March 10, 2000

 

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ALEXANDRE CHORIN
BERKELEY, CA ó For more than 30 years, Dr. Alexandre Chorin has worked to develop computational methods for solving problems in fluid mechanics, with the hope that they will eventually lead to an understanding of the most difficult problem of applied mathematics -- the problem of turbulence.

What makes the turbulence problem so compelling, in addition to its practical importance, Chorin says, is that the basic equations that describe turbulence are well known and simple, yet their solutions are incredibly complex and the computing power needed to find them transcends any imaginable computer.

Chorin, who holds a joint appointment as a senior scientist at the U.S. Department of Energyís Lawrence Berkeley National Laboratory and a professor of mathematics at UC Berkeley, has developed a number of methods for studying fluid flow since the mid-1960s. This work has been absorbed into the mainstream of research, yet it has not been sufficient for solving the problem of turbulent fluid flow. Chorin and his collaborators continue to explore new ways of tackling the problem, and their work continues to be acknowledged as critically important.

Most recently, the Society for Industrial and Applied Mathematics and the Applied Mathematics Society honored Chorin earlier this year with the 2000 Norbert Wiener Prize, one of the highest distinctions in applied mathematics. The prize citation states that Chorin received the prize "in recognition of his seminal work in computational fluid dynamics, statistical mechanics, and turbulence. His work has stimulated important developments across the entire spectrum from practical engineering applications to convergence proofs for numerical methods..."

Chorin will discuss his work in an April 5 lecture entitled "How to Put the Guesswork Back into Computing." His talk will begin at 4 p.m. in the Wheeler Auditorium on the UC Berkeley campus.

From the 1960s to the present day, Chorin has led and inspired applied mathematicians everywhere to tackle the most difficult real-world problems and to make full use of the combined power of advanced computers and sophisticated mathematical analysis. In the process, he has done more than anyone else to create and shape the important discipline of computational applied mathematics.

Chorin points out that his recent work has been done jointly with a number of collaborators, noted mathematicians as well as rising-star students, and that his work incorporates their many contributions. Yet his leadership is also widely acknowledged; in addition to the latest prize, he is in his third year as Chancellorís Professor of Mathematics and has just been elected by the UC Berkeley Academic Senate as one of two Campus Research Lecturers.

And although he is gratified by the recognition of his past efforts, he is more interested in the work at hand: the new method of optimal prediction which he hopes will contribute to the solution of the turbulence problem, and scaling methods that explore hidden structures in turbulence through the analysis of experimental data.

Turbulence is the seemingly random behavior of a fluid which occurs all over nature, in the atmosphere, the oceans, the interior of stars and inside internal combustion chambers. Being able to accurately compute turbulence has a wide range of applications, from predicting tornadoes to designing aircraft, from improving industrial processes to building more efficient boilers. As an engineering student with a love of cars and airplanes, Chorin became interested in turbulence as it affects lift (the force that keeps aircraft aloft) and drag (the resistance of air to the motion of cars and planes). In short, turbulence has the power to make vehicles run -- or not run.

Turbulence provides Chorin with a problem that combines his interest in physics, mathematics and computation. Turbulence is extraordinarily mystifying, he says. It looks like a problem in physics, but it isnít. "We have known the equations of motion for nearly two centuries and in principle they contain all the information one needs, except we donít know how to extract answers from these equations," Chorin said. "The physicists have done their job; the mystery is mathematical."

In fact, Chorin said that scientists today are better able to explain the structure of stars than accurately predict turbulence in a liquid flowing through a pipe.

One may think that the way to solve the problem is through computation, as is the case with many other problems in fluid motion; but trying to determine the structure of a turbulent flow in a typical application may take as many as 10-to-the-power-80 mathematical operations -- assuming that the chaotic nature of the solution does not interfere too much and that a way can be found to gather and understand the output from such a computation.

The problem totally defeats any present-day supercomputer, as well as any currently conceivable computer, Chorin says. It is too complex, too long. However, it would be extraordinarily interesting to find a way to tame the complexity.

What makes the problem so complex, he says, is that it is a multi-dimensional problem, and that it contains many different scales of motion, like a weather map, which contain patterns that cover entire continents and patterns that affect small regions, maybe a street corner. All these motions are coupled and affect each other; none can be neglected when one tries to understand what is happening.

Chorin and his collaborators, including Grigory Barenblatt, who also holds a joint appointment at Berkeley Lab and UC Berkeley, have come up with two ideas which they believe could lead to some progress.

The first is a new attack on the problem of scaling -- the uncovering of relations between motion on different scales in a problem that contains many scales. Using new mathematical tools and recent experimental data, Chorin and Barenblatt discovered new scaling relations, some of which overthrew earlier assumptions that had been widely used in engineering and aeronautics work since the 1930s.

Scaling laws are examples of general, useful relations that can be used and understood even without a detailed solution of the equations of motion.

The second idea Chorin and his collaborators (in particular, Prof. Ole Hald of the UC Berkeley Mathematics Department, Prof. Raz Kupferman of Jerusalem, and Drs. Anton Kast and Doron Levy of Berkeley Lab) have been developing goes under the name of "optimal prediction."

The equations that describe turbulence are general principles, asserting such obvious truths as the conservation of mass and energy. What one tries to do on the computer is extract from these equations detailed properties of flows in specific situations -- for example, the weather on a given day or the flow in a specific combustion chamber. This goal is unachievable. However, before one undertakes a calculation one knows a lot of "prior information": general mathematical, physical and statistical principles provide information about what kinds of solutions are likely to appear, and scaling laws and sheer experience provide other hints.

The idea in optimal prediction is to lower one's sights: settle not for a full, certain solution, but for the most likely solution that can be found, given the prior information and a preset, maybe modest, amount of computation. This approach, which may have wide applicability in other complex problems, is being tried on turbulence, and while it has not yet succeeded, Chorin is hopeful.

This is clearly a work in progress, but if successful, optimal prediction could prove useful in research areas ranging from biology to economics -- any complex problem with some data, but far more unknowns than can be conventionally calculated.

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