 
נושאים בבעיות הפוכות
Topic in inverse problems 
03724025  

מדעים מדויקים  

The problem of superresolution (SR) is to extract the fine details of a signal from incomplete and inaccurate measurements of its spectrum. It is an inverse problem of contemporary interest, arising in many fields: spectral estimation, sampling, direction of arrival and source localization, imaging beyond the diffraction limit, inverse scattering, exponential data analysis, to name a few. A major challenge for applied mathematics is to establish computational limits of resolution, and develop optimal inversion algorithms.
The course will introduce the students to mathematical theories of superresolution, covering both classical techniques as well as modern approaches. List of topics (some of them time permitting):
Wellposed and illposed problems, conditioning.
Resolution in inverse problems: bandlimited systems, classical resolution limits. Computational vs. instrumental SR. Importance of priors.
Superresolution by analytic continuation. Apriori stability estimates, regularization theory, truncated SVD as a universal method. Discretization. Iterative methods. Introduction to superoscillations.
Superresolution with sparsity: parametric deconvolution of spike trains by algebraic techniques. Fundamental computational limits. Connections to harmonic analysis. Vandermonde matrices on the unit circle. Highresolution algorithms. Sampling beyond the Nyquist rate. Optimizationbased methods.
Inverse scattering and inverse diffraction problems.
The course will include a theoretical/computational project, possibly leading to an M.Sc. thesis.