Interaction

Winter 2007

Numbers: The most basic building blocks

L.A. Cicero

Jeremy Kozden, left, and colleague David Gleich, who both had interdisciplinary undergraduate careers, are among the ICME graduate stdents working on athe Computational Consulting website. One of their biggest projects thus far was helping the Library of Congress with a massive digitization project.

You may not know it, but it is likely you need the services of a computational mathematician. Lucky for you, you’re at Stanford.

The Institute for Computational and Mathematical Engineering (ICME) was launched in 2005 after several previous incarnations. Said its director, Peter Glynn, “It’s almost hard to think of a part of the university that is not impacted by computational math.”

Engineering has always had two pillars: theory and experimentation. Computational mathematics—the result of the dizzying increase in computers’ ability to compute—has now created a third pillar, uniting the other two. Modeling and simulation are now possible to such a degree that they play a role equal to that of theoretical math and hands-on experimentation.

“We’re interdisciplinary; we do research that’s usable and that creates links between engineering and math,” said Margot Gerritsen, a faculty member in the School of Earth Sciences who is on ICME’s steering committee. “So it’s not that we’re spreading out; we reach out. We’re almost like a service department.”

It was in that spirit that Gerritsen worked with students to set up a Computational Consulting website (http://icme.stanford.edu/consulting/csquared/). Questions come from all over: graduate students, professors, industry professionals, even the Library of Congress. Consultations also can be face-to-face, said third-year students Jeremy Kozdon and David Gleich, who pointed out that they don’t always solve people’s problems, they mostly just put them on the right track.

L.A. Cicero

One of the petroleum engineer Margot Gerritsen's previous projects involved developing a computer code capable of tracking massive internal waves that begin on the ocean floor.

ICME’s origins lie in the Scientific Computing and Computational Mathematics (SCCM) program, which began in 1988. The story is not a simple one. The Department of Computer Science (CS) at Stanford was founded in 1965 by people who were primarily mathematicians. At that time, there were few CS departments in the country, and graduate students came from a variety of disciplines. Over time, CS as a discipline grew closer to electrical engineering than to mathematics, as a result of which the department moved into the School of Engineering in 1986.

“Originally,” said Walter Murray, professor of management science and engineering, “computers were designed and used by mathematicians to compute. But at some point, computer science became a subject in and of itself, devoted to the essence of the computer, and math was no longer a big part of its core. So, non-CS people were coming to Stanford to study computational math in the Computer Science Department, where they faced comprehensive exams in subjects such as hardware and artificial intelligence in which they had little interest and no knowledge.”

In other words, he said, Stanford was losing graduate students who wanted to focus on computational mathematics. So SCCM was established in large part as a place where graduates from a variety of disciplines could study. For over a decade, the program produced stellar master’s and doctoral students.

But according to Murray, it was always a struggle. The program relied mainly on faculty volunteers and it was under-funded, a challenge to even a mathematician.

So John Hennessy, who at the time was dean of the School of Engineering, formed a committee to figure out a long-term solution. Several were proposed, including folding the program back into CS. An appendix to the committee’s report, written by Murray, proposed that computational mathematics form a department of its own in which the teaching of mathematics to engineers at both the graduate and undergraduate levels would be added to the research agenda of SCCM. The idea found support among a broad range of faculty members, and, after conversations with Hennessy’s successor as dean, Jim Plummer, SCCM was disbanded in 2004-05 and morphed into ICME.

Engineering’s backbone

Everyone involved in the process agreed that the intellectual integrity of computational math called for a degree of institutional independence.

“Computational math’s importance to modern engineering has increased incredibly in recent years,” said Gerritsen, whose Stanford PhD is in SCCM. “Breakthroughs in engineering often originate with mathematical modeling, with fundamental breakthroughs in computational math. It is the backbone of computer-based engineering. You can see it in advances in bioengineering, the development of new tools for surgery, in visualization, in modeling medical problems. There’s a symbiosis between pure and applied math, and at ICME we strive to safeguard the math.”

Gerritsen is a faculty member in the Department of Energy Resources Engineering, which loans her out to ICME every fall to teach CME 200, a course on matrix algorithms for graduate students in engineering and earth sciences. She is an expert in fluid mechanics in subsurface oil reserves, specifically in enhanced oil recovery, which means more efficient and ecological ways of getting oil out of the ground. To do that, she creates models of the fluid flow in underground oil deposits.

