1. ANNUAL PROGRAM ORGANIZERS

Petter E. Björstad of University of Bergen (Informatikk) was one of the main organizers for the 1996-1997 annual program on "Mathematics in High Performance Computing." He was also a co-organizer in the IMA workshop on "Parallel Solution of Partial Differential Equations," June 9-13, 1997. Below is is report:

Summary

I had the great privilege to participate in the HPC year from the early planning phase until its conclusion in the summer of 97. This note is not intended to report on the complete year, many events of considerable significance have been left out. Rather, I will try to say just a few words about some of the aspects that characterize IMA and that, in my opinion, contribute to its unique atmosphere and stimulating environment.

Planning

It shall also be noted that it was properly recognized that an upgrade of the IMA local computer facilities as well as the local network would be required. Similarly, agreements on joint sessions with the supercomputer center and proper access to supercomputer cycles were made.

Organization

Each workshop was still small enough to foster a free flowing exchange of ideas, a typical size being about 30 people. The format is right, clearly derived from years of experience at the IMA.

Practical Matters

Workshops and Tutorials

Post Doctoral Programme

Science

Life after the IMA year

Mitchell Luskin from the University of Minnesota, School of Mathematics was one of the organizers for the year on "Mathematics in High Performance Computing." He was also one of the organizers for the April 28-May 2, 1997 workshop on "Grid Generation and Adaptive Algorithms." and the June 9-13, 1997 workshop on "Parallel Solution of PDE." He reports:

During the academic year 1996-97, I was an organizer for the IMA program on High Performance Computing. I also was an organizer for two workshops; Grid Generation and Adaptive Algorithms, April 28-May 2, 1997, and Parallel Solution of PDE, June 9-13, 1997. In addition, I organized the weekly Numerical Analysis Seminar in which many of the IMA visitors participated.

I was involved in continuous informal interactions with the IMA visitors. Several of the IMA post-docs worked on research projects that followed my research program in microstructure which will be described below. Matthias Gobbert and Andreas Prohl developed and analyzed a discontinuous finite element for microstructure computations, and Martin Kruzik worked on a project to compute microstructure by relaxation.

Advances in the understanding of material microstructure are playing an important role in the development of active materials: shape memory, magnetostrictive, and ferroelectric materials. During the past several years computational methods for the equilibria of martensitic crystals based on elastic energy minimization has been developed by Luskin in collaboration with students, post-docs, and co-workers. The invariance of the energy density with respect to symmetry-related states implies that the elastic energy density is non-convex and must have multiple energy wells. Microstructure then occurs as the fine-scale spatial oscillation between these symmetry-related states to attain the lowest possible energy state. During the 1996-97 IMA program on High Performance Computing, Luskin developed and analyzed computational strategies that cope with the complexity of microstructure in situations that were formerly far beyond computational capability, and he and his co-workers showed how these concepts apply also to theories of micromagnetics and magnetostriction.

During the 1996-97 year, Luskin (jointly with Bo Li) considered multi-well energy minimization problems modeling martensitic crystals that can undergo either an orthorhombic to monoclinic or a cubic to tetragonal transformation. They obtained results for the approximation of a laminate with varying volume fractions. They constructed energy minimizing sequences of deformations which satisfy the corresponding boundary condition, and they established a series of error bounds in terms of the elastic energy for the approximation of the limiting macroscopic deformation and the simply laminated microstructure. Finally, they gave results for the corresponding finite element approximation of the laminate with varying volume fractions.

In another project, Luskin (with Li and Riordan) analyzed a class of piecewise linear, nonconforming finite element approximations of a simply laminated microstructure which minimizes the nonconvex variational problem for the deformation of martensitic crystals which can undergo either an orthorhombic to monoclinic (double well) or a cubic to tetragonal (triple well) transformation. Discontinuous, nonconforming elements are appealing for problems with microstructure because the admissible deformations have more flexibility to approximate oscillatory functions. This project extended the stability analysis and numerical analysis to a class of nonconforming elements that have been used successfully to compute microstructure.

The IMA program on the Mathematics of High Performance Computing got off to an active and hand-on start with tutorials on The Message Passing Interface Standard (MPI) on September 9--11, 1996 and High-Performance Fortran (HPF), September 11--13, 1996. MPI and HPF are software tools which enable parallel computing. The IMA was fortunate to have some of the developers of these tools present the tutorial (Gropp and Lusk for MPI and Koelbel and Schreiber for HPF). The post-docs were active participants in the tutorial and several utilized these tools extensively in their research.

In addition to the workshops, the post-docs and visitors were actively involved in a weekly Post-Doc seminar (which was run by the post-docs) and the weekly Numerical Analysis Seminar which is run by the School of Mathematics in collaboration with the IMA. Two of the program organizers (Bjørstad and Luskin) were in residence during the entire year and ensured continuity in organization.

Among the highlights of the program was the tutorial on PDE Software held during April 21--25, 1997. This tutorial featured presentations by eight leading developers of software for PDEs. The presentations included an overview of the packages, a demonstration of the software, carefully prepared exercises, and a general discussion on the last day. The visitors and post-docs actively learned to use the software on the prepared exercises, as well as tried to utilize the software on problems of research interest. The final day featured a stimulating and energetic discussion among the developers and the participants on the different approaches and future directions.

It was very exciting to see at this tutorial that many of the concepts on adaptive error control, grid generation, and iterative solvers that have been developed on a theoretical basis during the past decade are now being used in general purpose software. The following workshops on Grid Generation and Adaptive Algorithms, April 28 -- May 2, 1997, and Parallel Solution of PDE, June 9--13, 1997, continued the discussion of grid generation, adaptive methods, and iterative solvers and offered a view of future directions for PDE software.

Many of the long-term visitors made significant contributions to the post-doctoral program. Rolf Rannacher (in residence during the spring program) gave a series of four of lectures on ``Computational Methods for the incompressible Navier-Stokes Equations.'' the numerical solution of the Navier-Stokes equations. Petter Bjørstad assisted many of the visitors and post-docs in the development of parallel versions of their codes. For instance, he guided post-doctoral scientist Brian Suchomel in the development of a parallel version of his code using MPI for message passing and the additive Schwarz Domain Decomposition algorithm to solve an elliptic pressure equation. Mitchell Luskin introduced several post-docs (Gobbert and Kruzik) to computational problems in materials science.

