Approximation methods II

Winter term 2017
Lecture
Tuesday, 8:15 - 9:55, MN 55
Friday, 8:15 - 9:55, MN 55
First Lecture: October 17, 2017

Exercises
Wednesday, time and room will be scheduled

Participants:
The lecture is suitable for students in mathematics, physics and computer science.

Assumptions to finish the lecture successfully:
There will be oral examinations (20 minutes) in February/March 2018.

Assumptions to get admission for the examination:
Attendance of the exercises, oral presentation of two exercise solutions, and 50 percent of the achievable points for homework.
With this lecture + exercises you can obtain 9 ECTS points.

Content:
This lecture Approximation Methods II is concerned with kernel based meshless methods in multivariate approximation.

1. Introduction

2. Kernels
2.1 Basics
2.2 Positive Definiteness
2.3 Inner Products
2.4 Duality
2.5 Native Space
2.6 Reproducing Kernel Hilbert Spaces
2.7 Kernels for Orthogonal Expansions
2.8 Native Spaces of Mercer Kernels
2.9 Finite Case
2.10 Kernels for Univariate Sobolev Spaces

3. Optimal Recovery
3.1 Optimality Properties
3.2 Lagrange Reformulation
3.3 Calculation
3.4 Regularization

4. Conditionally Positive Definite Kernels
4.1 Splines
4.2 General Case
4.3 Inner Products
4.4 Optimal Recovery
4.5 Projectors
4.6 Native Space

5. Practical Observations
5.1. Lagrange Interpolatio
5.2. Interpolation of Mixed Data
5.3. Error Behavior
5.4. Stability
5.5. Uncertainty Principle
5.6. Scaling
5.7. Practical Rules
5.8 Sensitivity to Noise

6. Error Analysis
6.1 Sampling Inequalities
6.2 Univariate Case
6.3 Example: Univariate Splines
6.4 Univariate Polynomial Reproduction
6.5 Norming Sets
6.6 Multivariate Polynomial Reproduction
6.7 Moving Least Squares
6.8 Bramble–Hilbert Lemma
6.9 Globalization
6.10 Error Bounds

7. Construction of Kernels
7.1 General Construction Techniques
7.2 Special multivariate Kernels
7.3 Translation–Invariant Kernels
7.4 Global Sobolev Kernels
7.5 Native Spaces of Translation–Invariant Kernels.
7.6 Construction of Positive Definite Radial Functions
7.7 Conditionally Positive Definite Kernels
7.8 Examples
7.9 Connection to Hilbert spaces
7.10 Characterization of Native Spaces
7.11 Connectiont to Sobolev Spaces

Literature:
R. Schaback: Approximationsverfahren II (http://num.math.uni-goettingen.de/schaback/teaching/Appverf_II.pdf)
M.D. Buhmann. Radial Basis Functions, Theory and Implementations. Cambridge University Press, 2003.
H. Wendland. Scattered Data Approximation. Cambridge University Press, 2005.