[Vortraege] Nachtrag zu den Vortragsankündigungen dieser Woche (KW20)

[Dekanat für Mathematik]

Sehr geehrte Fakultätsmitglieder,

anbei ein Nachtrag zum Mathematischen Kolloquium am Mittwoch, 13. Mai, 
16:15 Uhr bis 17:00 Uhr, C 209

Univ.-Doz. Dr. Massimo Fornasier (RICAM, Linz)
„Sparse Approximation and Optimization in High Dimensions (Approximation 
und Optimierung in hoeheren Dimensionen)“

Solutions of certain PDEs and variational problems may be characterized 
by "a few significant degrees of freedom", and one may want to take 
advantage
of this feature in order to design efficient numerical solutions. 
Examples of such situations are ubiquitous: adaptive solution of PDEs, 
degenerate PDEs for image processing, crack modelling and 
free-discontinuity problems, viscosity solutions of Hamilton-Jacobi
equations, digital signal coding/decoding, and compressed sensing. In 
the first part of the talk we review the role of variational
principles, in particular L1-minimization, as a method for sparsifying 
solutions in several contexts. Then we address particular applications
and numerical methods. We present the analysis of a 
superlinear-convergent algorithm for L1 minimization based on an 
iteratively re-weighted least-squares method. An analogous algorithm is 
then applied for the efficient solution of a system of degenerate PDEs 
for image recolorization in a relevant real-life
problem of art restoration. We give a short description of a few 
advances in the use of this sparsity promoting algorithm for certain 
learning
theory problems. This introduces us to other algorithms for performing 
efficiently L1 minimization, based on projected gradient methods and
subspace-correction/domain-decomposition methods, and their 
modifications for free-discontinuity problems.
The second part of the talk addresses the issue of embedding 
compressibility in numerical simulation, and in particular the use of 
adaptive strategies for the solution of elliptic partial differential 
equations discretized by means of redundant frames. We discuss the 
construction of wavelet frames on bounded domains and the optimal 
performances of adaptive solvers in this context. We conclude the talk 
with a vision of the prosepectives in this field and
several new open questions.


Mit freundlichen Grüßen
Margit Honkisz