Graduate Education for Computational Science and Engineering

SIAM Working Group on CSE Education

1. Introduction

Computation is now regarded as an equal and indispensable partner, along with theory and experiment, in the advance of scientific knowledge and engineering practice. Numerical simulation enables the study of complex systems and natural phenomena that would be too expensive or dangerous, or even impossible, to study by direct experimentation. The quest for ever higher levels of detail and realism in such simulations requires enormous computational capacity, and has provided the impetus for dramatic breakthroughs in computer algorithms and architectures. Due to these advances, computational scientists and engineers can now solve large-scale problems that were once thought intractable.

Computational science and engineering (CSE) is a rapidly growing multidisciplinary area with connections to the sciences, engineering, mathematics and computer science. CSE focuses on the development of problem-solving methodologies and robust tools for the solution of scientific and engineering problems. We believe that CSE will play an important if not dominating role for the future of the scientific discovery process and engineering design.

It is natural that SIAM, as the society whose aim is to foster the computational and applied mathematics which is at the core of CSE, should play a role in the growth and development of this new discipline. The objectives of this report are to attempt to define the core areas and scope of CSE, to provide ideas, advice and information regarding curriculum and graduate programs in CSE, and to give recommendations regarding the potential for SIAM to contribute.

2. Definition of CSE

2.1. What is it?

CSE is a broad multidisciplinary area that encompasses applications in science/engineering, applied mathematics, numerical analysis, and computer science. Computer models and computer simulations have become an important part of the research repertoire, supplementing (and in some cases replacing) experimentation. Going from application area to computational results requires domain expertise, mathematical modeling, numerical analysis, algorithm development, software implementation, program execution, analysis, validation and visualization of results. CSE involves all of this.

One point we would like to emphasize in this document is that CSE is a legitimate and important academic enterprise, even if it has yet to be formally recognized as such at some institutions. Although it includes elements from computer science, applied mathematics, engineering and science, CSE focuses on the integration of knowledge and methodologies from all of these disciplines, and as such is a subject which is distinct from any of them.

2.2. What is it not?

2.3. Research in CSE

Research in CSE involves the development of state of the art computer science, mathematical and computational tools directed at the effective solution of real-world problems from science and engineering.

3. CSE as an Emerging Discipline

3.1. CSE in science and industry

Although some researchers have been doing what might now be called CSE research for quite some time, for a number of reasons we appear to be at a critical juncture in terms of the role being played by simulation in science and industry. Historically, simulation has been used as a qualitative guide for design and control, but has often not been expected to provide accurate results for realistic physical systems. Increasingly, simulation is being used in a more quantitative way, as an integral part of the manufacturing, design and decision-making processes, and as a fundamental tool for scientific research. Problems where CSE has played and is expected to continue to play a pivotal role include:

CSE tools are critical in the exploration of scientific areas such as astrophysics, quantum mechanics, relativity, chemistry and molecular biology, where experiments are difficult and expensive if not impossible [9], and in analyzing the reams of experimental data and developing models in emerging areas such as:

3.2. Enabling technologies for CSE

Growth in the expectations for and applications of CSE methodology has been fueled by rapid and sustained advances over the past twenty years of computing power and algorithm speed and reliability (see diagram below), and the emergence of software tools for the development and integration of complex software systems and the visualization of results. In many areas of science and engineering, the boundary has been crossed where simulation, or simulation in combination with experiment is more effective (in some combination of time/cost/accuracy) than experiment alone for real needs.

4. CSE Graduate Education

4.1. Educational Objectives

In addition to a background in mathematics and computer science, a CSE graduate must have a thorough education in an application area (engineering discipline or science). The CSE graduate's mathematical knowledge will be sufficient to model technological and scientific problems. Knowledge of computer science, and in particular numerical algorithms, software design and visualization, enable the CSE graduate to make efficient use of computers. A graduate knows how to find and exploit software (-packages) for a certain task. A CSE student performs interdisciplinary work in mathematics, computer science and an application area. A CSE graduate is trained to communicate with and team with an engineer or physicist and/or a computer scientist or mathematician to solve difficult practical problems.

4.2. Curriculum

4.2.1. Relevant background

We describe below the background that a well-prepared undergraduate will have had before entering a CSE program. It is expected that many students will be strong in several of the areas but may need to acquire expertise in other areas during the initial stages of graduate study.

