: W. Rheinboldt

PUBLICATIONS LECTURESHIPS SECTION ACTIVITIES PROGRAMS

4

THE MATHEMATICS OF SCIENTIFIC COMPUTATION

I. INTRODUCTION:

The Department of Defense, as one of the largest users of computation,perhap3 the largest, must be vitally concerned with the quality ofthat computation. In particular, there is great sensitivity in termsof the accuracy, efficiency, and cost associated with the scientificcalculations that DOD does in its own laboratories, in the design andoperational activities of the services, and in DOD-sponsored researchand development work. As dependencies on large calculations grow, thequality of computation becomes an increasingly important matter. Wehave seen quality considerations catch up on a number of U.S. privatesector technologies and products in a frightening fashion. We must besure this does not occur in a field as important to DOD as scientificcomputation.

Large computer application programs have a number of levels at whichone must be concerned. The software codes must be well-structured andmodular to allow for efficient building, merging, and repair (correctionand debugging). If possible, some form of either validation for cor-rectness or testing must be built into the programming as it is beingconstructed. The algorithms chosen to be used for the unraveling ofthe equations and for their solving must be as close to optimum as pos-sible from the point of view of minimum use of machine instructionsand optimal transfers of data from large storage to fast storage andthen to the execution units.

Before any of these can be done, however, there must be adequate assur-ance that the physical and mathematical problems have been correctlyposed; that, in fact, there are solutions and something of their nuali-tative character is understood. And, at the very base of a useful eal-culation, the phenomena itself, whether one of physics and chemirryor of the operation or design of systems, or a logistics proble*, ustbe satisfactorialy understood and formulated.

Part of this whole process lies within the scope of the mathematicalsciences and part lies without. Those activities which depend on thelinguistic aspects of programming or on the architectural concepts andstructures of the operating systems or the hardware are the realm ofthe computer scientist even for very large application programs. How-ever, the mathematical parts--the problem formulation and its quali-tative analysis, the translation into numerical methods, and the selec-tion of an optimum algorithm--all depend heavily on the background thatthe linguistic and architectural components contribute.

In this spirit, we propose a new initiative in mathematical researchthat takes as a major goal the improvement of large scale scientificcomputing. There have been notable endeavors of this kind in pastyears; we do not mean to work in uncharted territory:

The Fast Fourier Transform began in an attempt to solve a problemin low temperature physics. A thorough understanding of its uses,

4 '/

the formulation of problems to take advantage of its capabilities,a search for best algorithms, and their efficient programming onmany machine oystems has followed. It is now a sine qua non insignal processlng and has revolutionized data handling and analy-sis.

The nonlinear partial differential equations of transonic flowhave been a mathemal!cal and computing target for over thirty-five years. Increased knowledge of the physical phenomena, under-standing of the qualitative behavior of solutions, new numericalapproaches and algorithms that take advantage of these and ofmachine structure have now produced fuel-saving airfoil designsthat are appearing on military and commercial aircraft.

Linear programming came out of W.W. II naval logistics. It hasbecome a money-saving and labor-saving mathematical tool, anessential part cf military logistics and operations and of design.The improvement of large software codes has gone hand-in-handwith a deepening understanding of the mathematical structure,complexity, and limitations of linear programming algorithms.In very recent years, a new approach, the elliptic method and anew estimate on the linear nature of the simplex method, havestimulated fuvi7)er mathematical study.

Examples such as these abound; they show very well that the most effec-tive of the large sore packages for important scientific calcula-tions are firmly based on well thought-through mathematical analysisand numerical methods.

II. AREAS OF RESEARCH:

Scientific computation depends on discrete approximations to the equa-tions that describe the physical phenomena. Asymptotic or otherperturbation approximations often provide invaluable insights into thenature of nonlinear behavior and are of great help in designing ef-ficient numerical methods.

The origins of methods for finding approximate solutions of an initialor boundary value problem can often be found in the mathematical analy-sis of the system. For example, the finite element method for solvingelliptic boundary value problems draws heavily upon the variationalframework in which the basic issues of existence, uniqueness, continuitywith respect to data, and regularity of solutions are addressed. Thisinteraction between the approximation methods and the underlying analy-sis is an important component of the whole study.

