Thank you very much for the invitation and for organizing such a nice seminar in these

kind of, let's say, crazy times and, yeah, keeping some kind of normality here.

I realized that my background actually fits quite well.

Maybe I have one more.

And so thanks everyone for joining and I hope families are safe and stay safe in the next

couple of months.

So I will talk about the inverse of transport.

So I was kind of happy when I thought that Gabriele is going to give the first talk because

I'm like, he's going to set the scene.

And so, yeah, thank you very much, Gabriele, for this excellent talk.

It kind of saved me a lot of work.

However, this kind of talk is now, as Martin already said, going to focus on the inverse

aspect of the problem.

And this is a problem that I started to work with, while he was still back at Warwick University.

He is now at Caltech.

And today I'm going to present upon request what the first outcomes of this project are.

So the question that I'm going to address is the following.

So it's a rather general problem.

So if we have some kind of input criteria and we know that I have a problem where the solution,

the output, depends on some kind of cost and optimal cost criterion, the question at hand

is what if I know the transformation from the input to the output, but I have no idea

or only partial ideas about the underlying cost criteria?

And we have already seen that there are several applications where you can kind of think about

that.

The first one is optimal transport, where it goes back to a very general problem that

we have two probability densities and we want to find the optimal transportation map according

to the cost criteria.

So we could, for example, know or observe the transportation map, but we don't know

the cost criteria.

This has applications in data science, but one could also go back and you can just have

the question of a linear assignment problem, which in general also fits into the optimal

transportation problem, but also applications, as I said, in matching or assignment problems

in economics following this criteria.

So the general outline of the talk is then I will give you a couple of problems where

you will run into what I call now inverse optimal transportation problems.

So the first one is actually some work which was done by Gabriel Pave and co-workers.

The second one also goes back to French, the French optimal transportation, it's the well-known

marriage market problem.

And then the third application is a problem that I have worked on, it's on international

migration problems.

And so the beginning, I will kind of set the scene and give you an idea what kind of problems

have been looked at in this context, what are the applications before turning to actually

what is the forward problem.

Since I'm going to talk about inverse optimal transport, I have to define what is the forward

problem and then looking in the last part at the corresponding inverse problem.

And we're going to solve this inverse problem using the Bayesian framework and then I will

present you some of those results.

Okay, so we've already had a very beautiful introduction to optimal transport, but I will

just kind of reiterate, I will set my notation here.

So my setting I will always consider two discrete probability vectors, P and Q, and some given