The parabolic Anderson model with Pareto potential on critical Galton-Watson trees conditioned to surviveVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

We prove that the solution to the parabolic Anderson model with Pareto potential on a critical Galton-Watson tree conditioned to survive with an offspring distribution in the domain of attraction of a stable law localizes with high probability in one single vertex for time going to infinity. The proof relies on the Feynman-Kac representation of the solution and spectral estimates of the Anderson Hamiltonian. This is joint work with Eleanor Archer (Tel Aviv University).

Exponential decay of correlations for O(N) spin systems for arbitrary N(using zoom) (Parkring 11, 85748 Garching-Hochbrück)

The Spin O(N) model is a classical statistical mechanics model whose configurations are collections of unit vectors, called spins, taking values on the surface of a N -1 dimensional unit sphere, with each spin associated to the vertex of a graph. Some special cases of the spin O(N) model are the Ising model (N = 1), the XY model (N = 2), and the classical Heisenberg model (N = 3). Despite the fact that it is a very classical model, there remain important gaps in understanding, particularly in the case N > 2. This talk will present a new recent result about exponential decay of correlations for arbitrary (non-zero) values of the external magnetic field and arbitrary spin dimension N>1, extending previous results which are only valid for N=1,2,3. Our proof is probabilistic and employs a new representation of the model as a system of “coloured" random walks which is of independent interest.

Oberseminar : A variational inequality for the derivative of the scalar play operatorPasscode 101816 (https://tum-conf.zoom.us/j/96536097137 , 0 (using zoom))

We show that the directional derivative of the scalar play operator is the unique solution of a certain variational inequality. Due to the nature of the discontinuities involved, the variational inequality has an integral form based on the Kurzweil-Stieltjes integral.

Testing relevant hypotheses in functional time series via self-normalization(using Zoom, see http://go.tum.de/410163 for more details) (Parkring 11, 85748 Garching)

In this paper we develop methodology for testing relevant hypotheses in a tuning-free way. Our main focus is on functional time series, but extensions to other settings are also discussed. Instead of testing for exact equality, for example for the equality of two mean functions from two independent time series, we propose to test a \textit{relevant} deviation under the null hypothesis. In the two sample problem this means that an $L^2$-distance between the two mean functions is smaller than a pre-specified threshold. For such hypotheses self-normalization, which was introduced by Shao (2010) and is commonly used to avoid the estimation of nuisance parameters, is not directly applicable. We develop new self-normalized procedures for testing relevant hypotheses and demonstrate the particular advantages of this approach in the the comparisons of eigenvalues and eigenfunctions.

Optimal Control Problems for the Cahn-Hilliard-Navier-Stokes SystemVirtuelle Veranstaltung (Boltzmannstr. 3, 85748 Garching)

In this talk I will address various optimal control problems related to the evolution of two dimensional bounded domain. I quote [1], [2], [3], for more details about the model. We prove the existence of an optimal control and establish the Pontryagin maximum principle, which gives the rst order necessary conditions of optimality. We also prove the unique solvability of the corresponding adjoint system and characterize the optimal control in terms of the adjoint variable. Furthermore, we establish the second order necessary and sucient optimality condition for the optimal control problem.

References [1] P. Colli, S. Frigeri, M. Grasselli, Global existence of weak solutions to a nonlocal Cahn-Hilliard-Navier-Stokes system, Journal of Mathematical Analysis and Applications, 386(1) (2012), 428-444. [2] S. Frigeri, M. Grasselli, P. Krejci, Strong solutions for two-dimensional nonlocal Cahn-Hilliard- Navier-Stokes systems, Journal of Differential Equations, 255(9) (2013), 2587-2614. [3] S. Frigeri, C. G. Gal and M. Grasselli, On nonlocal Cahn-Hilliard-Navier-Stokes systems in two dimensions, Journal of Nonlinear Science, 26(4) (2016), 847-893.

Large Deviations of a Spatial Cycle Huang-Yang-Luttinger Loop Soup(using zoom) (Parkring 11, 85748 Garching-Hochbrück)

It is conjectured that the emergence of Bose-Einstein condensation in interacting Bose gases should correspond with the emergence of "infinitely long" cycles in an interacting loop soup. The Huang-Yang-Luttinger (HYL) interaction in the Bose gas is an approximation of the hard-sphere interaction, and here we will relate it to a similar looking interaction on the loop soup. Using large deviation techniques we will derive various important properties of this interacting loop soup and relate them to known properties of the HYL-interacting Bose gas. In particular, we will derive a discontinuous "condensation density" for the loop soup and find a regime in which the large deviation rate function aquires distinct simultaneous minimisers.

