This is a joint work with a former master student, Vicente Okito, who is now in France,

Bordeaux, working under the supervision of Professor Toussaint that I see here in the

room. We have been interested in inverse problem, but not only in inverse problem,

but also try to combine this with a multi-scale. So we are going to try to retrieve information

under the effect of a micro-scale. Let me explain what that means. I have put here this figure,

because I think that it's a very nice metaphor to feel what is an inverse problem. So we have

a region or a domain or a material which is stimulated, for example, by a wave which is

coming from abroad, from an external wave, which we know. This wave interacts with the material,

with the medium, and the direct problem consists in knowing which is the resulting wave of this

interaction. So this is shown on the left figure. The inverse problem is something very similar,

but now what we know is the in-going wave, the outgoing wave, so we know which was the reaction

of the material under the stimulation, and we want to recover the medium. So at least it's

proper. We have no information about, we want to know who is the medium. Knowing which is the

in-going wave and which one is the resulting of this interaction between the in-going wave and

the medium. So it is normally inverse problems are quite challenging, because it's not easy to

recover to know which is the medium, knowing just information about boundary conditions or about

source term. Okay, we will do that in the context of heat conduction. So we will work with the

classical heat equation in a bounded domain of Rd. So we will assume that our domain omega is

fulfilled by a conductivity material with conductivity tensor, say, a of x. I will put

their epsilon because our conductivity material will oscillate also. We will be able to distinguish

a very fine scale inside the material, other than the macroscopic scale which will be of the size of

omega. So in omega we will have or we'll be able to distinguish two scales, one of the size of omega

and the second one which we call it, we will denote it by epsilon and which is going to be much smaller

than the other one. But in a first step, let us assume that epsilon is fixed. It is small but fixed.

So in practice we have only one scale but there are oscillations inside the medium. The medium can

oscillate at the scale epsilon, at the fine scale epsilon. So we are going to stimulate our domain

omega by a source energy or extended source F and under this stimulation the medium will attain a

temperature u epsilon which is a function of T and x which is the solution of a classical heat

equation. We tensor a epsilon of x and this is our unknown, a epsilon of x and what we know or what we will assume

that we know in order to determine a epsilon are measurements of the flux of u epsilon on the boundary d omega

and we will also assume that we know the solution u epsilon at a given time t equal theta. So here we have two main

information. We know the flux of u epsilon for different values of t and we know the solution at a given time t

equal theta and we would like to recover a epsilon. Now this information on the fluxes of u epsilon will be encoded in the

classical digital map. Well in this case let us call it source to flux map operator, the classical Stelkopf-Pancaré operator

which given F it maps the flux of the solution u epsilon over all over the boundary or just in a subpart gamma 0 of the boundary.

So we have here two big amounts of information. The first one is going to be this Stelkopf-Pancaré operator for different values of

t. We know it for t between tau 1 and tau 2 and we will know also the solution for a given time t equal theta and we want to recover a epsilon.

This is a very ambitious problem because there are plenty of unknowns. You can see in a epsilon in principle there are d square functions unknown.

So this is a little too much. So the first simplification we are going to do is assume that epsilon is symmetric. So in that case we will reduce our number of unknowns to d multiplied by d plus 1 over 2.

Even in that case the problem is extremely ambitious. So this is a starting point in trying to inverse a multiple coefficient operator.

In mathematics classical inverse problems just are able to inverse one coefficient. So in some way we are out of scope with this problem.

So what I need to do in order to start solving such a difficult challenge is to introduce several hypotheses on the set of tensors epsilon.

So for this problem I have much more questions than answers. So please excuse me for that.

I will not be disappointed if I have to introduce several hypotheses on the tensor epsilon in order to obtain at least one first result.

So before continuing the study of this problem let us consider two practical considerations in our inverse problem.

The first one is that when you measure the fluxes you need to stimulate the medium with the force F.

Normally this force F is distributed all over capital omega. But in practical situations you can access only a sub-domain small omega.

So in the following we will assume that F will be always compactly supported in a small domain small omega included inside capital omega.

On the other hand in experiments you cannot always measure the fluxes of the solution all over the boundary of omega just in one part of the boundary.

So here there are two practical considerations that we will carry on in all what follows.