Madame la Directrice, dear colleagues, dear students, ladies and gentlemen,
Stanley Osher is Professor at the University of California at Los Angeles. His discipline is numerical analysis and he has contributed to reshape this discipline.
Professor Osher's bibliography contains almost 200 papers. Summarizing them in less than 15 minutes is really not such an impossible task. Indeed, while Stanley Osher has dealt with just one problem, it just happens that this problem is actually the main problem of numerical analysis:
That is : how to represent, with a computer, continuous and infinitely accurate phenomena?
This problem has been a central concern for many mathematicians ever since computers were invented. The English mathematician Alan Turing, who invented computers, didn't exactly glorify computers. His model, the Universal Machine, is just a bit smarter than a typewriter. As you know, the computer obeys a finite and fixed set of instructions. The computer obediently reads sequences of bits. The computer obediently writes sequences of bits. That's all. The computer is just a reliable "bean counter".
All the same, Alan Turing, who is also the founder of artificial intelligence, planned in 1956 to simulate all natural phenomena with computers. Alan Turing was the first to write the partial differential equations modelling pattern formation in biology. And he was also the first to try them on a computer.
That a computer could be used to simulate natural phenomena was, and still is, far from obvious. Another founder of computer science, Howard Aiken wrote in 1956:
If it should turn out that the basic logics of a machine designed for the numerical solution of differential equations coincide with the logics of a machine intended to make bills for a department store, I would regard this as the most amazing coincidence that I have ever encountered.
To understand the challenge that this question still represents, we can take as an example the aliasing phenomenon, which appears in numerical television and in most digital video disks.
Think of what happens on a computer screen when you see a moving object. The screen is a discrete set of pixels, so the moving object has to jump from pixel to pixel. Like a frog! But, the physical motion is continuous. How is it possible to represent this continuous, almost infinitely precise motion on a finite grid? And how is it possible to represent very thin scale physical phenomena on the same rough grid? This problem has upset many mathematicians. It is the "discrete curse" of numerical analysis.
Stanley Osher found five solutions to the "discrete curse". He found ways to handle very precise geometric objects on the computer. We mathematicians call them curves, singularities, corners, junctions, and surfaces. In physics, they're called interfaces, shocks, fluids, bubbles, drops, multi-materials, patterns, crystals, thin films, self-intersecting wave fronts, and pulsating detonation waves. In biology and medicine they are simply called membranes or organs.
All in all, we are naming everything that has a shape or that takes a shape!
The numerical answers invented by Osher are very simple formulas called schemes or methods: Here are the main ones: "essentially non-oscillatory schemes, shock capturing schemes, threshold dynamics, total variation minimization and the level set method."
I'll explain one of these discoveries. But I should like first to mention that Stanley Osher has never practiced what we used to call "applied mathematics". This term, "applied mathematics" might be seen as a discipline of the second rank. "Applied" this would induce an inferiority complex!
Quite on the contrary, however, Professor Osher's numerical representations have changed for the better some mathematical misconceptions about basic mathematical objects such as surfaces and functions. Several very influential pure mathematicians of our time, Craig Evans, Mete Soner, Ennio De Giorgi, Yves Meyer, Pierre-Louis Lions,Yoshikazu Giga have worked directly on Stanley Osher's ideas.
Something else notable about Stanley Osher: his great skills in turning young researchers into accomplished inventors. Several, now famous, professors and engineers have been his students and collaborators.
The talent of these researchers is shown in the spectacular numerical simulations done by Ronald Fedkiw, professor at Stanford, which you see on this screen. These simulations show various implementations of the now-famous level set method invented by Osher and Sethian twenty years ago. This revolutionary method gives us a way to simulate the changes of the most intricate surfaces with high accuracy. The idea of the level set method is a Columbus' egg. All surfaces, even the most intricate ones can be represented by a simple function, easy to calculate. This function is the distance of every point of a Cartesian 3D grid to the surface. Using this method, the moving surface is represented implicitly as the zero level set of a function on a fixed grid. It is the simplest of all possible discrete representations. It is akin to the famous Newton method for solving equations.
Now, in contrast with Newton, who invented differentiation, the numerical analyst Stanley Osher is not fond of differentiation. He told me once: "never differentiate a function!" This is the main idea of the level set method: to replace a complicated derivative by its simple primitive.
By the level set method, fine characteristics of the surface evolution such as its topological changes and its very large velocities can be simulated accurately and effortlessly, by simple schemes on the distance function. This has immensely simplified the life of almost everyone doing numerical simulation.
Let me conclude.
Contrarily to the widespread idea, mathematicians weren't put on earth to make people's lives and studies uncomfortable. Stanley has in fact made life easier for tens of thousands of engineers, scientists, and doctors. Eventually, a longer life for their patients too! Thus, we are very happy to tell him "thank you"!