Approximation theory covers the theoretical aspects of how continuous objects (functions, operators,...) in infinite-dimensional spaces can be replaced (with quantifiable error) by objects described by finitely many parameters (e.g., polynomials, splines, finite-rank operators,...), it is interrelated with numerical analysis, optimization, and complexity theory. The founding members of this subdiscipline of analysis are Chebyshev (the Chebyshev polynomial T_n(x) solves the problem of best approximating the monomial x^n by polynomials of degree < n on [-1,1]) and Weierstrass (any continuous function on [a,b] can be approximated with arbitrarily small error by an algebraic polynomial). More recent chapters deal with multivariate piecewise polynomial approximation (splines, finite elements), wavelets and frames, neural network approximation, nonlinear approximation algorithms, and so on. This is a graduate course which will be organized into roughly three parts as follows: First, I will cover some classical aspects of approximation theory (the notion of best approximation, the interplay between approximation error for functions and their smoothness, etc.). To keep it simple, I will do this for functions of one variable, but on the way touch several types of approximation schemes, to show the great variety of subfields and problems. In the second part, I will switch to multivariate approximation, and its relation to numerical discretization schemes. In the last part, I attempt to give a glimpse into the fascinating world of nonlinear approximation schemes, with buzzwords such as best $n$-term approximation, greedy algorithms, adaptivity. Background from real and functional analysis is helpful but restricted to a minimum, background from numerical methods and/or optimization would help to motivate some of the problems to be discussed but is not necessary - I will introduce a few model problems myself. The aim of the course is to do a few things in sufficient detail, rather than touch everything a little. It is wise to come to class, as easily accessible textbooks covering the topics are not available. Basic references (covering most of the first part) are given below, more sources will be mentioned in class. Another feature of the course is that students can choose between final exam or final project paper. This might fit individual needs better (e.g., you could benefit for another project/subject area by studying a specific topic in more depth in form of a project paper). I am also open to offering oral exams (however, the midterm is in written form).
| Instructor of Record: | Peter Oswald |
| Email: | p.oswald@jacobs-university.de, |
| Phone: | 200-3179 |
| Office hours: | Tuesdays 2-3:30pm (or by appointment) in Research I, 113 |
| Consultation/Tutorial: | Tuesdays 3:30-4:30pm in Research I, 113 |
| Lectures : | Mo 11:15-12:30am and We 14:15-15:30am in West Hall 4 |
A midterm exam will be given, probably during the week after the Fall reading days. You have to choose between final exam and final project paper by Nov. 1st. Exact dates and details about the exams will be given in class at a later time.
Only valid and timely excuses (as regulated by Jacobs University policies, and confirmed by the registrar) will be accepted if you miss exams or final project paper deadlines.
| Homework: | 40% |
| Midterm Exam: | 20% |
| Final Exam or Project Paper: | 40% |
| Cutoff score: | 95% | 90% | 85% | 80% | 75% | 70% | 65% | 60% | 55% | 50% | 45% | 40% |
| IUB Points: | 1.0 | 1.33 | 1.67 | 2.0 | 2.33 | 2.67 | 3.0 | 3.33 | 3.67 | 4.0 | 4.33 | 4.67 |