1. Mathematics and computer science

2. Application area: basic knowledge in an application area such as

4.2.2. Core areas for graduate curriculum

1. Mathematics and computer science

We note that in many institutions there is much redundancy and overlap between courses for example in numerical analysis or applied mathematics being taught in various engineering departments and mathematics. It may be advantageous to have a single core track, perhaps adding special sections for discipline-specific material if that is deemed to be necessary. This has the obvious advantage of cost-effectiveness, enabling even relatively small institutions to start a CSE program. Perhaps even more importantly, it gives the students a common educational foundation for the more advanced courses, and exposes them to other students and faculty with a wide variety of interests in computer science, mathematics, science and engineering.

2. Application areas. It is absolutely essential that interdisciplinary collaboration be an integral part of the curriculum and the thesis research. Courses should include projects and presentations whenever possible. A CSE graduate should have working knowledge in an application area like

Interdisciplinary collaboration can be accomplished via participation in a multidisciplinary research team and/or internship at a National Laboratory or in industry.

4.3. Graduate Degree Programs and Models

There is a list of graduate degree programs that are in place or in progress.

Two general models for the organization of CSE graduate degree programs have emerged. In the first model, a graduate degree is awarded in the new discipline of CSE. Often in this model the CSE program resides within an existing department, usually mathematics or computer science. In the second model, graduate degrees are awarded in the traditional disciplines of mathematics, computer science, science and engineering, with an area of specialization of CSE. The CSE programs residing in different departments usually share a core curriculum and a basic set of standards for graduation with the CSE specialization. Here we describe several existing programs which illustrate the two models.

4.3.1. MS/PhD in CSE

CSE at Stanford University. The Scientific Computing and Computational Mathematics Program at Stanford University was established in 1987 and is a graduate degree (M.S. and Ph.D.) awarding unit comprised of faculty from a variety of departments, including Mathematics, Computer Science, Operations Research, Statistics, Chemical Engineering, Mechanical Engineering and Electrical Engineering. It was established in recognition of the need for graduate-level training at the intersection of the disciplines of mathematics and computer science, which at the same time draws on applications of fundamental scientific or technological importance.

The core courses, which all M.S. and Ph.D. students must complete, are two year-long sequences, one in Numerical Analysis the other in Methods of Mathematical Physics. In addition, both M.S. and Ph.D. student must complete a year of courses in a focused application area, a year of courses in Computer Science (Parallel Computing or Data Structures and Algorithms are common choices) and further courses in Applied Mathematics and Numerical Analysis. The choice outside the core sequences is very broad, allowing the flexibility desirable in a Program of this scope, while the core sequences ensure a sound intellectual basis for the Program and a commonality among the students.

Ph.D. work falls into two broad categories: theoretical studies of computational algorithms for the solution of problems in applied and computational mathematics, and the development and application of novel software for the solution of problems of scientific or engineering significance. Students in the former category typically work with faculty from Computer Science, Mathematics, Operational Rerearch or Statistics while those in the second category work with advisors rooted in specific application domains; occasional joint supervision of research also occurs and is encouraged.

Of the Ph.D. graduates of the Program a significant proportion (roughly 45%) have moved on to academic positions, typically in applied and computational mathematics environments. The remainder are distributed throughout government labs and industry.

CSE at University of Texas Austin. At the University of Texas at Austin, CSE is known as CAM (Computational and Applied Mathematics) but all six departments of the College of Engineering are strong contributors to the program. There are a number of special features of the CAM program at Texas that are noteworthy:

4.3.2. MS/PhD in traditional area, with specialization in CSE

CSE at University of Illinois. The purpose of the CSE Option at the University of Illinois is to foster interdisciplinary, computationally oriented research among all fields of science and engineering, and to prepare students to work effectively in such an environment.

Students electing the CSE Option become proficient in computing technology, including numerical computation and the practical use of advanced computer architectures, as well as in one or more applied disciplines. Such proficiency is gained, in part, through courses that are designed to reduce the usual barriers to interdisciplinary work. Thesis research by CSE students is expected to be computationally oriented and actively advised by faculty members from multiple departments.

CSE is administered by a Steering Committee composed of one representative from each participating department and chaired by the Director of CSE. All faculty members affiliated with CSE have regular faculty appointments in one of the participating departments. CSE is sponsored by the College of Engineering, but it is not limited to academic departments within the College. Departments involved include Aeronautical and Astronautical Engineering, Atmospheric Sciences, Chemical Engineering, Civil Engineering, Computer Science, Electrical and Computer Engineering, Materials Science and Engineering, Mathematics, Mechanical and Industrial Engineering, Nuclear Engineering, Physics and Theoretical and Applied Mechanics.

