Subscale Capturing and Its Numerous Applications

		       Professor Stanley Osher
	       University of California at Los Angeles

			      Abstract

In 1987, together with J.A. Sethian, we devised a new numerical procedure
for capturing fronts and surfaces. The method uses a fixed (Eulerian) grid
and finds the front as a particular level set (moving with a velocity
which originally was assumed to depend on the local geometry, but which
is now much more generally defined) of a scalar function which solves a
time dependent partial differential equation. The method has since 
been applied to an enormous number of topics, ranging from computer vision to
computational fluid dynamics to materials science and beyond. The technique
handles topological merging and breaking, works in any number of space
dimensions, does not require that the moving front be written as a function,
captures steep gradients and cusps, and is relatively easy to program.
The advection algorithms are often intimately connected with shock capturing
techniques devised in CFD. Also, the modern approach of PDE based image
processing is related to the above procedure. 

We shall describe the methods, present some new applications, and discuss
theoretical justification.