Talks

Stochastic optimization techniques have become an essential tool for training of
neural networks. One prominent algorithm is stochastic gradient descent
(SGD). Under smoothness and convexity assumptions one can show
convergence of SGD to a minimizer. However, the analyses of variants of
SGD require different techniques. In this talk, we look at recent
advances in modelling SGD by a continuous-time process defined by a
stochastic differential equation to obtain a unified framework. In
particular, we motivate the connection between the discrete and
continuous process and investigate in what sense they convergence to one
another. Further, we present examples of how the continuous-time model
behaves in practice.

Zoomlink:
https://tuhh.zoom.us/j/84729171896?pwd=ODArbForaUxMM3Q3VTJsNG1kaVNYQT09

Describing aspects of physical phenomena by forming abstract mathematical models is a common practice in scientific work: the mathematical formalism allows for permeation of the mathematical model as a means of creating insights and knowledge over the described real-world phenomenon.

In this talk, I will present how the topics of my dissertation contribute to the theory of popular mathematical models ranging from quantum physics to mathematical fluid mechanics.

In particular, you will find out

(I) how to classify periodic potentials of discrete Schrödinger operators with respect to the applicability of the finite section method,
(II) how to prove final-state observability for time-dependent diffusion problems, and
(III) how to improve the regularity of weak solutions to the Navier-Stokes equations on rough domains.

Link to slides:
https://math.fabian-gabel.de/talks/fabian_gabel_dissertation_pres.pdf

Link to video recording:
https://youtu.be/_2W-b-vXeZE

At the Large Hadron Collider, the interaction of subatomic particles with matter lead to severalmillions of collisions every second. For each collision, upto thousands of particles are producedfollowing stochastic processes. The accurate description of these particles require thousands ofvariables, which leads to large data sets with high dimensionality. The interaction of particleswith the detectors (like Compact Muon Solenoid) are best simulated with the GEANT4 software.Alternatively, less precise but faster simulations are sometimes preferred to reach higher statisticalprecision. We present recent progresses of refinement of fast simulations with Machine Learningtechniques to enhance the quality of such fast simulations. We demonstrate the use of adversarialnetworks in the context of jet simulation using the Wasserstein distance metric. The architectureconsists of opposing networks, Refiner and Critic. A Refiner refines the distribution of the energyof the jets obtained with the fast simulation. The Critic is used to effectively differentiate betweenthe distributions of refined energy and the distribution obtained by the GEANT4 simulation. Weapply the technique to jet kinematics, when the response is close to Gaussian, first on toy data setsand then on realistic data sets