Many members of the engineering faculty teach ICME courses, said Glynn, the Thomas W. Ford Professor in the School of Engineering. Some take on extra classes, while others do it within their normal load. (Like all interdisciplinary programs at Stanford, ICME cross-lists its courses.) Plummer “is very aware that this problem needs to be addressed,” Glynn said, adding that he hoped a mechanism can be found to allow faculty, including those from the School of Humanities and Sciences (home to the Statistics and Mathematics departments), to get full credit for teaching ICME courses.

But on the other hand, he said, there are advantages to the institute’s in-between position.

“Computational math at Stanford is quite unique,” he said. “We take advantage of what the university is good at. And as an institute, we have the ability to take advantage of our strengths in many, many departments. The asset here is the recognition that computational math cuts across so many parts of the university. An entity like ours has the potential to be flexible and rapidly adapt to the new applications that drive the discipline.”

As with other interdisciplinary centers or institutes at Stanford, leaders envision themselves somehow serving every part of the university and thus do not necessarily wish to be confined to one school. Thus Glynn, whose operations research PhD is from Stanford, is thinking very broadly.

“We want to be the go-to place,” he said. “The biggest challenge is how to reach everyone, and we’re working hard to address that. Right now it’s mostly by word of mouth, but a year or two from now we’ll be further along.”

Gerritsen thinks in similar terms, saying her dream is that ICME develop into something along the lines of the Woods Institute for the Environment, a place with links to virtually every department and discipline on campus.

The question of billets

In fact, when ICME got off the ground, it stepped right into the vanguard at Stanford with a plan for split billets similar to those at the Woods Institute.

When Charbel Farhat arrived at Stanford from the University of Colorado, he was supposed to have an appointment split between ICME and the Department of Mechanical Engineering. However, things turned out differently.

“We considered very carefully the appointment of joint faculty between institute and department,” Plummer said. “It had never been done before. But there was pushback at all levels, and eventually we said, it’s just too hard. Stanford still hadn’t figured out how to do that. So we backed away.”

Though the Woods Institute ended up being the pioneer of joint billets, “we had tested the waters,” Plummer said. “They took our idea of joint appointments and said, ‘We’ll try it.’”

As with many good ideas, ICME’s plan foundered on its novelty. The split billets vanished, and the institute’s leaders put their energy elsewhere, which they say is reaping excellent results.

“Stanford is transitioning toward an interdisciplinary model of teaching and research, and at some point we’ll have the mechanisms for making these appointments,” Glynn said. “In the meantime, we are focusing on building a world-class program that fully leverages all the opportunities that already exist here.”

For Gerritsen, “the most important thing is to show we can create a research vision. At the moment, we’re going for very big research grants that will allow us to attract more researchers. Later on, we can return to the billeting discussion.”

The institute has around 35 affiliated faculty members. They hail from computer science, mechanical engineering, energy resources, mathematics, statistics, aeronautics and astronautics, electrical engineering, civil engineering and management science. As Glynn said, there are few areas that couldn’t benefit from the assistance of a computational mathematician, and the field embraces such disparate areas as national security and ports, fluid dynamics and public policy.

“Every discipline has what physicists call the ‘grand challenge problems,’ and our faculty work on those,” Glynn said. “Since the advent of the computer, the human imagination has been highly creative in developing new problem structures that require ever more computational ability. To address those problems, we need high-end expertise.”

The students who choose to enter such a dynamic, challenging and extensive field are, to quote Murray, “terrific.” ICME has around 100 graduate students in the master’s and doctoral programs. This year’s crop of around 25 master’s students come from eight countries and have backgrounds in bioengineering, computer science, physics, applied math and aerospace.

“Because ICME addresses a much wider range of issues than engineering departments, students must be protected in the first and second year,” Murray said. “They need more time to figure out what they’re doing. We don’t want them to arrive here knowing what they want to do.”

Kozdon, one of the students who operates Computational Consulting, studied physics and computer science as an undergraduate. He said that when he got his bachelor’s degree he did not know of a graduate program that would allow him to continue in both fields. By chance, he saw a journal article that mentioned Stanford’s new program.

“It’s tailored for the non-mathematicians, aimed more at engineers, scientists, economists,” he said. “And because we’re an institute, not a department, I can work with anyone. We don’t have to work for affiliated faculty. That’s the biggest appeal, the number of opportunities to do what I want.”