CSE is a cooperative effort among the participating departments. Each participating department has its own specific requirements for its CSE option, but all are similar and share the common goals of the program. Details are available by accessing the CSE website. [3]

Upon satisfying the degree requirements of the student's graduate department and the CSE requirements, the student is awarded a CSE Certificate signifying successful completion of the CSE Option. Students must first be admitted to one of the participating departments before enrolling in the CSE Option.

All CSE courses are cross-listed with several of the participating departments. CSE instruction includes both core and advanced courses. A sampling of CSE courses includes numerical methods, parallel programming, computer architecture, combinatorial algorithms, data structures, software design, computer methods in civil engineering, computational solid and fluid mechanics, finite element analysis, computational aerodynamics, computer simulation of materials, parallel numerical algorithms, scientific visualization and grid generation.

CSE at Purdue University. The CSE Program at Purdue University is an interdisciplinary graduate program that started in Fall 1995. It offers specializations in Computational Science and Computational Engineering in 17 departments across the schools of Purdue. Specializations are offered at both M.S. and Ph.D levels.

The CSE Program at Purdue provides students with the opportunity to study a specific science or engineering discipline along with computing in a multidisciplinary environment. The aim of the program is not to produce a student with parts of two degrees, but rather a student who has leaned how to integrate computing with another scientific or engineering discipline and is able to make original contributions in both disciplines. The expected course load and examinations for students in this program are roughly the same as for M.S. or Ph. D degrees in other disciplines at Purdue, with approximately one third of the course load and examinations from computing and two thirds in the student's home department. For students whose home department is Computer Science or Computer Engineering, one third of the course load will be from outside department application areas.

Specializations at the M.S. and Ph.D. level are offered by Agricultural Economics, Agronomy, Biological Sciences, Chemistry, Computer Sciences, Earth and Atmospheric Sciences, Electrical Engineering, Food Science, Industrial and Physical Pharmacy, Mathematics, Mechanical Engineering, Medicinal Chemistry and Pharmacology, Nuclear Engineering, Pharmacy Practice, Physics, Psychological Sciences, and Statistics. Students must be admitted both to one of these departments and to the CSE program. The degree is awarded in the home department with the specialization Computational Engineering or Computational Science indicated on the transcript. The program is administered by the CSE Graduate Committee with representation from participating departments. CSE requirements are tailored to the home department's requirements. Students are expected to have a strong interest in computation and its application to science and engineering. Their undergraduate training is expected to have given them a strong foundation in several areas of science, engineering and computing [12].

4.3.3. CSE programs in Europe

We discuss CSE programs in Europe separately from those in North America because of the differences in structure of the educational systems, although there is also much in common.

In Europe there are strong activities in CSE. For example, a curriculum was started in 1997, at the Royal Institute of Technology in Stockholm, Sweden. Interdisciplinary application oriented and problem solving curricula that take into account computer simulation have been introduced into the U.S. in recent years under the label Computational Science and Engineering (CSE). Related curricula called "Industrial Mathematics" or "Technical Mathematics" have been introduced earlier at various places, e.g., in Linz and in Kaiserslautern.

Diploma (M.S.) in CSE at ETH. At ETH Zurich, W. Gander (Computer Science), M. Gutknecht and R. Jeltsch (Mathematics) took up the discussion on CSE in 1995. They were supported by the Executive Board of the ETH Zurich which nominated a committee with two more members, W. van Gunsteren (Chemistry), and K. Nipp (Mathematics). In April 1996 this committee worked out a curriculum relying on existing courses at ETH, a cost neutral approach without a need for additional personnel. The concept for the curriculum CSE was accepted by the Executive Board ETH in July 1997 and the program began in October 1997. Instead of the usual 4.5 years (9 semesters) of diploma studies at ETH, the CSE curriculum takes only 2.5 years and builds upon knowledge acquired in the first two years of a classical discipline. Candidates for the CSE curriculum must take two years of basic studies in Mechanical or Electrical Engineering, Computer Science, Chemistry, Mathematics, and Physics at ETH, or elsewhere. Depending on their first studies the students have to close the "gaps" between those first two years of studies and the model undergraduate studies for the CSE curriculum. CSE graduates are able to work on interdisciplinary problem solving in an application area making use of knowledge in mathematics and computer science.

Outline of the Curriculum

 

Hours of Work Per Semester

Short Description

Core Courses

37

computational mathematical methods, knowledge in computer science, important application areas, case studies

Field of Specialization

10

work in one computational application area

Elective Courses

12

in addition to the field of specialization and core courses, also in a computational area

Two Term Papers

12

application oriented, computational, at least 180 hours of work

'Gaps'

12

basics for computation and applications

 

83

 

The core courses are mandatory. One field of specialization is to be chosen.