The workshop on Grid Generation and Adaptive Algorithms during April 28--May 2, 1997 was one of the two major workshops of the Spring focus on Parallel Computational Mechanics. Grid generation is a common feature of many computational tasks which require the discretization and representation of space and surfaces. The approximation of the equilibrium states and dynamics of continua require that the continua be represented by a grid. The workshop speakers discussed how the geometric complexity of the physical object or the non-uniform nature of the solution variable make an unstructured grid desirable. Since an efficient grid requires knowledge of the computed solution, many of the workshop speakers spoke on how to construct grids that are adaptively computed with the solution.

Some of the themes discussed were the development of a posteriori error estimators (Bank, Berzins, Dupont, Le Veque, Nochetto, Rannacher, mesh generation schemes including refinement, coarsening, and mesh movement (Bank, Bern, Mukherjee, applications to degenerate, singular, or heterogeneous problems (Baker, Le Veque, Nochetto, Oden, Shephard), and software (Bank, Le Veque).

Parallel computers are only easily programmed with regular data structures, so the development of adaptive grid generation algorithms which can utilize parallel computers effectively has posed a great challenge. Approaches to repartioning and load balancing were discussed by several workshop speakers (Biswas, Flaherty).

The workshop on Parallel Solution of PDE held during June 9--13, 1997, was the other major workshop of the program on Parallel Computational Mechanics. Parallel computers offers the promise of greatly increased performance and the routine calculation of previously intractable problems. This period of concentration and the workshop promoted the development and assessment of new approximation and solution techniques which can take advantage of parallel computers. Techniques discussed included domain decomposition methods (Brenner, Cai, Chan, Dryja, Fischer, Hoppe, Keyes, Pavarino, Widlund) and multigrid methods (Bastian, Rannacher, Xu). Applications discussed included fluid dynamics, solid mechanics, and semiconductor simulation).

2. WORKSHOP ORGANIZERS

Robert Schreiber from Hewlett-Packard, Inc. (Hewlett-Packard Labs) was a member of the organizing committee on "Mathematics in High Performance Computing." He also co-organized the workshop on "Algorithms for Parallel Processing, September 16-20, 1996" along with the special IMA Short Course: High-Performance Fortran (HPF) held on September 11-13, 1996.

He writes the following preface for the proceedings where he is one of the editors:

The Workshop on Algorithms for Parallel Processing was held at the IMA September 16 - 20, 1996; it was the first workshop of the IMA year dedicated to the mathematics of high performance computing. The workshop organizers were Abhiram Ranade of The Indian Institute of Technology, Bombay, Michael Heath of the University of Illinois, and Robert Schreiber of Hewlett Packard Laboratories. Our idea was to bring together researchers who do innovative, exciting, parallel algorithms research on a wide range of topics, and by sharing insights, problems, tools, and methods to learn something of value from one another. As the remarkable ensemble of papers contained herein should show, we seem to have succeeded in creating ample opportunity for such exchanges. The papers offer a wide-ranging tour of recent developments in the very rapidly growing and changing field of parallel algorithms. They cover the following general areas:

* models and mechanisms for parallel machines (Chapters 1-4),

* discrete and combinatorial algorithms (Chapters 5-7),

* mathematical issues in parallelizing compilers (Chapter 8),

* parallel algorithms for matrix computation, differential equations, random number generation, and Fourier methods (Chapters 9-14),

* new parallel computer systems and software (Chapters 15-16).

We hope that readers will find this collection as enjoyable and informative as we have.

Raymond Johnson of the University of Maryland (Mathematics), Fletcher Jones of (IBM (Research Division), and James Turner of Rensselaer Polytechnic Institute (Computer Science) have the following joint report:

Preparing for opportunities was the theme of a workshop on Minorities and Applied Mathematics-Connections to Industry, held October 4-6, 1996 at the Institute for Mathematics and its Applications (IMA), University of Minnesota.

Approximately sixty invited minorities in mathematical sciences attended the workshop. Of these, forty were Ph.D. students from mathematical sciences departments in North America; the other twenty participants represented a range of professional experience from postdoc to senior scientist. Also attending were Avner Friedman, Director of the IMA; Robert Gulliver, Associate Director of the IMA; and Barry Cipra, a science writer.

The workshop was arranged by the IMA Director and the organizers to provide an atmosphere in which minority graduate students could hear about the research and careers associated with applying mathematics to real- world problems. The real-world problems presented to students involved mathematics at all levels from elementary to technical, and showed students the need to communicate across disciplines with scientists and engineers having sophisticated mathematical training. Students began that communication process (listening and speaking) at this workshop. Their careers will be enhanced by this type of exposure; they were encouraged to seek similar opportunities at their home institutions and in their home regions. Each graduate student attending the workshop agreed to return to their home institution and make a presentation about the workshop, so that its benefits would not be limited to those who attended.

The composition of the workshop-- a relatively small group, mostly graduate students-- was based on the model used at the workshop for women at the IMA in February, 1996 and was intended to create a comfortable and relaxed environment. The workshop contained four components:

1. Overview talks by senior participants about their technical work and career experiences;

2. Technical talks about the applications of mathematics associated with various real-world problems;

3. Focused small-group discussions charged to produce action items for colleges and universities, government laboratories, funding agencies and professional organizations;

4. An after-dinner talk by Earl Barnes of Georgia Tech describing how the discovery of Karmarkar's algorithm affected IBM's business strategy and the relationship between further developments in the mathematics and changes in strategy by IBM and its competitors.

The minority mathematics community is small, and the workshop was the first opportunity for many of the students to network with minority professionals sharing their interest in mathematics.

The technical talks were uniformly of high quality, and covered a range of applications, including manufacturing of semi-conductors, microstructure of materials, design of a chemical vapor deposition reactor, mathematical problems arising in biology including freezing of tissues for biomedical engineering, dynamics of proteins in aqueous solutions, transport of solutes across cell membranes and reconstruction of images in tomography. The mathematics involved included wavelets, Markov processes, optimization, partial differential equations and computer models. (Abstracts of all the talks are in Section IV below.)

The small-group discussion sessions were also modelled on the program held for women in February. Each group included about twelve people, typically eight graduate students and four senior mathematicians. One or two people served as coordinators to assure that everyone had a chance to speak and to assure that the group covered all relevant topics. One person was designated as recorder to prepare notes of each group's discussions. Another member of each group was asked to present the group's recommendations at the final assembly of all workshop participants. Student volunteers introduced speakers after the morning session, providing another chance for them to practice their communications skills.

The organizing committee was extremely pleased with the workshop. Participants were so enthusiastic that one of their primary suggestions was a request to meet again to see how people had carried out the suggestions made to them. They wanted to use another meeting to practice skills suggested at this workshop, where graduate students would give more of the talks, and would receive advance help in order to make maximum use of the conference.

