abstract:
In this talk we study the stable solution of illposed equations
Ax = y
where A : X → Y is a linear operator between Hilbert spaces X and Y . If this equation
is illposed, then the solution does not depend continuously on the data
y. In order to stably solve the equation, stabilization methods have to be applied. The most widely used stabilization technique is Tikhonov regularization
Τy(x) := Axy^{2} + αR(x) → min.
Here R is a penalty functional and α > 0 is the regularization parameter.
In the recent years  motivated by the success of compresses sensing  the
use of l^{1} penalty
R(x) = x_{l1} := ∑_{λ ∈ Λ}<φ_{λ},x>^{ }
became very popular.(Here (φ_{λ})_{λ ∈ Λ} is some basis of X.) In this talk we study Tikhonov regularization with l^{1} penalty term. In particular, we show that the range condition (ran(A^{*}) ∩ R(x^{+}) ≠ ∅) together with a certain injectivity condition allows linear error estimates (convergence rates) between the solution
x^{+} and minimizers of Τ_{α,y}. Moreover, we show that the range condition is even necessary for this linear convergence rate. This is talk is based on joint work with M. Grasmair and O. Scherzer.
M. Grasmair, M. Haltmeier, and O. Scherzer. Necessary and sufficient conditions
for linear convergence of l^{1}regularization. Reports of FSP S105  \"Photoacoustic Imaging\" 18, University of Innsbruck, Austria, 2009.
