Numerical Analysis: Solving Linear Systems in Higher Dimensions

This lecture focuses on numerical analysis and optimization, specifically addressing the resolution of linear systems in higher dimensions. The instructor begins by discussing the importance of quizzes and exercises related to finite elements and the upcoming topics. The lecture progresses to Chapter 11, where the instructor introduces the concept of solving a large linear system, transitioning from one-dimensional to two-dimensional problems. The physical context is illustrated using examples such as elastic membranes and drums, emphasizing the mathematical modeling involved. The instructor explains the mathematical formulation, including the Laplacian operator and boundary conditions. A finite difference method is introduced for numerical resolution, detailing the grid setup and approximation techniques. The lecture culminates in the formulation of a linear system, discussing the matrix representation and the significance of the Cholesky decomposition for solving these systems efficiently. The instructor highlights the computational complexity and memory considerations when dealing with high-dimensional problems, setting the stage for future discussions on iterative methods and their applications in machine learning.