A major challenge is the development of comprehensive theories for thenonlinear problems arising for elliptic problems (nonlinear elasticity,the static semiconductor equations), parabolic problems (nonlineardiffusion and transport, combustion), and hyperbolic problems (wave

motion on ocean surfaces, ground motion, shocks, entropy inequalities).A study of the questions of well-posedness and regularity is of funda-mental importance in laying the groundwork for sound methods for theapproximation of solutions. Support of this area is currently inade-quate in view of the importance of these problems.

There are many other areas in which more analysis is needed in compu-tational mathematics: solution-adaptive methods need considerableattention at present. These methods use information about approximatesolutions developed at initial stages to refine and modify the frame-work of the approximation itself. The adaptation should be incorporatedinto the approximation process at a fundamental stage and should bedone in an optimal way. Research into the developement of approximatemethods for problems with different time and length scales and problemswith singularities or roughness in coefficients or data requires moreanalysis. Another challenging area arises from parameter dependentproblems. The question of solution stability typically involves exami-nation of the solution manifold, which for many complex problems isaccessible only by computation. The comparison of the solution manifoldof the discrete problem with that of the original problem is an importantincompletely explored question. These and many other research areasrequire a closer tie between the computational methods under develop-ment and the analytical tools applicable to these problems.

A related topic in which further research is needed to support theeevelopment of computational mathematics concerns implementation of thecomputational methods by a computer architecture. With the advent of"vector," "array," "systolic," and "parallel" processing, it is evidentthat the architectural issues in the development of computationalmethods are becoming increasingly important. In fact, we must be ableto resolve the interplay between the methods and the computers on whichthey are to be implemented in order to be succes-ful. Thu3, we shouldconsider computation as a path from a partial differential equation toa discretionization method to a method for solving the resulting linearalgebra, then the "mapping" of these to the computational process on acomputing engine. We need to address the relative merits for scientificcomputing of massively parallel architectures, systolic architectures,data-flow machines, machines with reconfigurable communication net-works, computers with a few, powerful processors. These machines mustbe matched to the computational tasks for which they are best suited.

III. FUNDING MODE:

We propose that this initiative, The Mathematics of Scientific Compu-tation, be managed as a coordinated program through the three servicesmathematical sciences basic research groups. At present, each of thesegroups has a very small base program that covers some of these topics.These efforts require substantial broadening in terms of particularproblem coverage and technique development. These base programs areinadequate in terms of the problems they can cover; we are asking fora major expansion of these efforts. In addition, there are excellent

researchers in the fields we have mentioned who are not now supportedand who are capable of making excellent contributions.

We propose that funding for this research initiative be, for the mostpart, concentrated at a number of research centers. A few of theseexist at present at which there are a large enough group of researchers,post-docs, and graduate students plus the necessary computation facili-ties to provide in-depth study on one or several of the problem areas.One purpose of funding in a focussed fashion is to establish other cen-ters of excellent research work in these computation fields so that, asnew application topics arist in DOD technological requirements, thesegroups will be available. Unlike other work in mathematics, the prob-lems, the methods, and the implementation of trials and tests of codesdo require group activity.

It is our estimate that at a full scale strength of twenty-five people(five researchers, five post-docs, and fifteen graduate students) ateach center would cost on the order of .9-1.1 million dollars per year.We propose the establishment of six new centers in the next three years.These would not be established with block "center funds" but rather asa set of separate research contracts aimed at particular applicationsor technique development problems. Thus, the contracting and reviewingprocedures would be the ones usually employed with 6.1 funding.

Finally, recent reviews of the 6.1 research funding by external groups(the Bennett Committee of the DOD-University Forum; the David Committeeof the NAS/NRC) have concluded that the base programs in the mathemati-cal sciences, while generally and seriously underfunded, are of highquality and represent a solid contribution to DOD. For this reason weare requesting that the program described here be a new effort, compli-menting and adding to research now underway.

