## Applied Mathematics Colloquium

The Applied Mathematics Colloquium provides an opportunity for the various applied mathematics teams (lead by Prof. Dr. Michael Dellnitz, Prof. Dr. Balázs Kovács, Prof. Dr. Sina Ober-Blöbaum and Jun.-Prof. Dr. Thomas Berger) at Paderborn University to give talks and hold joint discussions on different topics related to applied mathematics. External members are also welcome to give talks. The Colloquium is usually held on Thursdays. However, depending on the organizing team, the time and place may vary. Possible times are as follows: Thursdays at 11.15 am, 12.15 pm or 4.15 pm.

## UPCOMING TALKS

**Title**: A new perspective on spectral methods

**Abstract**: In this talk we describe a new perspective on spectral methods for time-dependent PDEs. Basically, given an equation evolving in a separable Hilbert space, a spectral method is no more than a choice of an orthonormal basis. The choice of such a basis is governed by a raft of considerations: stability, speed of convergence, structure preservation and the ease of numerical algebra.

Focussing on a single space dimension, we distinguish between two cases: T-systems and W-systems. T-systems are defined on L2(R), can be characterised completely and possess a tridiagonal, skew-Hermitian differentiation matrix: this renders linear algebra very easy. W-systems act on L2(a, b), where (a, b) ⊂ R, and are defined directly from orthogonal polynomials. In their case the differentiation matrix is semi-separable of rank 1, again yielding itself to rapid linear algebra.

We describe the state of the art with both types of systems, with an emphasis on their approximation-

theoretic features. Time allowing, we will mention a generalisation to Galerkin–Petrov-type methods and to multivariate setting.

**Title**: Can neural networks solve high dimensional optimal feedback control problems?

**Abstract**: Deep Reinforcement Learning has established itself as a standard method for solving nonlinear optimal feedback control problems.

In this method, the optimal value function (and in some variants also the optimal feedback law) is stored using a deep neural network. Hence,

the applicability of this approach to high-dimensional problems crucially relies on the network's ability to store a high-dimensional

function. It is known that for general high-dimensional functions, neural networks suffer from the same exponential growth of the number of

coefficients as traditional grid based methods, the so-called curse of dimensionality. In this talk, we use methods from distributed optimal

control to describe optimal control problems in which this problem does not occur.

## PREVIOUS TALKS

**Title**: Trajectory optimization in space mechanics with Julia

**Abstract**: TBA

**Title**: The linearly implicit two-step BDF method for harmonic maps into spheres

**Abstract**: After recalling the notion of harmonic maps into spheres, we discuss two variational formulations of the corresponding Euler–Lagrange equations. The second variational formulation leads easily to a linearization of the nonlinear equation. Subsequently, we focus on the gradient flow approach and recall known results for the linearly implicit Euler method, namely, energy decay (stability) and constraint violation properties. Our contribution concerns the application of the linearly implicit two-step BDF method to the gradient flow problem. We devise a projection-free iterative scheme for the approximation of harmonic maps that is unconditionally energy stable and provides a second-order accuracy of the constraint violation under a mild, sharp discrete regularity condition. The considered problem serves as a model for partial differential equations with holonomic constraint. For the performance of the method, illustrated via the computation of stationary harmonic maps and bending isometries.

The talk is based on joint work with Sören Bartels and Christian Palus (Albert-Ludwigs-Universität Freiburg).

**Title**: Error Analysis for the Stochastic Koopmann Operator

**Abstract**: Koopman operator theory has emerged as a powerful modeling approach for complex dynamical systems arising in physics, chemistry, materials science, and engineering. The basic idea is to leverage existing simulation data to learn a linear model that allows to predict expectation values of observable functions at future times. Though the resulting algorithm, known as Extended Dynamic Mode Decomposition (EDMD), is conceptually quite simple, its underlying mathematical structure (the Koopman operator semigroup) is very rich, and can be used for different purposes including control, coarse graining, or the identification of metastable states in complex molecules and materials.

**Title**: Efficient Protocols for Simple Dynamic Distributed Systems

**Abstract**: Population protocols and related models allow to study the dynamics of distributed systems consisting of a vast number of simple and identical agents. The standard model assumes a complete network of n-agents modeled as simple finite state machines. Pairwise interactions between agents happen either adversarially or in a randomized way and cause the agents to update their respective state, depending on their own state and that of their interaction partner.

Despite their simplicity, population protocols can solve fundamental distributed problems like leader election, majority, or consensus problems. My talk will start with a general introduction into the topic and then dive into some of our recent and ongoing work on how to efficiently solve consensus-related problems in population protocols (and variants).

**Title: **Modelling opinion dynamics under the impact of influencer and media strategies

**Abstract: **Online social media are nowadays an integral part of people's everyday life that can influence our behaviour and opinions. Despite recent advances, the changing role of traditional media and the emerging role of "influencers" are still not well understood, and the implications of their strategies in the attention economy even less so. In this talk, we will propose a novel agent-based model (ABM) that aims to model how individuals (agents) change their opinions (states) under the impact of media and influencers. We will show the rich behavior of this ABM in different regimes and how different opinion formations can emerge, e.g. fragmentation. In the limit of infinite number of agents, we will derive a corresponding mean-field model given by a PDE. Based on the mean-field model, we will show how strategies of influencers can impact the overall opinion distribution and that optimal control strategies allow other influencers (or media) to counteract such attempts and prevent further fragmentation of the opinion landscape.

**Title**: The origin of Fourier series

**Abstract**: TBA

**Title**: Time-dependent electromagnetic scattering from dispersive materials

**Abstract**: This talk discusses time-dependent electromagnetic scattering problems in the context of dispersive material laws. We consider the numerical treatment of a scattering problem in which a retarded material law, for a causal and passive homogeneous material, determines the wave-material interaction in the scatterer. The resulting problem is nonlocal in time in the interior of the scatterer and is posed on an unbounded domain. Well-posedness of the scattering problem is shown using a formulation based on time-dependent boundary integral equations, which is fully formulated on the surface of the scatterer. Discretizing the boundary integral equation by convolution quadrature in time and boundary elements in space yields a provably stable and convergent method that is fully parallel in time and space. Under regularity assumptions on the exact solution we present error bounds with explicit convergence rates in time and space. Numerical experiments illustrate the theoretical results and show the effectiveness of the method.