The core courses are: theory and numerical techniques of differential equations, optimization, parallel computing, computational quantum mechanics, computational statistical mechanics, software engineering, and visualization/graphics. The specialization fields are: physics of the atmosphere, computational chemistry, computational fluid dynamics, control theory, robotics and computational physics.

Further information is available at the ETH CSE website [6].

International Master Program in Scientific Computing at the Department of Numerical Analysis and Computing Science, Royal Institute of Technology (KTH), Stockholm, Sweden. KTH, founded in 1827 and the largest of Sweden's six universities of technology, has extensive international cooperation both in research and education. About 30% of the regular students spend one year of their studies at some university abroad and the KTH Master Programs were started to strengthen internationalization. The programs are given in English with tuition free of charge. A limited number of grants from the Swedish Institute is available to cover living expenses.

The International Master Program in Scientific Computing, hosted by the Department of Numerical Analysis and Computing Science, was initiated in 1996. It is open for students from all over the world with a BSc/BEng or equivalent with a very good background in mathematics and knowledge in numerical methods. Applicants are also required to be experienced in programming and to have good overview of at least one application area relevant to scientific computing, such as fluid dynamics, solid mechanics, or electromagnetics. The duration of the Program is one and a half years, with one year of courses and projects and half a year of thesis work. The first class was enrolled in 1997 with eight students. The program now attracts more than 100 applications, and 20 students from more than 10 countries were accepted to enroll autumn 1999.

In the first week at KTH, the last before the official term starts, the students are given an intensive course in the computer environment and MATLAB and C programming. After this immersion the term proper starts with lectures, labs, and project work. The regular courses are:

First semester

Second Semester

Third semester

The thesis work is started at the end of the second semester and completed during the third semester. Throughout the entire program there are projects within the courses and projects running through several of the courses.

Requirements for Graduation: To obtain a Master Degree in Scientific Computing at KTH the students must pass written exams on the six compulsory courses and at least three eligible courses. All labs and projects shall be completed and the thesis shall be presented orally in a seminar and as a written technical report.

More information about the program can be found on the web page [5].

Impact of educational structure for CSE education in Europe vs. North America. In North America every child goes to high school and graduates at the age of 18. After that most continue with a college education, and some continue their studies at a university. Selections are done by the colleges and universities themselves.

In Europe the selection of university students is often done at an earlier stage. Though there are great differences in the evaluation process, in principle schools of different intellectual level exist in parallel. To enter a school of higher level, an entrance exam has to be passed.

As an example we consider Switzerland. Nine years of school are mandatory. We distinguish three levels:

Level 1. Primary School: Nine school-years from age 7-16. Children who remain in this school will typically become factory workers or do some other less intellectually challenging work.
Level 2. Secondary School: The entrance exam takes place after the first four or five years of primary school. The school also terminates at the age of 16. Typically a graduate of secondary school will continue with a 3-4 year apprenticeship and become a craftsman like a car mechanic, an optician or a bookkeeper.
Level 3. Gymnasium: In secondary school (at the age of about 13) another exam can be taken to pass into the Gymnasium. This school takes another six years and finishes with an exam, called Matura, at the age of 19. Passing the Matura means being mature for an university and every university in Switzerland will accept the person as a student.

The Matura level differs from country to country in Europe. Only about 18% of the children in Switzerland pass the Matura. The European mean is about 30% with exceptions like France with about 50% doing the "Baccalaureat."

ETH Zurich offers a Diploma after 4.5 years of studies. This title is roughly equivalent to a M.S. degree in North America.

Persons who have completed secondary school and an apprenticeship have the possibility to take an entrance exam to one of the "Fachhochschulen" ("Universities of Applied Sciences"). These institutions may be best compared with colleges and their diploma after three years of studies is generally accepted to be equivalent to a North American B. S. degree.

4.3.4. Discussion: structure of CSE graduate programs

It is clear that two major models for structuring graduate programs in CSE have emerged, and that both have proven to be successful at various universities. When considering the creation of a new program, the model to be chosen and its chances for success will be highly dependent on local academic strengths and political considerations. There is much in common in terms of basic academic content between the two models.

Comparing the two models directly is difficult. We offer a few observations. The degree in CSE may allow a more focused and cohesive program, since this program is often implemented within a single department such as mathematics or computer science, or within a separate academic unit. On the other hand it may be difficult to achieve depth within this type of program, especially for the application areas, or to distinguish this program from applied mathematics.

Since CSE is an emerging area, employment possibilities for students with the CSE degree are not well documented. There is a strong feeling that the current climate is highly favorable toward interdisciplinary work in science and engineering. One argument in favor of the degree in traditional disciplines, with specialization in CSE, is that employers and in particular universities and industry may be more open to this more traditional degree structure.

