6 EC
Semester 1, period 1, 2
5334WAVE6Y
Theoretical aspects: The course contains an abstract mathematical introduction on the topic of wavelets and their general construction using the concept of multiresolution analysis. Based on two concrete examples, being the classical Haar wavelet and the less know and somewhat less trivial Stroemberg wavelet, the more abstract concept of multiresolution analysis is introduced, and together with it the scaling function. It is shown how wavelets can be constructed from the multiresolution analysis, and hence from the scaling function. Apart from revisiting the Haar- and Stroemberg wavelets, this also leads to several types of wavelets, such as periodic wavelets, Meyer's wavelets, spline wavelets, and Daubechies' compactly supported wavelets. Special attention is paid to their decay, their vanishing moments and their smoothness.
Practical aspects: Apart form the theoretical aspects, we will pay attention to applications of wavelets in both linear and nonlinear approximation theory and signal processing. In particular we discuss the wavelet transform, the discrete wavelet transform and the fast discrete wavelet transform, and their implementation.
Historical aspects: We will also spend some time on the historical development and the status of wavelet research in mathematics, highlighted by the Abel Prize awarded to Yves Meyer in 2017, Ingrid Daubechies' membership of the Royal Netherlands Academy of Arts and Sciences, and the Fast Fourier Transform being named in the Top 10 algorithms in the 20th century.
Context: The vast topic of Wavelets and Multiresolution Analysis is situated both in abstract and in applied analysis. Their existence and mathematical properties are worth studying on their own account, nonetheless they find applications in many fields, particularly in signal processing (wavelets were used in the recent detection of gravitational waves). Wavelets are well-suited to analyze irregular and non-smooth signals (such as from geophysics, finance, etc). Wavelets are also used in ingenious approximation methods for solutions of PDEs.
A Mathematical Introduction to Wavelets, by P. Wojtaszczyk. London Mathematical Society. Student Texts 37.
Matlab, Python
Hand-outs by the lecturer, slides.
The course is a lecture course. However, in weeks 4 and 12, students are given the opportunity to work on assignments that they should hand in. The format of handing in is a zoom-recorded video-presentation about the assignments (which explains the ``presentation/symposium'' box above being ticked).
Activity | Hours | |
Tentamen | 3 | |
Self study | 165 | |
Total | 168 | (6 EC x 28 uur) |
This programme does not have requirements concerning attendance (TER-B).
| Item and weight | Details |
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Final grade | |
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1 (100%) Tentamen |
The final individual exam counts for 70%, whereas each of the assignments counts for 15%. The score for the individual exam should be at least 5 out of 10. The minimum score for each assessment and for the exam is 1 out of 10.
In the event of a resit, the results for the assignments will still be valid, with the same weights. The assignments do not have a standard resit or second chance. In case of exceptional circumstances that hinder a student to do the assignment, the Study Adviser should be informed before the assignment deadline. The Student Adviser may then, after judging whether the request is reasonable, ask the lecturer to provide an alternative for the missed assignment for that student.
Inspection of assessed work is possible on appointment with the lecturer.
The assignments may be made and handed in with at most one other fellow student, not necessarily the same for both assignments. An assignment consists of a recording (ideally made in Zoom) of the student(s) explaining and demonstrating (running in clear view) their own written computer programmes, discussing the results and answering questions posed in the assignment text. The length of such a video recording is typically 20-25 minutes.
The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl
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