The primary value of workshops like this is the students' exposure to people like themselves with interests like theirs, who have accomplished what they are striving to accomplish. All mathematicians are members of many communities--minorities, women, men, analysts, geometers, topologists, applied mathematicians. Workshops like this do not substitute for the specialized meetings of those communities; they serve to demonstrate the existence of a minority mathematics community which is not visible to students isolated in their graduate programs.

The meeting was valuable because minority mathematicians have an unparalleled opportunity. Mathematics research and education are rapidly changing. The minority mathematics community did not prosper under the old model; there is a willingness in our community to consider other models of preparation for a career in research. This workshop showed that minority students are eager to prepare themselves for twenty-first century opportunities.

Mathematical problems arising in industrial applications typically embody complicated, interdisciplinary issues of formulation, analysis and solution. Minorities in mathematical careers are often attracted to areas in which their results can have a societal impact. There are many opportunities provided by real-world problems for high-quality research, contributions to practical results, and rewarding scientific careers. The purpose of the weekend workshop is to show examples of people and problems from industrial settings and to develop a set of concrete action items that individuals and agencies can carry out and help minority scientists at all levels and in varied environments become involved with industrial problems.

The first goal will be achieved through technical talks by selected participants chosen based on their success with real-world problems. The collection of action items will build on suggestions received at earlier workshops.

Breakout Group Recommendations

Each breakout group was asked to discuss the following topics:

1. Undergraduate and graduate education in mathematical sciences
   • Transition from undergraduate to graduate work;
   • Curricular issues;
   • Uses of technology.

2. Preparing for opportunities
   • Bridging the gap between academia and industry;
   • Breadth of training;
   • Role of interdisciplinary work;
   • Role of internships;
   • Entrepreneurs and the global economy.

Although each group took a slightly different perspective on the main issues, many common elements were cited. Means of overcoming difficulties faced by students at the transition points (undergraduate to graduate, graduate to work) were subjects of numerous suggestions. Since members of the minority community frequently work in isolation, most of the recommendations were actions for individuals to undertake to prepare themselves better. The key to increasing the number of minority mathematicians is individual initiative on the items discussed below.

The main recommendations from all breakout groups are listed here in four groups; recommendations for faculty and students, recommendations for students, recommendations for academic mathematical sciences departments and recommendations for the professional societies. (Some recommendations are listed under more than one heading.)

A. Actions for everyone

1. Get connected; have and use e-mail and internet access

2. Make departmental presentations about this workshop; invite students from other departments

3. Take advantage of computer center resources (C, C++, software packages, LATEX, UNIX, EXCEL)

4. Encourage the Department to invite speakers who can give talks about applications of mathematics (and to make contacts with local industry)

5. Attend seminars in other departments

6. Contribute information to Internet sites for minorities (information about internships, co-ops, programs, etc.)

7. Become aware of other minority/professional organizations

8. Look for internships and summer appointments in industrial settings

9. Keep in contact with mentors

10. Set up a Web site on the Internet containing:
    1. Profiles of minority industrial mathematicians
    2. Names and research areas of minority graduate students who are working toward the Ph. D. degree; thesis topics when completed
    3. Profiles of minority businesses
    4. Listing of available internships
    5. Profile of tools needed for successful graduate experience (C, C++, Hypertext, GAMS, presentation skills, LATEX)
    6. Profile of conference abstracts, speakers, e-mail addresses, as in AARMS at http://www.wam.umd.edu/~rlj/aarms.html ; contribute to such sites
    7. Information on how to subscribe to minority e-mail lists
    8. Current sources of minority scholarships
    9. List of industrial and academic mentors
    10. Support for budding entrepreneurs, such as information on how to get started, information about others interested in starting businesses and information about writing proposals and reviewing proposals
    11. Information about getting involved with "virtual" companies

11. Use the Web to foster Applied Math team projects
    1. Identify hot areas in applied mathematics
    2. Recruit students
    3. Encourage students to form teams around these areas
    4. Support students planning and executing their chosen project
    5. Encourage student presentations on their projects at conferences

B. Actions for students

1. Set a goal and remain focused on it

2. Spend some time learning how to learn mathematics; take responsibility to prepare yourself

3. Explore academic offerings of other departments to broaden research opportunities: take a computer course; perhaps minor in some area of engineering, science, business, etc.

4. Develop facility with written/spoken language

5. Get computational experience; learn a computer language and how to apply it to your problem

6. Request a cross-departmental math modeling class with strong industry involvement

7. Start an interdisciplinary journal club (students getting together to read articles from journals) or a graduate student seminar

8. Get involved with a project involving applications or integrating math with other disciplines

9. Make contact with other (minority) graduate students for possible collaboration on research-seek out a "like-minded" group

10. Discuss with advisor "what lies ahead"

11. Always keep your resume in mind
    1. Go to conferences (for example, SIAM conferences including SIAM's Diversity Day during the SIAM meeting at Stanford, July 14-18, 1997) and take a leadership role
    2. Prepare for conferences by reading abstracts, deciding on talks you will attend and contacting authors of articles in which you are interested
    3. Do things inside and outside of school to make yourself more marketable
    4. When working on a project, always think about what part or extensions will be publishable

12. Make an all-out effort before graduating and looking for a job
    1. Network at every opportunity-- attend seminars, attend conferences, e-mail authors of articles
    2. Contact all your mentors and professors as you near completion of your M.S. or Ph.D. degree, asking them to get the word out that you are close to graduating
    3. Ask professors and mentors to send recommendations; those based on personal contact are particularly important
    4. Send letters/resumes "out of cycle" when the majority of letters/resumes are least likely to come (this is less effective in academe than in industry)
    5. Always follow up contacts
    6. Continually update your resume
    7. Stay aware of current events to facilitate conversations during job interviews
    8. Call ahead to determine which areas of research are of interest to the company with which you are interviewing-- meet industry halfway by showing them you are a good match with their needs

C. Actions for academic mathematical sciences departments

1. Organize student-to-student forums conducted by graduate students for undergraduate student math majors to talk about the transition to graduate school

2. Have a "strategies to get a job" seminar (for undergraduates and/or for graduate students). Invite employers of all types-- community colleges, four-year colleges, industry and government representatives

3. Recognize and support students who plan to enter the job market with a B.S. or a M.S. degree

4. Forward all job listings to all graduate students at all levels

5. Offer a math modeling class where students can work on problems from industry-- expose students to working in teams and learning how to approach problems

