Okay, so welcome everyone to this year online seminar. We have Professor Martin Berger from

the chair of Applied Mathematics 1 here at FRO and he will be speaking about adaptive

computation of sparse solutions. Please, Professor Berger, you have the floor.

Thanks a lot. Yeah, so I was, ah, it does not work to go on. Yeah, okay. So, yeah, I

was asked to talk about this topic today. It's a joint work mainly with Alexandra Kulluri,

who was a postdoc when I was in Munster and has now two half positions in Finland, I guess,

for the summer and in Basford winter. And also with Pia Heinz, who was a PhD student

a few years ago, then went to Bosch and is now in Hannover at the Technical University

or University of Applied Sciences. And in the second part, I will mention about a related

work recently with Pratim Pataturi and Uncle Berner at the DLR in Berlin and also Benjamin

Mormsten, who is actually, I think today has his last, his last day at FRO, was involved

there. So what we are interested in is, yeah, mainly motivated by imaging. There are some

other problems where you have similar issues. So we look for the resolution of some spike

or I would say delta peak structure. And this can be either in space or in time or in frequency

or maybe also in some, in some continuum or in some basis in an infinite dimensional Hilbert

space. So like in a wavelet basis or something similar. But mainly I will talk about really

a continuum. So I could think about either R1 as time or frequency or R2 or R3 for imaging.

And some applications that, yeah, we are particularly working on is one, the localization of structure

in super resolution microscopy. So in super resolution microscopy, you now get to resolutions

really where you see single proteins. So which are really like single points in the image.

Another interesting thing is peak detection in spectroscopy, in particular when it gets

from chemistry or physics towards biology, where you have very complicated spectra and

it's not so easy to pick some peaks by hand. And another maybe interesting problem is the

localization of brain activity with electric or magnetic recordings on the brain surface.

So it's like the typical brain activities at a certain time, but it's like a combination

of several, it's not really delta peaks, it's really a dipole. So it's really like a divergence

of the delta peak times a vector, but you can still formulate the source localization

is a estimation of some delta peaks. Okay. And as I said, like also in spectroscopy,

which is common, if you have only few or very few of them, like one, two, three that are

very pronounced, you can do some simple methods. So in spectroscopy people may pick peaks by

hand from the data or in eGMEG, this is called dipole fitting. So in very simple standard

experiments where you only have one or two sources in the brain, you could just make

an ansatz of a sum of two or linear combination of two delta peaks and then find the locations

and the coefficients explicitly. Okay. But if you get to more relevant things, then you

get into very complicated structures. So for example, you would have in microscopy or in

spectroscopy, you have the convolution of several or linear combination of several delta

peaks with some kernel. And then it's not so easy to find the peaks. And in brain activity,

if you think about real relevant diseases like epilepsy, you have a lot of brain activity

at the same time, and then you cannot just pick a few of them. And just as a side comment,

there is a paper by Bach and CoArthur who also relate this to the training of neural

networks. So if you want to train compact sparse neural networks with as few entries

as possible, kind of you have similar problems here. Okay. So let me start with the simplest

model and also the most frequent one in applications is just the convolution. So we have a forward

model where we have measured a function f, which is the convolution of some kernel G.

You can think of some Gaussian to make it easy with a density in this case, typically

even a non-negative density, but for my analysis that does not play a role. So in some cases,

you can also have signs in applications. And if you think for, look for spikes or peaks,

then as I said, you look for a mu that is a linear combination of delta peaks at certain

unknown locations psi L. Okay. And the total number, capital L of these peaks is also an

unknown. So you see it's a complicated problem to estimate. Okay. What you can do is an appropriate