Good. Continue. So I am the control now?

Yes, you are.

You are the boss.

Okay. So first I want to thank you, Enrique, for your kind words. And I thank also Marius

for the organization. The crash course gave me this morning to operate Zoom.

So I will close the movie. I have several movies. I may not have a chance to show all of them

because the talk today will be mainly theoretical. So we are going to do some mathematics.

But mathematics with high potential for applications. So, okay, I have the control now.

Transfinite interpolation, that's something for some of you might be new. It was new for me the

first time I heard about this through my co-author, André Garron. André Garron is a professor of

mechanical engineering. He is interested in fluid structure interaction and all the numerical

software that goes along with such problems. So it starts with interpolation.

So the first thing that comes to my mind when I talk about interpolation is the Lagrange

interpolation. In fact, this interpolation is not due to Lagrange but to Waring. And it was not,

does not appear in any of the work of Lagrange except in the lectures he gave in 1795

at the École Normale. And I'd like to quote Lagrange. The method of interpolation is one of

the most genius and useful that astronomy possesses. And you echo that with Newton more than 100 years

before. We said about the problem, it ranks among the most beautiful of all that I could wish to

solve. That gives some nobility to the problem of interpolation. Now, just to fix the notation,

let me recall what the interpolation means for a finite number of points. So the notation,

you have the Euclidean space, scalar product, the dot, the norm, and the set of points at which

the data is given will always be denoted E in the stock. The function which assigns value to

each point will always be denoted small f. And we are interested in finding a continuous

interpolation that is a function on the whole space Rn such that it coincides with f on the set of

point E. Okay. Now, transfinite interpolation is when you go from finite set E to an infinite set

of points. But you want what is interesting are the sets which are structured. And by a structured

set, I mean a manifold. So we have the example here of a curve gamma, for instance. That's a structure

infinite set. In finite element methods, usually those curves are approximated by polygons in this

way. And you can imagine if you interpolate now on E, if you want to use finite interpolation,

you would need many, many points to model the curve. And same thing for the other curve.

And same thing if you're doing finite element methods, you will need many points, but here you

can restrict to the corners of the set. Now, this problem is not really new from the mathematical

point of view. And it has been solved by Enrich Teetzer in 1915. So it's really a continuous

extension of a continuous function defined on a cross subset of a metric space.

TJ proved the existence. And a few years after, Osduff gave an explicit expression

for the interpolation. The DE is the distance function right here.

So you could say that's it. You don't have to go young. But people in numerical analysis,

well, they want some interpolations which are easy to compute, which are economical and so forth.

I've never seen anyone implementing the formula of Osduff so far. But that's an open question.

Now, there are, when you look at this in this way, for instance, if you have an open domain

and you have a function which is defined on boundary gamma, when you solve the Laplace

equation with the Dirichlet boundary problem, you construct an interpolation of the function

F from gamma to omega. You can also use a vector function F. And then you can solve, for instance,

the equation of three-dimensional elasticity. And in fact, this has been used, for instance,

in shape optimization by the school of Thierry Nys and in imaging in Paris by Trouvé, the group

of imaging. In engineering, this is the pseudo-solid method. But of course, there is no explicit

formula for the solution of the vector PDE. And you have to solve a finite element problem

of big size and compact matrices. So that's not very desirable. If you can avoid it,

so much the better. There is another approach. It is the second layer potential via integral

equations. And probably this was the inspiration behind the transfinite mean value interpolation