# Lecture SS 24 Wissenschaftliches Rechnen II

## Scientific Computing II

### 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 for function and data representation:

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 kernel based methods in machine learning will be addressed in the lecture.

The second part of the lecture covers nonlinear dimensionality reduction / manifold learning for the analysis of high-dimensional data. The mathematical foundation of algorithms and numerical schemes will be described, and examplary applications presented. Studied dimensionality reduction methods include principal component analysis, IsoMap, Diffusion Maps and Variational Auto-Encoder.

### Prerequisites

Lectures on numerical mathematics in addition to Algorithmische Mathematik I+II are recommended, but not required. In particular knowledge in numerical linear algebra and numerical optimization is useful for the algorithmic aspects. Due to the data analysis topic statistic knowledge is required, i.e. the contents of Algorithmische Mathematik II, more stochastics knowledge can be helpful, but is not assumed. 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.