6. Make the modeling class interdisciplinary by cross-listing it with other departments

7. Encourage students who want to take courses outside the Mathematics Department

8. Invite speakers from industry to talk about real-world problems
   1. Contact graduates who work in industry
   2. Set up an Advisory Committee with invited representatives from local industry to provide another source of speakers

9. Improve advising for graduate students; some groups even suggested development and use of a placement exam

10. Offer support to students other than teaching assistantships; research internships in industry would prepare students to begin industrial careers as teaching assistantships encourage them to pursue teaching

11. Be aware of students in other disciplines, such as EE, who take lots of mathematics, as sources of double majors and graduate students

12. In industry, mathematics departments should explain their usefulness to the company; in academe, mathematical sciences departments should explain their usefulness to allied departments

D. Actions for professional societies

1. Encourage student participation at meetings
   1. Organize events for students
   2. Support students' attendance at society meetings (as is done by the Society for Mathematical Biology)

Lawrence David Davis from Tica Technologies, Inc. was one of the organizers for the workshop on "Evolutionary Algorithms" held on October 21-25, 1996. He reports:

In my estimation, the 1996 workshop on "evolutionary algorithms" yielded a number of important benefits. It brought together representatives of the superconductor community, applied mathematics community, and evolutionary algorithms community in a congenial and constructive interchange. I believe a number of positive results will follow from this. I describe several below.

Many of the interactions between representatives of different fields are leading and will lead to beneficial results. For example, some of the applied mathematicians became better acquainted with techniques for using evolutionary algorithms for real-world applications and intend to add the evolutionary techniques to their problem-solving machinery. Several of the parallel computing speakers discussed techniques for parallelizing algorithms at a meta-level. (This approach is not one typically followed in the evolutionary algorithm field, where hybridization of algorithms tends to occur in the operator set.) The result of this is likely to be renewed experimentation with hybridization techniques in the evolutionary algorithm field. Finally, some of the insights gained by members of the evolutionary algorithm field with regard to minimizing evaluation time appeared to be of benefit to members of the supercomputing field.

In addition, the workshop brought together representatives of different branches of the evolutionary algorithm field. Again, the result was a cross-fertilization of ideas that I expect will result in applied and theoretical work for several years to come.

The IMA did a wonderful job in creating the environment for the workshop, in promoting it, and in structuring it so that there was extensive time for offline interaction. I greatly enjoyed attending the workshop, and benefited tremendously from it.

Michael Vose from the University of Tennessee (Computer Science) was one of the organizers for the workshop on "Evolutionary Algorithms" held on October 21-25, 1996. His report follows:

The Evolutionary Algorithms workshop assembled a variety of people with differing viewpoints. The field is not mature, still enjoying youthful exuberance (and occasional excesses). I believe the workshop facilitated communication between many of the various camps in the evolutionary computation field, encouraging mutual cooperation, and challenging each to translate and integrate within their own perspectives the principles and conceptions of the others.

The workshop also provided opportunity for people who infrequently have the opportunity to meet with each other - but who do belong to the same subgroup of the field, and who do have similar outlooks or techniques - to share and discuss ideas.

As for specific contributions made by myself or by by other visitors to the IMA, I feel comfortable about commenting on my own part.

My point of view is that it makes sense, within the vast collection of evolutionary algorithms and variants, to focus attention on a simple, commonly used type, to then formalize it as a mathematical object, and to study it with the aim of proving theorems about its behavior.

That puts me in the lunatic fringe of the Evolutionary Algorithms crowd for various reasons ... not the least of which is that the vast majority are interested in solving some specific problem or class of problems (which are usually NP complete), and their goal is to find a usable heuristic. Any heuristic - however obtained - that seems to work, at least better than what was tried before, is the answer; proof, to most, is at best an irrelevant concept.

Although the paper I submitted for the IMA proceedings focuses more on explaining the mathematical framework than anything else, the message I tried to get across in my talk contained the following basic ideas:

1. Standard GA theory, from the perspective of logical rigor, leaves much to be desired.

2. There is an alternative, which, though more mathematically demanding, has more solid foundations, more predictive power, and is capable of supporting nontrivial and provably correct analysis.

3. That alternative points towards fixed points of an underlying dynamical system, among other things, as important objects in the theory.

4. Besides asymptotic results (i.e., proven theorems), there is empirical evidence gathered from extensive computer runs verifying the claim that the alternative theory can make qualitatively correct assertions about the behavior of genetic search as used in practice.

I hope you will be able to paraphrase parts of my comments to serve your purposes. The workshop was quite enjoyable, and I thought it a tremendous success. Best Regards.

Darrell Whitley of Colorado State University (Computer Science) was one of the organizers for the October 21-25, 1996 workshop on "Evolutionary Algorithms." He writes:

The support of the IMA for the Workshop on Evolutionary Algorithms made it possible to bring together for the first time a significant number of the top researchers working in the area of Evolutionary Algorithms, including the fields of Genetic Algorithms, which developed in the United States in the mid-70's, and Evolution Strategies, which developed in Germany in the mid-70's. These two communities worked in almost total isolation until 1990; in the last 5 years or so the Evolution Strategies community has been fairly active in integrating Genetic Algorithm based concepts into their work. But knowledge of Evolution Strategies on the part of the larger US community and even other European communities has been slower to develop. By bringing together key individuals in these communities in one place in a relative small forum (50 individuals) commonalities and differences in the two approaches were made much clearer. Even though I have been one of perhaps a half dozen individuals in the Genetic Algorithms community to interact with the Evolution Strategies community since 1990, I learned new things at the workshop that surprised me and deepen my understanding of the two approaches. It is somewhat symbolic that the "Fathers" of these two fields, John Holland and Hans Paul Schwefel, had only been briefly introduced once before and were able to interact for the first time at this workshop.

The workshop also did a great deal to clarify the current state of the theory in Evolutionary Algorithms; the existing theory might be characterized as belonging in at least two different major camps. There is a high level macro-theory that looks at the processing of "building blocks" and "schemata" that are shared by many good solutions when searching a problem space. There is also a low level micro-theory that builds exact Markov models of the search process. The macro-level theory of course ignores certain detail; the micro-level theory involves working with detailed models that unfortunately growth exponentially as a function of problem size. (The size of the models are larger than the search space itself.) I think that the IMA workshop helped both sides in this debate to present their cases and to move toward common ground.

