Zuse Research Colloquium

The Fast Newton Transform (FNT) addresses the computational bottleneck that arises in solving high-dimensional problems such as 6d Boltzmann, Fokker-Planck, or Vlaslov equations, multi-body Hamiltonian systems, and the inference of governing equations in complex self-organizing systems. Specifically, the challenge lies in numerically computing function expansions and their derivatives fast, while achieving high approximation power. The FNT is a Newton interpolation algorithm with runtime complexity O(N n m), where N is the dimension of the downward closed polynomial space, n its degree and m the spatial dimension. We select subgrids from tensorial Leja-ordered Chebyshev-Lobatto grids based on downward closed sets. This significantly reduces the number of coefficients, N « (n+1)^m, while achieving optimal geometric approximation rates for a class of analytic functions known as Bos–Levenberg–Trefethen functions. Specifically, we investigate l^p-sets, where the Euclidean degree (p=2) turns out to be the pivotal choice for mitigating the curse of dimensionality. Furthermore, the differentiation matrices in Newton basis are sparse, enabling the implementation of fast pseudo-spectral methods on flat spaces, polygonal domains, and regular manifolds. Numerical experiments validate the algorithm’s superior runtime performance over state-of-the-art approaches.

In this talk model predictive control (MPC) is utilized to stabilize a class of linear time-varying parabolic partial differential equations (PDEs). In our first example the control input is only finite-dimensional, i.e., it enters as a time-depending linear combination of finitely many indicator functions whose total supports cover only a small part of the spatial domain. In the second example the PDE involve switching coefficient functions. We discuss stabilizability and the application of reduced-order models to derive algorithms with closed-loop guarantees.

This is joint work with Behzad Azmi (Konstanz), Michael Kartmann (Konstanz), Mathia Menucci (Stuttgart), Jan Rohleff (Konstanz), and Benjamin Unger (Stuttgart).

Solving a polynomial system, or computing an associated \gb basis, has been a fundamental task in computational algebra. However, it is also known for its notorious doubly exponential time complexity in the number of variables in the worst case. In this talk, I present a new paradigm for addressing such problems, i.e., a machine-learning approach using a Transformer. The learning approach does not require an explicit algorithm design and can return the solutions in (roughly) constant time. This talk covers our initial results on this approach and relevant computational algebraic and machine learning challenges.

Many real-world systems are composed of agents whose interactions result in a collective swarm behavior that may be complex, unexpected, and/or unintended. We highlight intriguing cases of interplay between the micro-scale behavior of agents and the macro-scale performance of the swarm, with a particular emphasis on heterogeneous systems composed of different types of agents, such as: traffic flow (the role of automation/connectivity on the energy footprint of urban traffic flow), mixed human/robotic groups (transportation of supplies to a disaster area), and biological systems (schools of fish and colonies of penguins). We particularly show how behavior interpretable as ‘equitable’ or ‘altruistic’ is possible to arise from pure survival-of-the-fittest objective functions