MINNESOTA CENTER FOR INDUSTRIAL MATHEMATICS

Early in the Fall semester, Fernando Reitich, Associate Director, and Fadil Santosa, Director, took stock of the center in order to evaluate its impacts and identify areas where it could improve. We were pleased to discover that MCIM continues to play an important role in the graduate education of the students in our department. We estimate that nearly one quarter of our graduate students have taken advantage of the Center's internship program. Over 30 MS thesis which are closely linked to the students' internship project have been completed. Indeed, several of the recent PhD thesis topics have their origin in industrial internships.

It can be said that not only our students benefit from the internship and industrial collaborations, our industry partners also attest to the value they have gotten in their interaction with the Center. Indeed, many of the companies we have worked continue to take our students for internships year after year.

The impact of the center can be measured also by the broadening of the mathematical experience our students are exposed to. Beyond the internships the Center and the IMA jointly sponsors the Industrial Problems Seminar. In this series, a speaker from industry makes a presentation about his or her research and explains how mathematics are used to answer important questions that arise. Many of these issues are in product development, while others arise in the service sector. Nearly all our speakers spend time discussing with students in an informal setting about job opportunities and careers in industry.

Research in the department has also been enriched by the Center's contacts with industry. The accompanying article by Bob Gulliver, Fadil Santosa, and Jing Wang, gives an account of a research project that was started during an internship, and eventually lead to the filing of a patent.

A report on the Center, which goes deeper into the activities and accomplishments of the Center, and includes candid testimonials from our former students and industry contacts, is available upon request. Please send email to mcim@math.umn.edu to ask for a copy.

OPHTHALMIC LENS DESIGN: SEEING BETTER THROUGH MATHEMATICS

By Robert Gulliver, Fadil Santosa and Jing Wang.

In the summer of 1999, two representatives from a company that makes ophthalmic lenses, made a cold-call to Fadil Santosa, and arrange to talk to him. Fadil was puzzled about why a local lens company that he has never heard of would want to talk to him. It turned out that the company was looking for help in their lens development program, and wisely realized that sophisticated mathematics was the key to their quest. Fadil subsequently found out that they had talked to engineering professors on campus who told them that they should talk to the mathematicians.

Nevertheless, the result of this initial conversation was to set up an internship for Jing Wang, then third year graduate student in Mathematics, for the following summer. At the end of summer of 2000, Jing came back and told Fadil about the project he worked on, and his interests in further developing his research experience into a thesis topic. After many hours of discussion, it was clear that the problem was basically that of surface design, and involves a lot of geometry. It was also clear that we needed to recruit a geometer, so they convinced Bob Gulliver to be coadvisor to Jing.

The design of progressive ophthalmic lenses, which are worn by many adults over 40 years old, call for creation of an optical element whose power vary smoothly over different portions of the lens. More negative correction is usually needed in the far distance-viewing zone, and less is required in the near distance-viewing zone. Geometric optics is an accurate theory for describing the light bending phenomena. But upon further simplification, the problem can be reduced to that of prescribing mean curvature, which is proportional to power. While there are many ways to achieve the desired power distribution, for the purpose of minimizing aberration, it is desirable to have small principal curvature difference. In a nutshell, we want to create a surface over the xy-plane such that where P(x,y) is the prescribed progressive power distribution over the lens. The curvature difference is referred to as astigmatism, and is undesirable. If P is not constant, it is impossible to make zero everywhere. Thus, a compromise is necessary. The game is to balance how well the prescribed power is met with how small one can make the astigmatism in critical viewing areas. This "game" can be played by devising a cost function and optimizing it.

While this idea is not new, what is new in our approach is the idea to linearize the problem around a base surface and look for a perturbation. The resulting problem is a fourth order elliptic partial differential equation. We also devised an accurate and efficient numerical method to solve the problem. Jing defended his thesis in August 2001, and is now a postdoc at the IMA. The work we did resulted in two papers published in international applied mathematics journals. 

A screen shot from a lens design tool. Displayed on the left is the designed surface. Parameters of the lens are entered into the form on the right panel. A lens is produced almost instantaneously when the "Design the Lens" button is pushed.

When we showed our results to our contacts in the lens industry, we were encouraged to market the method. We then got in touch with Jim Hildebrand at the University of Minnesota’s Patents and Technology Management Office about our method. Jim hired a lawyer and we started working with him in writing a patent application. This in itself was a very interesting and educational process for us.

Jim is now busy marketing the technology to several companies. We are hopeful that it will start paying rolayties. By the way, the income distribution is as follows. A third goes to the University, a third goes to the patent authors, and the remaining third is divided between the department and the IT Dean’s office. While it’s too early to know if we’ll ever make money off this patent, the experience has been truly exhilarating.

We have learned a lot through our interactions with companies in the effort to market our technology. More questions and problems arise that need further research. These are challenging and interesting problems on which we hope to continue to work. Who knows, the next pair of glasses you buy, may be designed by us!