5. Broadening the CSE Student Experience

It is absolutely essential that interdisciplinary collaboration be an integral part of the curriculum and the thesis research. There are a number of ways in which this can be achieved.

The NSF National Partnership for Advanced Computational Infrastructure (NPACIs) in the U.S. has programs in place to assist in CSE education. Two excellent online news magazines provide up-to-date summaries of the activities of the NPACIs: http://www.npaci.edu/npaci/online/ and the National Computational Science Alliance (NCSA) http://www.ncsa.uiuc.edu/. The focused education programs are summarized through the cooperative EOT-PACI (Education, Outreach and Training) Web Site for both PACIs http://www.eot.org/. There are Research Experience for Undergraduates (REU) opportunities coordinated through the EOT-PACI and details for applying are provided.

6. Role of SIAM

Recommendations regarding the potential role and contributions of SIAM in CSE education and research include:

1. Define the core areas and scope of this field.
2. Outline ideas for curriculum.
3. Examine SIAM's existing journals and determine whether there is a place for CSE research as we have defined it.
4. Hold conferences on CSE.
5. Establish a special interest group (SIAG) on CSE.
6. Create an electronic CSE Bulletin Board that would include the following: discussion forum; pointers to graduate degree programs; infrastructure for universities, government and industry to post internship opportunities and for students to post resumes; infrastructure for universities, government and industry to post job opportunities and for job seekers to post resumes
7. Publish information of use for CSE education, for example articles specifically oriented to teaching in CSE programs or courses.
8. Publish CSE textbooks.
9. Publish CSE research books.

Acknowledgments

We would like to thank our colleagues, Andrew Stuart and Gene Golub of Stanford University, Lennart Edsberg of KTH, Peter Arbenz and Kaspar Nipp of ETH Zurich, the SIAM Education Comittee and the SIAM Council and Board for their input and advice, and Laura Helfrich of the SIAM office for her assistance in creating this report.

Appendix. The SIAM Working Group on CSE Education

The SIAM Working Group on CSE Education was formed in November 1998 to study the recent developments in CSE Education and to give recommendations regarding SIAM's potential role. The Working Group was comprised of: Prof. Linda Petzold (University of California Santa Barbara, Chair); Prof. Uri Ascher, University of British Columbia; Prof. H. Thomas Banks, North Carolina State University; Dr. James Crowley, SIAM; Prof. Walter Gander, ETH Zurich; Prof. Leslie Greengard, Courant Institute of Mathematical Sciences, New York University; Prof. Michael Heath, University of Illinois at Urbana-Champaign; Prof. Andrew Lumsdaine, Notre Dame University; Dr. Cleve Moler, The Mathworks, Inc.; Prof. Tinsley Oden, University of Texas Austin; Prof. Robert Schnabel, University of Colorado Boulder; Prof. Kris Stewart, San Diego State University; and Dr. Anne Trefethen, Numerical Algorithms Group (NAG).

References

[1] Accelerated Strategic Computing Initiative (http://www.lanl.gov/projects/asci/).

[2] Chemical and Engineering News, May 1997.

[3] Computational Science and Engineering, University of Illinois at Urbana-Champaign (http://www.cse.uiuc.edu/links/index.html), MS/Ph.D Option handbook, M. T. Heath, Director, 1998-99.

[4] Colorado School of Mines, Robert J. Kee website (http://engineering.mines.edu/faculty/detail/?fid=72).

[5] Computational Science and Engineering, KTH Stockhom (http://www.nada.kth.se/kurser/master/index-eng.html).

[6] Computational Science and Engineering, ETH Zurich (http://www.cse.ethz.ch).

[7] International Union of Pure and Applied Chemistry (IUPAC) (http://www.chem.qmw.ac.uk/iupac/).

[8] National Workshop on Advanced Scientific Computing, National Academy of Sciences, James S. Langer, Chair, 1998.

[9] W. J. Kaufmann III and L. L. Smarr, Supercomputing and the Transformation of Science, Scientific American Library, 1993.

[10] Rensselaer Polytechnic Institute, Dept. of Biomedical Engineering, Robert L. Spilker website (http://www.rpi.edu/~spilker/).

[11] University of Colorado, Dept. of Chemical Engineering, Victor H. Barocas website (http://spot.colorado.edu/~chemeng/faculty/barocas.html).

[12] Purdue University, CSE website (http://www.cse.purdue.edu/).

[13] W. Schiehlen, editor, Multibody Systems Handbook, Springer-Verlag 1990.

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