Okay, thank you very much, Leon. Thanks a lot, Daniel, for the really nice introduction.

And thank you everyone for switching who switched on the cameras. It's nice to see you. And

actually, Josh, thanks a lot for advertising it in Cambridge. I can see lots of Cambridge

people here. Okay. Well, okay. So let me actually just start by thanking my group, my research

group, without whom nothing that I'm going to show you today would have been possible.

So they are doing some tremendous work, and I would like to acknowledge them. They're

great. And I would also like to acknowledge all the funders from government, from private

trusts and from industry who are supporting our research. And then, and I only do this

advertisement because it's topic is connected to the topic of this talk. I'd like to call

to anyone who is looking for a postdoc that we are looking for postdocs in mathematics

to join a relatively large research program. We have, or we are starting with the University

of Bath and with University College London on the mathematical foundations of deep learning.

So please do get in touch or I'll try to later paste this link in the chat if you're interested.

Have a look at this website where you can also find a link to the postdoc positions.

Okay. So what I would like to do in this presentation is to convey a couple of messages. First one

is I would like to convey really my fascination for deep neural networks and the capabilities

of deep neural networks that I have gathered, I would say in the last couple of years, and

in particular their promise and their potential for helping us to solve inverse problems for

two reasons. One is that they are very, very good in capturing structure and capturing

pattern information and data. And we can use that to our advantage to produce, to help

us produce highly accurate solutions to inverse problems. Accurate actually both qualitatively

as well as quantitatively as I will show you later. And the second reason is that for all

of us who have been solving inverse problems in the past with kind of maybe traditional

or statistical mathematical approaches, all of those will appreciate that mostly these

type of solutions are very, very costly to computationally achieve because they boil

down to very large scale non-convex, variational problems or non-linear PDEs that we need to

solve. And that is definitely in contrast to this an advantage of deep neural networks,

which once they're trained, it's an explicit application of linear and non-linear operations.

So once they're trained, they are cheap. I mean, apart from always a kind of memory bottleneck

that you have depending on how large your inverse problem is. So that's really the main

point I would like to convey. The other one is that deep learning doesn't work as a hammer

for inverse problem in the sense that you can't use off the shelf deep learning solutions

and just make them work for inverse problems if you really are interested in solving them

for practical problems where you need mathematical guarantees to get robust solutions and to

get solutions which generalize to the problem you're interested in solving to not just the

tiniest small data set that you're training with, but that actually have the potential

to solve your problem in general. So you need to combine, therefore, deep neural networks

with mathematical and statistical modeling. And in contrast to that, and this is where

all of us are called to change things, most of the approaches that are out there are actually

lacking mathematical scrutiny. And so this is something we need to change. And one attempt

to go, let's say, in the right direction, and I emphasize it's really one attempt, is

to combine deep neural networks with so-called variational models. And I will show you what

variational models are in a moment. And all of what I'm going to tell you about draws

a lot of the information and the knowledge that we have gained in writing this review

paper a couple of years ago with Simon Erich, Peter Maas, and Uso Nugtem.

So what is an inverse problem? And I'm mainly interested in inverse problems where the solution

is an image, so inverse imaging problems. So an inverse problem always appears when

what you're interested in is not directly what you're measuring, but the data is an

indirect measurement of the image, in our case, that we want to reconstruct. So just

putting this in one equation, y are my measurements, u is the image that I want to reconstruct.