A concept that was discussed at the workshop includes the so-called "No Free Lunch" result which shows that all search algorithms are the same when their performance is averaged over all possible discrete functions. Debate on this topic has motivated me to do new theoretically work proving that there are interesting and potentially useful ways to partition the set of all possible discrete functions such that some search algorithms are indeed better than others. The results in fact have practical implications for problem representations and improving search methods.

Another area where there was some real progress was in the area of communication between theorist and practitioners. A wide range of applications were presented. In some cases, theoretically motivated methods worked quite well. In other cases, practitioners used ad hoc methods to obtain better performance than could be achieved with theoretically motivated methods; but at least some individuals on both sides went away with a better appreciation of the successes and failures of current theory. I believe that the workshop will help to change what practitioners say about the current state of theory in the field.

Some very surprising and impressive applications were presented at the workshop, including data decomposition, deriving control parameters for modeling systems using partial differential equations and even new best know solutions to bin packing problems.

Overall, I found the workshop not only useful for bring people together and strengthening the community, but also as a learning experience.

Dianne P. O'Leary from the University of Maryland (Computer Science) was a member of the organizing committee on "Mathematics in High Performance Computing." She also co-organized the workshop on "The Mathematics of Information Coding" held on November 11-15, 1996. She writes:

A workshop on Information Coding, Extraction, and Distribution was held at IMA November 11-15. There were approximately 30 attendees. The speakers included Michael Orchard, Robert Gray, Ahmed Tewfik, Jorma Rissanen, Julia Abrahams, Paul Siegel, Gil Strang, Cynthia Dwork, George Cybenko, Chris Atkeson, Dianne O'Leary, Geoff Davis, Eric Metois, Duncan Buell, and Manfred Opper. Topics ranging from the mathematics of coding theory to the practicalities of copyright preservation for Internet resources drew spirited discussion and interaction among experts in diverse but related fields. George Cybenko and Dianne O'Leary made plans to collaborate on the study of iterative methods for matrix approximation.

Tamar Schlick from New York University (Courant Institute, Mathematical Sciences and Chemistry) is one of the organizers for the January 20-23 workshop on "Molecular Structure: Dynamics, Geometry, and Topology." She writes:

The workshop on macromolecular structure brought together, at the heart of winter, applied mathematicians and biophysicists. The talks covered many exciting current topics in molecular mechanics, molecular dynamics, global minimization of potential energy functions, fast electrostatics, parallelization of large-scale molecular codes, and much more.

Most exciting about the workshop was the cross-fertilization of ideas, as reflected by the many discussions throughout the lectures and during the meals and long breaks. The mathematical and biological scientists learned and exposed their colleagues to new information, new ideas, and future challenges. We all left the workshop feeling that the informal setting and small group size really worked to make the conference what it should be -- scientific exchanges of ideas -- rather than a continuous series of expositions by people who had been on the lecture circuit for a long time...

Scott Baden from the University California-San Diego (Computer Science and Engineering) was one of the organizers for the March 12-13, 1997 workshop on "Structured Adaptive Mesh Refinement Grid Methods." He writes:

After the formal talks had ended, a number of us (SAMR workshop participants) decided that there was enough interest to look into building standardized software support for structured adaptive mesh refinements. A web page will also be maintained.

Christoph Borgers from Tufts University (Mathematics) was one of the organizers for the workshop on "Computational Radiology and Imaging: Therapy and Diagnostics" held on March 17-21, 1997. He shares the following:

Margaret Cheney told Todd Quinto about a problem concerning singularity detection from sonar data. Todd was able to solve the problem, and even proposed a singularity detection algorithm based on his solution.

Robert Levine learned about the ongoing work on diffusion tomography. He talked to me about his interest in this subject, and I introduced him to David Boas, a researcher in diffusion tomography at Tufts University. Robert, David, Todd Quinto, and I have started a working seminar on the subject together since then.

I can think of a lot of points and ideas and references that I learned about at the workshop, or made other people aware of, but that is too detailed for this report. Several participants have made enthusiastic comments about the workshop.

Some examples:

1. Robert Levine made me much more aware of the importance of the beam selection (as opposed to beam weight optimization) problem in radiation therapy planning.

2. I provided Meldon Lodwick with a list of references on biological response models that have been proposed for the purpose of formulating the radiation therapy planning problem as a constrained optimization problem.

3. I learned about various approaches to the diffusion tomography problem: Michael Klibanov's and Simon Arridge's from their talks, and Sarah Patch's through conversations with her.

4. I pointed out to Yair Censor that constraints used in the radiation therapy planning problem often have strong "volume dependence:" It may be acceptable to give a large dose to a small piece of a healthy organ, but not to a large piece. It is not immediately obvious how such constraints could be incorporated into his approach to the radiation therapy planning problem. He thought about it a bit and came up with a suggestion that ought to be explored further.

5. I learned about some work by Lech Papiez on dose calculations in radiation therapy planning, and gave him references to the "Boltzmann/ Fokker-Planck splitting" method, which has been proposed in the Nuclear Engineering literature and appears to be closely related to Lech's ideas.

6. George Majda told me about his work on the Vlasov-Poisson equation. Some aspects of this work appear related to the dose calculation problem in radiation therapy planning. I visited George at Ohio State University in early May, and we discussed this subject further.

Frank Natterer from the Universitaet Muenster (Fachbereich Mathematik) was one of the organizers for the March 17-21, 1997 workshop on "Computational Radiology and Imaging: Therapy and Diagnostics." He reports:

The meeting brought together specialists from quite different fields. This resulted in many lively and often controversial discussions.

One example is regularization. While everybody agreed that some kind of regularization is necessary for virtually all medical imaging problems, some people contended that Tikhonov regularization is not sufficient. It does not permit to take into account properly the properties of the image/object to be reconstructed, nor does it reflect the role of the observer. These questions led to very lively discussions after the talks of Barrett, Herman and Johnson. A related controversial issue was assessment of image quality. Barrett and Herman pointed out quite clearly that naive distortion measures for images, such as norms, are insufficient. Statistical and psychological assessment methods such as ROC studies are required.

Another subject which was discussed quite controversially was the usefulness of seriously ill-posed problems in medical imaging, such as impedance and near infrared imaging. Some participants frankly questioned the practical value of such techniques, describing situations in which these methods are bound to fail. On the other hand, Arridge presented simulations and also reconstructions of measured data which demonstrate that near infrared imaging can produce useful pictures for neonatal heads. Isaacson impressed the audience with quite realistic patient studies. His talk demonstrated the potential and the limitations of imaging techniques based on seriously ill-posed problems.