Asymptotic spectral density of non-linear random matrix modelsvia ZOOM (Boltzmannstr. 3, 85748 Garching)

We compute the asymptotic empirical spectral distribution of a nonlinear random matrix model by using the resolvent method. Motivated by random neural networks, we consider the random matrix M = YY∗ with Y = f(WX), where W and X are random rectangular matrices with i.i.d. centered entries and f is a nonlinear smooth function which is applied entry-wise. We prove that the Cauchy-Stieltjes transform of the limiting spectral distribution satisfies a quartic self-consistent equation up to some error terms, which is exactly the equation obtained by Benigni and Péché (2019) with the moment method approach. The significant role of biases in neural networks has led us to extend the model by adding a random bias matrix, i.e., we investigate the case with Y = f(WX+B), where B is a random rectangular matrix of i.i.d. centered Gaussian random variables. We prove that the self-consistent equation approximately satisfied by the Cauchy-Stieltjes transform of the limiting spectral distribution is similar to that found in the bias-free case. This is joint work with Dr. Dominik Schröder.

Approximation of Functions with Tensor Networks(using zoom) ( , 0 kein Ort)

We consider two questions regarding approximation of functions with Tensor Networks (TNs): ``What are the approximation capabilities of TNs?'' and ``What is an appropriate model class of functions that can be approximated with TNs?''

To answer the former: we show that TNs can (near to) optimally replicate h-uniform and h-adaptive approximation, for any smoothness order of the target function. Tensor networks thus exhibit universal expressivity w.r.t. isotropic, anisotropic and mixed smoothness spaces that is comparable with more general neural networks families such as deep rectified linear unit (ReLU) networks. Put differently, TNs have the capacity to (near to) optimally approximate many function classes -- without being adapted to the particular class in question.

To answer the latter: as a candidate model class we consider approximation classes of TNs and show that these are (quasi-)Banach spaces, that many types of classical smoothness spaces are continuously embedded into said approximation classes and that TN-approximation classes are themselves not embedded in any classical smoothness space.

Simplified R-vine based forward regression (using Zoom, see http://go.tum.de/410163 for more details) (Parkring 11, 85748 Garching)

An extension of the D-vine based forward regression procedure to a R-vine forward regression is proposed. In this extension any R-vine structure can be taken into account. Moreover, a new heuristic is proposed to determine which R-vine structure is the most appropriate to model the conditional distribution of the response variable given the covariates. It is shown in the simulation that the performance of the heuristic is comparable to the D-vine based approach. Furthermore, it is explained how to extend the heuristic into a situation when more than one response variable are of interest. Finally, the proposed R-vine regression is applied to perform a stress analysis on the manufacturing sector which shows its impact on the whole economy.

Reference: Zhu, Kurowicka and Nane. https://doi.org/10.1016/j.csda.2020.107091

Multilevel estimators for models based on partial differential equations(using Zoom, see http://go.tum.de/410163 for more details) (Parkring 11, 85748 Garching)

Many mathematical models of physical processes contain uncertainties due to incomplete models or measurement errors and lack of knowledge associated with the model inputs. We consider processes which are formulated in terms of classical partial differential equations (PDEs). The challenge and novelty is that the PDEs contain random coefficient functions, e.g., some transformations of Gaussian random fields. Random PDEs are much more flexible and can model more complex situations compared to classical PDEs with deterministic coefficients. However, each sample of a PDE-based model is extremely expensive. To alleviate the high costs the numerical analysis community has developed so-called multilevel estimators which work with a hierarchy of PDE models with different resolution and cost. We review the basic idea of multilevel estimators and discuss our own recent contributions:

i) a multilevel best linear unbiased estimator to approximate the expectation of a scalar output quantity of interest associated with a random PDE [1, 2],

ii) a multilevel sequential Monte Carlo method for Bayesian inverse problems [3],

iii) a multilevel sequential importance method to estimate the probability of rare events [4].

[1] D. Schaden, E. Ullmann: On multilevel best linear unbiased estimators. SIAM/ASA J. Uncert. Quantif. 8(2), pp. 601-635, 2020

[2] D. Schaden, E. Ullmann: Asymptotic analysis of multilevel best linear unbiased estimators, arXiv:2012.03658

[3] J. Latz, I. Papaioannou, E. Ullmann: Multilevel Sequential² Monte Carlo for Bayesian Inverse Problems. J. Comput. Phys., 368, pp. 154-178, 2018

[4] F. Wagner, J. Latz, I. Papaioannou, E. Ullmann: Multilevel sequential importance sampling for rare event estimation. SIAM J. Sci. Comput. 42(4), pp. A2062–A2087, 2020