Hirsh CohenJune 11, 1984

For: Ad hoc mathematics advisory group to the DOD-University Forum.

50

New York UniversityAtntate unit taffy in she publa !mice

Courant Institute of Mathematical Sciences251 Mercer StreetNew York, N.Y. 10012Telephone: (212) 460-7100

December 4, 1984

Dr. Werner C. RheinboldtUniversity of PittsburghFaculty of Arts and SciencesDepartment of Mathematics and StatisticsAndrew W. Mellon Professor of MatilematicsPittsburgh, PA 15260

Dear Werner:

I enjoyed very much the workshop on ComputationalMathematics. I was very impressed by the quality and diversityof the participants. I feel that the program might be Presentedto the Science Foundation as an initiative very much on the linesof the Experimental Computing Program of the Division of ComputerScience. That means that each grant would be for a 5 yearinitial period and there would be 3 or 4 new grants every year

particuntil a sufficient level is reached. Industrial ipationshould be sought. I see three natural points of contact:

1) Summer employment for graduate students withinindustrial research groups

2) 1 year or shorter visits by industrial researchers touniversity computational research groups.

3) Visits by members of university researchers toindustrial research groups

Obviously, industry is a good source of computers as gifts or atgreatly reduced prices.

Originally the NSF experimental computation initiative wasto be jointly with the Department of Defense but after theinception of the program the D.O.D. pulled out. Now is the timeI feel to approach the D.O.D. for joining this Prog

ntramp but byno means should the NSF part of the program be conting ie onD.O.D. participation.

Whatever plans are developed I would presume would besubmitted to the Advisory Board of the Mathematics Divis ion fortheir endorsement.

With

Pete

National Aeronautics andSpace Administration

Ames Research CenterMoffett Field, California 94035

NASA

1""A" D:200-1 March 7, 1985

Dr. Werner C. RheinboldtAndrew W. Mellon Professor

of MathematicsDepartment of Mathematics and StatisticsFaculty of Arts and SciencesUniversity of PittsburghPittsburgh, PA 15260

Dear Werner:

I appreciate receiving from you a draft of the Report of the Panel onFuture Directions in Computational Mathematics, Algorithms, and ScientificSoftware. Your study certainly is a timely one. Research advances inthese discipline areas are changing the way the country does business in awide range of technical areas. We at Ames have participated in that revo-lution for fifteen years and expect the same rate of progress to continuefor the foreseeable future.

I will be very interested to see how the Panel recommendations are receivedby I6F. While I realize it is outside your charter, I would be interestedto hear any ideas you and the Panel have on what additional steps Ames cantake to advance computational mathematics.

Best regards,

William F. Ballhaus, Jr.Director

52

The University of MichiganComputing Center1075 Beal Avenue

Ann Arbor, Michigan 48104

March 12, 1985

Professor Werner RheinboldtDepartment of MathematicsUniversity of PittsburghPittsburgh, PA 15260

Dear Professor Rheinboldt:

The enclosed description of computing facilities for mathematicalcomputation evolved from a series of messages exchanged betweenGregory Marks, Special Assistant to the Vice-Provost for Infor-mation Technology, Professors Ridgway Scott, William Martin, andRichard Phillips, and me.

The "5M" criterion was developed within the College of Engin-eering and is basically a model of its current activity inworkstation design. For example, the College has had greatsuccess in using Macs as reduced function workstations attachedto Apollo rings with the computing capability obtained from theApollos.

Traditionally, floating-point performance and quality have beengiven little attention in comparison to the more general hardwarefeatures and software. Because the systems under considerationare specifically for mathematical computation, it is appropriateperhaps that the floating-point characteristics of the hardwarebe more carefully examined in this case.

We have taken the liberty of devoting at least some space to thesoftware systems for the workstations, but our specificationsremain quite incomplete.

I hope that our description will be helpful.

Sipcerely,

(_

Leonard J Harding

LJH:mft

Enclosure

cc: Professor Bernard Galler

53