# Lecture SS 21 Wissenschaftliches Rechnen II

## Scientific Computing II

The times of the lectures are changed from the initial announcement. Same day, but different times.

### Content

Scientific Computing is an applied discipline which deals with methods, techniques, and algorithms that are necessary to obtain advanced insight into processes which stem from engineering tasks or natural sciences.

The lecture focuses on two topics:

Kernel-based methods for function approximation Nonlinear dimensionality reduction/manifold learning

The first part of the lecture concerns kernel-based methods for function approximation in the form of positive definite kernels and radial basis functions. The lecture covers theoretical results on kernel-based approximation methods and their application, which includes machine learning, surrogate modeling, spatial statistics, boundary value problems, and finance. In particular machine learning will be addressed in the lecture.

The second part of the lecture covers nonlinear dimensionality reduction / manifold learning for data analysis. The mathematical foundation of the algorithm and the numerical schemes which can be applied will be described, and examples for applications given.

### Prerequisites

Lectures on numerical mathematics are recommended, in particular knowledge in numerical linear algebra and numerical optimization is needed for the algorithmic aspects. Due to the data analysis topic stochastics is required, i.e. the contents of Algorithmische Mathematik II, more statistics can be helpful. Any standard introductory textbook on these topics will suffice.

Knowledge of Module Wissenschaftliches Rechnen I is not required.

### Literature

- Lecture Notes on Kernel-Based Meshless Methods, Robert Schaback Link
- Kernel techniques: From machine learning to meshless methods, Robert Schaback and Holger Wendland Link
- Gaussian Processes for Machine Learning, Carl Edward Rasmussen and Christopher K. I. Williams Link
- Learning with Kernels, Bernhard Schölkopf and Alexander J. Smola, MIT Press
- Meshfree Approximation Methods with MATLAB, Gregory E. Fasshauer, World Scientific
- Nonlinear dimensionality reduction, John A. Lee and Michel Verleysen, Springer.