His colleague, David Gleich, had studied math and computer science, and he, too, wanted to keep doing both. Again, ICME allowed him to do so.

The institute’s leaders say they are very grateful for the generosity of the School of Engineering in making fellowships and other resources available to the incoming graduate students. But at some point, Glynn acknowledged, they’ll have to find alternative funding.

Building a large tool set

The financial protection for the students is essential. Right off the bat they are exposed to a multitude of concepts and courses, many of which may be entirely new to them. This year’s 10 incoming doctoral students all share one large office and, Glynn said, within a few weeks were working on problems together despite coming from different fields.

“At the basic level, they have the same skills, but it’s easier to learn as a group,” Murray commented.

The core courses cut across every methodological discipline, he added, and all their students can address a wide range of applied problems and know which of their many tools is the most appropriate. There are eight application areas, including aeronautics, computer science, mechanical engineering, statistics and a miscellaneous grab bag that reflects the rapidly changing nature of the field.

“We think they’ll be facing far more complicated problems in the future; the easy problems have all been solved,” Murray said. “The larger the number of models you’re exposed to, the more you can do. We give them a broad set of ideas so they’re equipped to find the appropriate concepts for applied problems. As they say, if you have a hammer, every problem is a nail. The larger your tool set, the bigger the picture of the world.”

ICME eventually will be on the ground floor of the new Engineering Center, symbolizing, Murray said, its usefulness to everyone. It is not yet clear which of the ICME faculty members will leave their current homes to move into the new building, but not all the core faculty will be able to move, Glynn said.

“Co-location is important, of course, but on the other hand, we have more than 30 affiliated faculty, plus all the loosely affiliated people, and we want to maintain that broad interaction among many people,” he said.

Aside from deciding where Stanford faculty will be located, the ICME leadership is working out the thematic research agenda for the coming years, which will also draw on visiting scholars.

“We envision that people will work together over a one- or two-year period, with focused talks and courses, and then they’ll disperse and take the knowledge back to their homes,” Glynn said. “This is a very exciting opportunity; people could change their entire research orientation on the basis of these encounters. Imagine a full year of meeting with people from biology, or wherever, and getting a complete understanding of the interface between the disciplines, and consequently reorienting one’s research. It would be like a mini-sabbatical.”

Such an opportunity, he pointed out, would be along the lines of what the Commission on Graduate Education proposed in its 2005 report as a means for encouraging cross-disciplinary collaboration. Indeed, Mark Horowitz, associate vice provost for graduate programs, suggested to the Faculty Senate in November that one of the next interdisciplinary summer institutes be devoted to computational math.

Making connections

Mathematician Gunnar Carlsson, who is affiliated with ICME, said he, for one, would embrace such a possibility. Thinking back on his own serendipitous path to engineering, he said he was lucky in knowing people who knew people.

“These things happen at Stanford because the atmosphere is good, but they happen at random. The question is, can we do more in a formal way? I think the [graduate] commission’s recommendations for faculty sabbaticals for cross-training is a fantastic idea. But we need to identify people and put them together with the right people. You can’t just browse ideas, you need focused browsing, you need people who can tell you what the important problems are.”

It is the usefulness of ICME that faculty members and directors return to continually, which gives them confidence that the program can only grow.

“It’s like computer science,” Gerritsen said. “At first it was not its own department either, but today, everyone knows it’s its own discipline. There are some people who say, ‘We’re engineers, we can develop computational algorithms ourselves, we don’t need that research.’ I ask them, ‘Why do we have a CS department? Everyone knows how to run a computer, how to write programs. But that doesn’t mean we don’t need a CS department to do the basic research.” ICME provides that basic research function, she said.

“At other universities, students sort of understand; they sort of apply. But we guarantee that Stanford will always have a good selection of fundamental courses at a very advanced level and that our students will be able to develop their own computational algorithms.”

Carlsson’s research direction took a decisive turn as a result of ICME. For a decade or so, he had been working on a pure problem regarding topology. He sensed there must be some application for the work, but he was unable to figure it out. Speaking about the problem one day to a friend in statistics, the friend recommended someone in psychology, who in turn suggested someone in computer science, and Carlsson was introduced to the world of computational math.

“I had been dreaming of this project for 15 years,” he said, “but it wasn’t until I spoke to people from engineering that I realized what it was about, what it could mean.”