-Invariantly Characterize Numerical Spacetimes

-Abstract: This topic will cover the fundamental concepts related to the invariants characterizing numerical spacetimes. Initially, I intend to elucidate the methodology employed in computing the electric and magnetic components of the Weyl tensor through the utilization of the 3+1 slicing formulation. And the presentation will include an exploration of the relationship between the Weyl scalar and the electric and magnetic components in terms of null tetrad basis. Finally, the Petrov classification and fundamental invariants are presented.

-Spectral Methods

-Abstract: First, theoretical introductions to interpolation and spectral expansion are given and a particular emphasis is put on the fast convergence of the spectral approximation. I will present different approaches to solve differential equations, limiting ourselves to the one-dimensional case, with one or several domains.

-Constrained Optimization and Penalty Method

-Abstract: In this topic, I will provide a brief introduction to constrained optimization and the penalty method, which transforms the original constrained problem into a single unconstrained problem. I will also discuss the linearized approximation of the feasible set, which helps derive the constraint conditions necessary for analyzing practical optimization methods. Additionally, unlike the quadratic penalty method, the penalty parameters in the augmented Lagrangian and exact penalty methods do not need to be infinite. However, the exact penalty method is non-smooth, which can be addressed by introducing artificial variables. Meanwhile, the augmented Lagrangian method also introduces additional slack variables when applied to constrained problems.