IMA Outcomes

The IMA serves as a scientific melting pot, bringing together researchers from widely different fields or traditionally disjoint communities within a broad research area. As long-term visitor Govind Menon expresses it, “The best thing about my stay at the IMA was the ‘organized random conversations’. Since this sounds paradoxical, let me explain what I mean. I say ‘random’ because I spoke with visitors who I hadn't met before and whose work I didn't know anything about. By ‘organized’, I mean that I would never have met these people if the organizers hadn't invited them.” These unanticipated interactions are a crucial ingredient in the excitement of the IMA and, as the following two examples demonstrate, can lead to important new interdisciplinary collaborations.

Homeostasis in computer security.

The language of computer security is highly biological—systems are infected by viruses and worms—and computer scientists look to biological systems for inspiration in combating such attacks. In fact inspiration flows both ways: biologists can learn about the immune system by studying some forms of artificial complex systems such as computer networks. In 1998, biomathematician Lee Segel and computer scientist Stephanie Forrest brought immunologists, mathematicians, and computer scientists together at the IMA for a period of concentration entitled Forging an Appropriate Immune Response as a Problem in Distributed Artificial Intelligence.

Forrest and her group had already experimented with mimicking immunological mechanisms in computer intrusion detection programs. But these programs, which discover invaders by identifying anomalous patterns of activities, are far from perfect: in a perpetually changing computer environment, false positives are inevitable. The limited repertoire of responses then available (most of which were severe and irrevocable), meant that most intrusion detection systems required a great deal of human oversight and/or exacted a large cost in system usability.

For Anil Somayaji, a Ph.D. student working with Forrest, inspiration came when he met Erin O'Neill, a Ph.D. student in immunology from Australian National University at the IMA. O'Neill explained that the immune system is primarily homeostatic rather than destructive. Homeostasis is the physiological process by which the internal systems of the body are maintained within normal ranges despite variations in operating conditions. The body achieves homeostasis, for example, through its reactions to temperature variations, physical and psychological stresses, and microbial invaders. For the latter, containment and regulated coexistence is often the best strategy: drastic countermeasures are rarely called for.

Basic flow control in pH.

An understanding of the immune system's role in biological homeostasis led Somayaji to modify the goal for computer security systems from one of destroying a perceived intruder when it is first recognized, to one of adapting the system to maintain stability. This idea blossomed into his Ph.D. project over the next several years. In his disseration, entitled Operating Systems Stability through Process Homeostasis, Somayaji developed a Linux kernel extension, called pH, that maintains system stability by slowing the response to system calls when abberant behavior is detected.
 

Somayaji's pioneering concept of process homeostasis is proving its worth in the computer industry. The ideas behind pH have been commercialized at Sana Security, founded in 2000 by Steven Hofmeyr, another participant in the IMA workshop, and a collaborator of Somayaji's. They have also been incorporated in the software program Virus Throttle released by Hewlett-Packard Co. in February 2005. Somayaji, now on the faculty of Carleton University, is currently researching the application of homeostatic approaches to managing network traffic.

Breaking down scientific barriers within mathematics.

Many of the complex physical phenomena in the world around us can be modeled mathematically by systems of partial differential equations (PDE). Most PDE systems are far too complex to solve with paper and pencil, so instead scientists and engineers employ algorithms which allow them to approximate the solutions on high performance computers. The development and certification of algorithms which can be used to provide efficient and accurate solutions to the immense variety of PDE which arise in applications is a crucial and difficult challenge has engaged many mathematicians, scientists, and engineers around the world for the better part of a century. Vast progress has been made—the speed up obtained in this period through improved algorithms is every bit as great as that obtained through faster computers—but the need for fast, robust, and flexible numerical methods promises to challenge researchers for decades to come.

Because of the number of researchers involved, the variety of methods employed, and the mature and highly technical nature of the field, ideas obtained by researchers working on one class of methods of numerical PDE often take a long time to percolate to other researchers working on different methods. Important results may not reach the scientists who need them most. The IMA was designed to break down just such barriers.

Figure of mesh elements
from paper of Brezzi et al.

In May 2004 the IMA hosted a workshop entitled Compatible Spatial Discretizations for Partial Differential Equations, designed to foster communication among diverse groups of researchers developing, applying, and analyzing numerical methods that mimic fundamental geometry properties underlying the PDEs. At that meeting Franco Brezzi of Pavia, one of the world's leading researchers in finite element methods, met Mikhail Shashkov and Konstantin Lipnikov of Los Alamos National Laboratory, top scientists in the area of finite difference methods (and, as such, members of a scientific community nearly disjoint from Brezzi's). Intense and sometimes heated discussions at the IMA led to a collaboration which quickly bore fruit. Their joint paper on the analysis of mimetic finite difference methods on unstructured curvilinear polyhedral meshes is featured in LANL's Mathematical Modeling and Analysis Research Highlights series, and has been submitted to SIAM Journal on Numerical Analysis. It is the first paper on finite difference methods in Brezzi's four decade career.

The collaboration between Brezzi, Shashkov, and Lipnikov, like others forged at the IMA 2004 workshop, continues. In response to the success of the workshop and the enthusiasm of the participants, the Centre of Mathematics for Applications in Oslo hosted a follow-up workshop on Compatible Discretizations for Partial Differential Equations in September 2005.