Modern Approximation Theory

What it is about

You will learn tools of modern approximation theory: s-numbers (such as approximation, Gelfand and Kolmogorov numbers), entropy numbers as well as methods and bounds for sparse reconstruction (e.g., Prony methods, Restricted Isometry Property, iterative hard thresholding, CoSaMP). In the end you can apply these concepts to tasks for the approximation of vectors, functions and operators and you will recognize fundamental lower bounds in sampling theory.

What you will be able to do

• Understanding of s-numbers (approximation, Gelfand, Kolmogorov)
• Knowledge of entropy numbers
• Familiarity with methods of sparse reconstruction (Prony, RIP, IHT, CoSaMP)
• Applying the theory to the approximation of vectors, functions and operators
• Foundations of information-based complexity and lower bounds in sampling theory

What the module consists of

• LecturesConveying the content through lectures, examples and discussion
• Übung/PraktikumExercise sheets with solutions for deepening understanding and self-check

Teaching method

• Lecture with examplespresents the concepts and motivates own analysis
• Discussion with studentspromotes understanding and critical engagement
• Practice problems with model solutionsenable deepening and verification of learning achievement