Mini-workshop: “Analysis, Numerics and Control”

Date: Tue. November 14, 2022
Organized by: FAU DCN-AvH, Chair for Dynamics, Control and Numerics – Alexander von Humboldt Professorship at FAU Erlangen-Nürnberg (Germany)
Title: Mini-workshop “Analysis, Numerics and Control”

11:00H
Title: Nonlocal complement value problem for a global in time parabolic equation
Speaker: Dr. Jean-Daniel Djida
Affiliation: Visiting Postdoctoral researcher from AIMS-Cameroon
• Slides

Abstract. In this joint work with Guy Fabrice Foghem Gounoue and Yannick Kouakep Tchaptchie, we investigate the existence and uniqueness of weak solution of a semilinear parabolic equation with double nonlocality in space and in time variables that naturally arises while modeling a biological nano-sensor in the chaotic dynamics of a polymer chain. In fact, the problem under consideration involves a symmetric integrodifferential operator of Lévy type and a term called the interaction potential, that depends on the time-integral of the solution over the entire interval of solving the problem. Owing to the Galerkin approximation, the existence and uniqueness of a weak solution of the nonlocal complement value problem is proven for small time under fair conditions on the interaction potential.

11:30H
Title: Breaking the curse of dimensionality with Barron Spaces
Speaker: Antonio Álvarez López
Affiliation: Visiting PhD Student from UAM, Autonomous University of Madrid (Spain)

• Slides

Abstract.Approximating an unknown function with arbitrary precision is one of the main tasks in Supervised Learning. Nevertheless, the error and complexity bounds are usually dependent on the dimension of the ambient space, whose typical values on this type of problems are extremely large. In this context, the term “curse of dimensionality” refers to the common situation when these relationships are exponential, deteriorating the performance of the model when the dimension increases.

Therefore, it seems very important to be aware of the characteristics of the functions that a particular Neural Network model is able to approximate efficiently. I will briefly introduce a classical simple Deep Learning architecture, the two-layer neural networks, in order to present a class of functions that they are able to approximate in the sense that optimal direct and inverse approximation theorems hold. Those function spaces are named the Barron Spaces, and they allow us to shed some light on the possibilities of overcoming the dimensionality problem and gaining a deeper understanding of the capabilities of these models.

12:00H
Title: Geometric guidance strategies for terminal angle and time control problems
Speaker: Ziqi Wang
Affiliation: PhD Student at FAU DCN-AvH Chair for Dynamics, Control and Numerics – Alexander von Humboldt Professorship (Germany)

• Slides

Abstract. The recently developed advanced guidance laws aim to arrive at the final position with a specified angle or time for better effectiveness. However, aspects such as feasibility, robustness, and implementation complexity of the constrained guidance laws are currently significant issues that attract more interest in research. In this work, a two-stage ITAG law based on the geometric property of the circle involute is designed. The proposed technique is easy to implement, in that it does not involve model linearization, time-to-go estimation, and numerical optimization routine. Additionally, the achievable range of different constraints is analyzed considering practical issues such as initial launch angles, initial line-of-sight angles, and acceleration limits.