Seminars in Numerical Analysis

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Seminar series in Numerical Analysis

Our seminar series is starting again, this year we're back on campus!

The Numerical Analysis seminars are a series of lectures to showcase and foster state-of-the-art research in the fields of computational sciences and engineering, mathematical modelling, and applied mathematics. These lunch lectures take place on a monthly basis, the dates are announced below. If you want to sign up for the monthly invitation, please sign up for our newsletter.

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15 December 2023 12:30 till 13:15

[NA] Stefan Kurz: Observers in relativistic electrodynamics

"We introduce a relativistic splitting structure to map fields and equations of electromagnetism from four-dimensional spacetime to three-dimensional observer's space. We focus on a minimal set of mathematical structures that are directly motivated by the language of the physical theory. Space-time, world-lines, time translation, space platforms, and time synchronization all find their mathematical counterparts. The splitting structure is defined without recourse to coordinates or frames. This is noteworthy since, in much of the prevalent literature, observers are identified with adapted coordinates and frames. Among the benefits of the approach is a concise and insightful classification of observers. The application of the framework to Schiff's ""Question in General Relativity"" [1] further illustrates the advantages of the framework, enabling a compact, yet profound analysis of the problem at hand.  
[1] Schiff, L. I. ""A question in general relativity."" Proceedings of the National Academy of Sciences 25.7 (1939): 391-395.
Consider two concentric spheres with equal and opposite total charges uniformly distributed over their surfaces. When the spheres are at rest, the electric and magnetic fields outside the spheres vanish. [...] Then an observer traveling in a circular orbit around the spheres should find no field, for since all of the components of the electromagnetic field tensor vanish in one coordinate system, they must vanish in all coordinate systems. On the other hand, the spheres are rotating with respect to this observer, and so he should experience a magnetic field. [...] It is clear in the above arrangement that an observer A at rest with respect to the spheres does not obtain the same results from physical experiments as an observer B who is rotating about the spheres."

19 January 2024 12:30 till 13:15

[NA] Jakob Zech: Nonparametric Distribution Learning via Neural ODEs

In this talk, we explore approximation properties and statistical aspects of Neural Ordinary Differential Equations (Neural ODEs). Neural ODEs are a recently established technique in computational statistics and machine learning, that can be used to characterize complex distributions. Specifically, given a fixed set of independent and identically distributed samples from a target distribution, the goal is either to estimate the target density or to generate new samples. We first investigate the regularity properties of the velocity fields used to push forward a reference distribution to the target. This analysis allows us to deduce approximation rates achievable through neural network representations. We then derive a concentration inequality for the maximum likelihood estimator of general ODE-parametrized transport maps. By merging these findings, we are able to determine convergence rates in terms of both the network size and the number of required samples from the target distribution. Our discussion will particularly focus on target distributions within the class of positive \(C^k\) densities on the \(d\)-dimensional unit cube \([0,1]^d\).

Organizers

D. (Deepesh) Toshniwal

A. (Alexander) Heinlein

In Conversation With

The accompanying interview series of the Numerical Analysis seminars.

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