A criticism which often came up was oversimplification and naive modeling, in particular in connection with ultrasonic and optical imaging. The modeling of noise and its propagation through the algorithm has almost never been discussed.

Jaeger stressed in his talk that image reconstruction and image enhancement should be considered as a single task. An example is the removal of artifacts from CT pictures which should be done on the data rather than on the image.

Sparr brought up Doppler imaging and the relevant mathematical apparatus which has been developed recently by Sharafutdinov. This talk impressively demonstrated how new mathematics leads to new imaging techniques. The role of mathematics in 3D imaging was the theme of the talk of Clack. A few years ago the underlying mathematics was understandable only to a few experts, but it is now almost as simple and elegant as in the 2D case. However, there is still need for algorithms which can handle the case of truncated projections.

Several approaches to the radiation treatment planning problem were discussed in the talks by Censor, Levine, and Lodwick. Levine stressed the importance of selecting a clinically manageable number of beams, an aspect of the treatment planning problem that is neglected in algorithms for optimizing the weights of pre-selected beams.

One of the mathematical tools which came up in many talks was the transport equation. It serves as a model for different modalities such as treatment planning, optical imaging, tomography including scatter, and emission tomography. It seems that a large part of medical imaging can be viewed as inverse problems for the transport equation.

Jeff Blaney was one of the organizers of the April 7-11, 1997 workshop on Mathematical and Computational Issues in Drug Design. Following is the panel discussion where he served as moderator:

This panel's challenge was to identify important problems and challenges in drug discovery that should be addressed within the next decade given steady, predictable improvement in computational power and perhaps less predictable improvement in algorithms and methods. What are the pressing problems and bottlenecks that may succumb to new computational and theoretical approaches?

Garland Marshall discussed the need for improved methods to deal with error analysis and propagation of errors in the drug discovery and optimization process. We routinely face many sources of error, experimental and theoretical, that complicate our efforts to interpret and predict structure-activity relationships. He also noted that we need to improve our ability to predict the structure of drug receptors. Despite the impressive increases in the ability of Xray crystallography and NMR to solve the structures of biologically relevant macromolecules, many critical receptors remain intractable. For example, 7-helical transmembrane, G protein-coupled receptors (7TM GPCRs) make up the largest single class of receptor targeted by today's drugs, yet no experimental structures are available due to the extreme difficulties in crystallizing transmembrane proteins. He also noted the immense informatics problem posed by the explosion of data available on the internet. We're "drowning in a sea of data," with only primitive tools to integrate, assimilate, and interpret the data. The explosion of genomics data is only beginning, due to the massive efforts behind the human genome project (the goal is to completely sequence the 3 billion bases in human DNA by 2005) and other sequencing efforts targeting a variety of microorganisms and pathogens.

Phil Portoghese also discussed the importance of 7TM GPCRs, noting that our only structural information comes from low-resolution experimental data and approximate homology models. The mechanism by which 7TM GPCRs transduce an extracellular signal across the membrane is poorly understood. He described experimental methods that can provide additional data to help modeling, such as recent work in making chimeric receptors that combine features of two different receptors in a single molecule. For example, an extracellular loop of the kappa-opiate receptor was transferred to the mu-opiate receptor, resulting in a new, kappa-selective, chimeric receptor. Current structures and models are not able to explain the difference between 7TM GPCR agonists (molecules that turn on an intracellular response) and antagonists (molecules that inhibit the intracellular response). For example, morphine-type molecules containing an N-CH₃ group are agonists at the mu-opiate receptor, but converting the N-CH₃ group to N-CH₂-cyclopropyl produces an antagonist. This is a very small structural change, yet it produces a dramatically different response.

Gordon Crippen noted that structure-activity relationships and computer-assisted drug design rely on the rather vague notion of molecular similarity as a central theme: similar molecules tend to have similar biological activity. However, we have many different ways of defining, measuring, and calculating similarity. He suggested that molecular similarity should be defined relative to the environment in which it is being measured, rather than as an independent property. He also addressed the critical problem of modeling oral bioavailability and drug distribution within the body. We have extremely limited control and understanding of how to alter chemical structure to achieve the desired properties (for example, a pill that is stable in the gut, reaches the target organ and receptors intact, has a long enough half-life to require only 1-2 doses per day, and does not produce toxic metabolites). Drug design and discovery is a complex, interdisciplinary problem: rather than focus on isolated problems, we should consider the entire system as a large graph depicting how drugs interact with the body. The nodes are organs, tissues, fluids, membranes, and other structures within the body. The edges are known or hypothetical pathways; sometimes there may be more than one path between a pair of nodes. Nodes are labeled by various capacities, edges by rates and permissions. Can we integrate known data into such a model?

Simon Kearsley provided a brief historical overview and a map to the future of modeling challenges, noting that modeling has already had a large impact in lead identification and optimization. Due to the advent of high-throughput screening, genomics, and bioinformatics, data mining and modeling are being used to help biologists get needed chemistry support (for example, by computationally identifying existing molecules in databases that validate biological hypotheses, which in turn creates demand for additional, optimized molecules). Current approaches have focused on the early stages of drug discovery by improving the odds through statistical and qualitative methods, for example, by helping to prioritize screening and assisting in the identification of new leads. Such methods include two-dimensional chemical substructure and similarity searching, and three-dimensional superposition and docking calculations. He suggested that the next challenge is in information science: extrapolating from data mining to "concept" mining and mining relationships in the data, going beyond improving the odds by studying case histories and anticipating decision points. He also mentioned that modeling approaches will need to head toward modeling cellular processes and relationships (similar to Gordon Crippen's ideas). This will require a canonicalization of text, and the building of regular vocabularies and thesauri to deal with the huge number of imprecise names and synonyms in biology. A formalized mathematical language will need to be developed for biology which can deal with a limited number of observable properties and data. He mentioned the need for technology to deal with very large databases and "data warehousing": for example, data integration, data "scrubbing," and multidimensional trend analysis. He also discussed the need for rule-generating algorithms that can deal with fuzzy logic and ambiguity. These methods will need to define and measure information content, including the subjectivity content, deal with noise and sparse data, and capture human expertise and knowledge.

There are clearly huge problems, challenges, and opportunities for drug discovery in the next decade. Most of today's methods focus on the early phases of drug discovery, which consume only a small fraction of the time and money (less than 20%) of the total path (10-12 years, $300-500 million) on the way to a marketed drug. Integrated approaches such as those suggested by Kearsley and Crippen have the potential to improve drug discovery further down the expensive and time-consuming development process, but will require increasingly sophisticated computational methods and information science technology.

