Seminar: Applied Optimization

FS 2022

The theme of this seminar will be Numerical Linear Algebra. We will start with a few lectures introducing important fundamentals. The core part of the seminar will then consist of presentations by the participants on individually assigned topics. Throughout the semester we will additionally include programming exercises in C++ such that all participants will develop a solid practical understanding of the covered methods.

Numerical Linear Algebra provides all essential tools to the numerical scientists, those using vectors and matrices. Those tools consist of practical algorithms with a design relying on functional analysis. This form of applied linear algebra is ubiquitous in the engineering world. However, although broadly used, few scientists know the tricks of modern linear algebra systems.

This seminar course is designed to reveal the secrets of the important methods:

  • QR factorization: (modified) Gram-Schmidt orthogonalization and Householder reflectors
  • LU decomposition: Gaussian elimination, pivoting and Cholesky factorization
  • Eigenvalues: Power iteration, Inverse iteration, QR algorithms and generalization
  • Iterative methods: Krylov methods (conjugate gradients), convergence, preconditioning and generalization
In addition to those methods, the Singular Value Decomposition (SVD) is an important topic of the seminar. SVD yields a valuable decomposition of general matrices. This representation enables studying the stability of all linear algebra methods. Above methods are applied in practical assignments to solve a 2D Poisson problem, using a discontinuous Galerkin P1 method.