Thanks to Enrique.

Oh, I got a message.

This thing is recorded.

Okay, that's fine.

Thank you, Marius.

And also thanks to Enrique for inviting me to give this talk.

It's actually very nice to meet the group in Erlangen and people in Germany and in other

places.

I just got a message from Caroline.

So see you, Caroline, online as well.

So I'm going to today give a bit of an overview talk.

So I will not get into too much detail.

I mean, I had a discussion with Enrique and also with Marius about this.

So I thought it was better to give an overview talk rather than getting into too much detail.

I'll be happy to stay around after the talk or we can exchange emails afterwards.

So I will talk about applications of fractional operators and I will touch on applications

of fractional operators in deep learning later on in my talk.

So this work is supported by National Science Foundation and also Air Force Office of Scientific

Research and Department of Naval Postgraduate School located in Monterey, California.

So why do we care about fractional operators?

So I will start with a very simple application.

Actually it's not but it's still one of the most challenging problems, especially from

the numerical point of view.

Let's say you are given a noisy image F and you are trying to reconstruct U which does

not contain any noise.

So that's what the energy functional in the first bullet represent.

We are trying to make sure that U is close to F and also we want to get rid of the noise

by adding some regularization which here is the total variation semi-norm, formally speaking

the absolute value of NABLA U.

So this model was introduced by Rudin, Osher and Patami in 1992 and it's still one of the

most widely used models in imaging science.

In image genoising also in other types of imaging applications.

But the challenge with this problem especially from numerical point of view is if you try

to write the Euler-Lagrange equations for this problem and they are non-linear and in

general.

So a couple of years ago jointly with Soren Bartels from Freiburg in Germany we replaced

this total variation semi-norm by delta S over 2 where this delta S is the fractional

laplacian.

S is between 0 and 1.

And now if you write Euler-Lagrange equations in this case what we arrive at is a linear

equation delta SU plus alpha U minus F equals to 0.

Since images are typically given on rectangular or square type domains so one can use actually

standard Fourier transform to solve this equation in MATLAB it's just five lines of code.

Very simple and it's not limited to one dimension or two dimension you can go to higher dimensions

as well.

So let me show you an example.

So let's say you start with a noisy image which is the second one that's given by F

let's say and then you solve this fractional PDE and in this case I randomly chose a parameter

S to be 0.4.

We get a reconstruction which is the third image that you see.