A.J. Hopfinger and W.J. Howe were co-organizers of the April 7-11, 1997 workshop on Mathematical and Computational Issues in Drug Design. Following is the panel discussion where they served as moderators:

The intent of the first panel discussion was to identify areas, within the broad range of fields represented at the workshop, in which there are particularly important problems that may be amenable to theoretical, mathematical, or computational advancement. The focus was on areas that are currently under study, as opposed to the second panel discussion which covered areas that one may anticipate will begin to yield to computational methods within 5-10 years or so. Given the breadth of the subject matter covered at the workshop, the issues presented during the panel discussions were necessarily only a small subset of a potential list of "important problems," and they were obviously biased by the particular areas of research interest of the panelists. Nonetheless, each of the issues presented by a panelist engendered considerable discussion among the workshop participants, and the issues discussed in the first panel session can be considered to be representative of problems currently under study by numerous scientists in the field.

The panel discussion was structured around brief presentations by each of the panelists on one or more issues viewed as particularly difficult or important for the furtherance of the field. Each presentation was then followed by discussion among the larger audience. After the four presentations were complete, additional topics and issues were put forth for discussion by workshop participants. Following is a brief summary of the issues presented by the panelists and other participants.

The approach taken by Graham Richards was to present the drug discovery process as a continuum -- an essentially linear process that begins with genes, and moves through proteins of therapeutic interest that are encoded by the genes, to small molecules (potential drugs) that are designed to modulate the function of the target proteins. At the genetic end, the problem most suited to computing (and particularly, supercomputing) is hunting for sequence similarities between known genes and the entire genome database. The vast amount of data from sequencing the human genome and those of pathogens means that this is an enormous computational task. A recent example of the types of discoveries that can be made from such data mining is the identification of the similarity between smallpox viral genes and some of the genes in the human immunodefense system. It is clear at this point that the rapidly increasing amount of genetic data that are becoming available, coupled with the increasing reliance of pharmaceutical companies on the selection of drug targets based on information contained in these databases, will translate to substantial opportunities for improvements in bioinformatics algorithms.

Moving along the continuum, the greatest scientific challenge remains the `protein folding problem' -- going from gene sequence (and therefore protein sequence) to protein structure. Sequence identification far outpaces experimental structure solution, so there is a need to predict structure from sequence. Folding must depend on sequence and physics, but new insights and methods are needed. This remains the "Nobel Prize" problem.

Moving from protein targets to the small molecules that are being designed to bind to them, it is noted that such molecules are increasingly being synthesized by combinatorial or highly parallel synthesis techniques. While such methods are able to generate molecules on a scale of tens to hundreds of thousands, there is a need (and a recent trend) to reduce the structure space somewhat by clever application of computational methods to achieve greater focus and to improve the odds of identifying active compounds. This also carries over to the need for more efficient methods for comparing and selecting molecules (with specified properties) from databases containing hundreds of thousands of "real" compounds, to billions of "virtual" compounds (currently existing databases), each of which can have many conformationally-accessible structures. It was noted in the workshop that organic structure-space, of molecular weight less than 500, has been estimated variously as from 10⁶⁰ to 10⁴⁰⁰ unique compounds.

Doug Rohrer's focus was in the "small molecule" region of the continuum. He noted that many of the current challenges in making sense of structure-activity relation (SAR) data relate to choices, and that a large number of choices are generally required. To name a few, these choices range from selection of compounds, generating 3D structures for each compound (the conformational flexibility problem), determining the most appropriate overlay of the compounds (alignment rules), defining a consensus multi-molecule binding model, and determining how the consensus features of the model can best be used in drug discovery and optimization (that is, now that we have a model, what do we do with it?).

The initial choice of compounds used to develop an SAR model typically encompasses a small number of highest-affinity compounds. However, as the project matures, the selection of additional compounds must include as much diversity as possible, in both the 2D (connectivity) and 3D (spatial) sense. The challenge is to evaluate this diversity quickly. It is important, as was mentioned numerous times during the workshop, to maintain a fast turn-around time on such studies in order to remain in the decision and design loops. Otherwise the chemistry will move ahead without it. Doug suggested that perhaps improved graph theoretic approaches could be applied at this stage.

Usually the compounds of interest can have numerous low energy conformations. The challenges here involve methods for exploring conformational space, followed by analysis to select a representative subset of diverse conformations. Several novel methods were presented at the workshop which may address this challenge. One involved reducing the 3D structures to a 2D format that retains more information than does a simple projection technique. Modifications of alphanumeric character recognition algorithms were then used to identify similar shapes. Another technique that was presented involved mapping the structures into pre-defined arrangements of spheres to classify the shape of the molecules. Such classification of the molecular shape could provide a starting point for the production of multiple molecule overlays to generate the SAR model.

Field-based similarity matching provides an excellent means of optimizing structural overlays. However, the speed of this process needs to be improved in order to be useful with larger numbers of molecules (greater than ten). In addition, information about the relative biological potencies somehow needs to be factored into this process, so that the good, the bad, and the ugly features of the overlay model can be identified.

Finally, methods to use the overlay features of the SAR model to derive new molecules algorithmically, to test hand-generated molecules against the model, or to optimize existing leads via suggestions from the model, would be the ultimate goals. Automated de novo design programs like GROWMOL might be a start, but the solution is still elusive.

Bill Dunn continued the discussion of challenges related to small-molecule design, and again, for the common situation where the 3D structure of the target (protein) is not known, leading to the application of indirect methods in the delineation of 3D quantitative structure-activity relations (QSAR). One difficulty with these methodologies (e.g. comparative molecular field analysis, molecular shape analysis, etc.) is the assignment of a receptor-bound conformation (conformational flexibility again) and an alignment for the ligands (overlay problem). This is generally done by trial and error, with some guidance provided by ligands with restricted degrees of conformational freedom. This is an extremely costly exercise that limits the utility of 3D-QSAR.

A general solution to this problem may be found by relaxing the conformation and alignment constraints of existing approaches, and by describing the active ligands with features that vary with conformation and alignment. The resulting arrays are third-order tensors that can be decomposed to yield the descriptor vector(s) most highly correlated with biological activity.

Dunn's group is exploring a general formalism for 3D-QSAR that includes various methods useful for the decomposition of 3-way arrays of conformation- and alignment-dependent feature data. They are also exploring novel methods of aligning sets of bioactive compounds, in such a way that leads to an unconstrained 3D-QSAR procedure.

