Okay, thank you, Philipp.

Thanks very much for the nice introduction and for having me here, for inviting me to

give a talk.

Can you hear me?

Yes.

Okay, good, good.

Okay, just to check.

So, it's a pleasure to talk about some research together with several collaborators here.

One is Daniel Rudolph from Göttingen.

The other collaborator is Claudia Schillings from Mannheim.

And the last one is also Philipp.

And we were investigating noise-level robust Monte Carlo methods for basin inference or

basin inverse problem with highly informative data where the posterior measure is then highly

concentrated.

Usually, informative data is, of course, a good thing in terms of inference or identification

problems.

But this high concentration of the posterior can also pose a serious challenge for naive

numerical methods.

And I will talk a bit about how you can derive robust methods, which also work quite fine,

even for very, very accurate or very large amounts of data.

So, let me introduce the basic setting, which we are considering, basin inverse problems.

We would like to infer an unknown x belonging for this talk to a finite dimensional space

Rd.

And we would like to infer it based on noisy observations of a measurable forward map from

Rd to Rk.

For these measurement noise, we assume here this additive model.

And suppose that the measurement noise is normally distributed with mean zero, particular

covariance matrix sigma, and here a scaling parameter n, a natural number, which steers

the level of the measurement noise in the data.

If we then have some prior information in terms of a prior measure, mu zero, which for

most of the time will be considered this mean zero Gaussian measure with C0 as covariance

operator or covariance matrix, we obtain by the basin approach a resulting posterior measure,

which we denote by mu n.

So, it depends in particular on this noise scaling parameter n.

And it's of this particular form.

So, you have here a normalization constant Zn, then e to the minus n phi.

That's basically the likelihood term phi corresponds for this Gaussian measurement

noise to this quadratic data mismatch function concerning the forward map G.

And then we have here again, the prior measure.

What we would like to compute or what we would like to do is to sample approximately from

the resulting posterior and compute posterior expectations of quantities of interest F.

And we would like to analyze or study this objective in the case of increasing precision

n tending to infinity in the data.

Now, I probably don't have to motivate the basin approach to inverse problems that much,

but let me introduce on the next slide a motivational example, which also introduces the basic model

problem, which we will consider in most parts of the talk.

So, this is now an example for uncertainty quantification and groundwater flow modeling.

We are here considering a deep geological repository for radioactive waste located in

the USA.

It's called WIPP site.