Talks

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

We discuss spectral inequalities and observability for the harmonic oscillator and more general Schrödinger operators with confinement potentials on the whole space. It turns out that the (super-)exponential decay of the corresponding eigenfunctions allows to consider sensor sets with a density that exhibits a certain decay. This, in particular, permits sensors with finite measure.

Nuclear fusion reactor design crucially depends on numerical simulation. The plasma can usually be modeled using fluid equations (for mass, momentum and energy). However, the reactor also contains neutral (non-charged) particles (which are important in its operation), of which both the position and velocity distribution is important. This leads to a Boltzmann-type transport equation that needs to be discretised with a Monte Carlo method. In high-collisional regimes, the Monte Carlo simulation describing the evolution of neutral particles becomes prohibitively expensive, because each individual collision needs to be tracked.
In this presentation, we overview a number of approaches that can alleviate the computational burden associated with the high-collisional regime. One option is to avoid simulating each invididual collision. In the limit of infinite collision rate, the law of large numbers dictates the approach of an advection-diffusion like particle behaviour, in which the accumulated effect of an infinite amount of collisions is aggregated in a Brownian motion (diffusion). To maintain accuracy and remove exploding simulation costs in high-collisional regimes, one can define hybridized particles that exhibit both kinetic behaviour and diffusive behaviour depending on the local collisionality [3].
Additionally, we can reduce the number of Monte Carlo particles that needs to be simulated via the multilevel Monte Carlo method[5]. Finally, one can also reduce the variance of the simulation by using an approximate fluid model for the neutral particles, discretized with a finite volume methods. This deterministic simulation can be used as a control variate, allowing the Monte Carlo simulation to focus on solely the deviation of the kinetic model with respect to the approximate fluid model.
References
[1] KukushkinA.S.,PacherH.D.,KotovV.,PacherG.W.,andReiterD.(2011)FinalizingtheITERdivertordesign:thekeyroleofSOLPSmodeling Fusion Eng. Des. 86:2865-2873.
[2] ReiterD.,BaelmansM.,andBörner,P.(2005)TheEIRENEandB2-EIRENEcodes,FusionSci.Technol.47:172-186.
[3] MortierB.,SamaeyG.,BaelmansM.(2019)Kinetic-diffusionasymptotic-preservingMonteCarloalgorithmsforplasmaedgeneutralsimulation.
Contributions to Plasma Physics, in press.
[4] Horsten N., Samaey G., Baelmans M. (2019) Hybrid fluid-kinetic model for neutral particles in the plasma edge. Nuclear Materials and Energy
18:201-207.
[5] Løvbak E., Samaey G., Vandewalle S. (2019) A multilevel Monte Carlo method for asymptotic-preserving particle schemes. Submitted. https://arxiv.org/abs/1907.04610.

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