Dave Doherty focused on problems in biological chemistry that are characterized by structural transitions that involve large-scale motions of portions of the structure. One obvious example of this is the folding of proteins. But there are other, less global, changes that also occur via large scale motions, and Doherty's statement of the problem of identifying these transitions was aimed at discovering cooperative or nonlinear effects that cause such large scale motions. Definitive answers to these questions would contribute greatly to our understanding of protein structure, and particularly to our ability to predict tertiary structure and its longer time-scale behavior.

The issues that he raised were intended to provoke thought among the mathematicians and drug designers about how further improvements in solutions to nonlinear mathematical systems might shed some light on current problems in drug design. To motivate such thought, an example was given of some of his studies that used direct molecular dynamics simulation to search for nonlinear motions that have long been postulated to be characteristic of the so-called "rotator" pre-melting phases in n-paraffin crystals (Doherty, D. C., Hopfinger, A. J., Phys. Rev. Lett. (1994) 72 (5), 661). These "sine-Gordon" soliton waves arise from the solutions to a set of nonlinear differential equations. The fact that these equations have an analytical solution (a traveling arctangent waveform) provides a unique opportunity to search for these cooperative motions using direct simulation. This work appears to have definitively identified these high amplitude, localized waves in the molecular dynamics simulations, even though there is no explicit term in the molecular mechanics force field that specifically gives rise to them -- which is indicative of the cooperative effects of a collection of multiple chains.

This is an exciting development in our understanding of cooperative effects in molecular phase transitions. So, to the attending mathematicians one might ask, "What are the current developments in solutions (analytical or simulated) to molecular dynamics equations for complex systems that may be of some use in our understanding of cooperative processes in biochemical systems?" And one might ask the computational chemists, "What are the important questions that might move us along to a better understanding of cooperative effects, through some combination of mathematics and direct simulation?"

After the panelists' presentations and the ensuing discussions, additional challenges were put forth by other workshop participants. It was noted that few of the issues discussed to that point related to what has become known as `direct methods' -- techniques such as docking, de novo design, ligand-binding scoring, and so on, that can be applied when the target protein is known in structural detail. Such detailed knowledge is becoming much more common, in part because of the rapid emergence of genomics-based targets. In many pharmaceutical companies, all new discovery programs are molecularly-based, and they follow the continuum outlined by Graham Richards. Genomics is used to select the target, which is then further developed through cloning, expression, folding, and purification of the relevant protein(s) to produce molecular (and generally, high throughput) assays of small-molecule activity. With protein on hand, structural biology (X-ray crystallography, NMR structure determination, antibody epitope mapping, and so on) can provide the structural detail needed for application of direct methods to the task of lead discovery, or at least to lead optimization. (An "epitope" is the surface region of another molecule that is recognized and bound by an antibody. A "lead" is a compound with the sought for bioactivity, but which requires further optimization, for example to improve its bioavailability or eliminate certain side reactions, in order to become a useful drug.)

A number of the important issues in this area were discussed in the main workshop presentations during the week. One of the most pressing issues was improved methods for dealing with the physics and physical chemistry of water, sometimes through explicit consideration of solvation effects during simulations or binding energy calculations, but more typically through continuum solvation models. It was stated by one of the participants that we can generally do quite well now in predicting geometries of potential drug molecules at a binding site, but not so well in accounting for their solvation effects on the binding energy or even for the intrinsic components of the binding energy. Accurate calculation of ligand/protein binding energies (including entropic considerations) was generally viewed as one of the most important current needs, as it pervades all of the direct-methods techniques for studying and predicting the activity of small-molecule analogs. It was also noted that, in order to remain within the design cycle, molecular modelers must choose their methods, in part, on the basis of speed, to provide quick turn-around of results. This places a greater burden on finding energy-based methods or scoring functions that are accurate and fast. The need for speed is most evident in the effective application of automated de novo design methods, where tens to hundreds of thousands of candidate structures must be generated, evaluated (by some form of scoring function which assesses goodness of fit to the target), and assigned a priority ranking. Of course, when opting for methods that are fast, one must always question whether the reliability of the results warrants the use of the faster methods.

Another participant went further and pointed out that many of the currently available methods typically produce huge lists of seemingly reasonable molecules. However, only a few can be chosen for synthesis. Ideally a free energy scoring function would provide a basis for ranking and selecting the structures. Despite much progress, this remains an unsolved problem. Although the list can be reduced, one cannot with confidence pick only the top ten or so compounds for further study. Until the perfect scoring function is developed one must sample the reduced list in some rational manner. Experimental design methods are needed to address this. This is an opportunity for closer collaboration between the computational chemistry community and the statistical and applied mathematics community.

Donald G. Truhlar from the University of Minnesota (Chemistry and Supercomputer Institute) was one of the organizers for the workshops on "Evolutionary Algorithms" and "Mathematical and Computational Issues in Drug Design." He together with W. Jeffrey Howe, Anthony J. Hopfinger, Jeff Blaney, and Richard A. Dammkoehler write the following foreword for the IMA Volume on "Rational Drug Design."

Drug research and discovery are of critical importance in human health care and are becoming increasingly expensive, while the need for new drugs is also increasing. Computational approaches for discovery and optimization of drug leads have proven successful in many recent research programs. (A "lead" compound is one with sought for bioactivity but which requires further optimization, for example to improve its bioavailability or reduce certain side reactions, in order to become a useful drug.) Methods for drug lead discovery and optimization have grown in their effectiveness not only because of improved understanding of the basic science -- the biological events and molecular interactions that define a target for therapeutic intervention -- but also because of advances in algorithms, representations, and mathematical procedures for studying drug processes. In order to promote the interaction of mathematicians, computer scientists, and chemists to further the progress in the field and alert researchers to the opportunities for interdisciplinary research, the University of Minnesota's Institute for Mathematics and Its Applications and the University of Minnesota Supercomputer Institute sponsored a Workshop on Rational Drug Design on the Minneapolis campus, April 7-11, 1997. The workshop was devoted primarily to mathematical and computational issues in drug design. This volume contains the proceedings of that Workshop.

The workshop brought together top researchers in computer-aided drug discovery, computational chemistry, mathematics, and computer science to present state-of-the-art research in both the science and the underlying mathematics and to identify new problems for possible collaborations. General subject areas of the workshop included receptor-based applications such as binding energy approximations, molecular docking, and de novo design; non-receptor-based applications such as molecular similarity, conformational analysis, and structural diversity; molecular dynamics simulations; and solvation issues related to partitioning of a solute between aqueous and nonpolar media. The workshop focused on the mathematical procedures and algorithms upon which the scientific applications are based.