Applied Mathematics Colloquium

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.