# Some approaches to sparse solutions of linear ill-posed problems

During the past two decades it has become clear that $lp$ spaces with $p in (0,2)$ and corresponding (quasi)norms are appropriate settings for dealing with reconstruction of sparse solutions of ill-posed problems. In this context, the focus of our presentation is twofold. Firstly, since the question of how to choose the exponent $p$ in such settings has been not only a numerical issue, but also a philosophical one, we present a more flexible way of (performing/achieving) sparse regularization by varying exponents. Rather than using norms or quasinorms, we employ F-norms on infinite dimensional spaces. Secondly, we approach the ill-posed problem $Au=f$ by appropriate discretization in the image space. We formulate the so-called least error method in an $l1$ setting and perform the convergence analysis by choosing the discretization level according to both a priori and a posteriori rules.
Convergence rates are obtained under source condition (usually) yielding sparsity of the solution.

Joint research with Kristian Bredies, Barbara Kaltenbacher and Dirk Lorenz

How can mathematics help us to understand the behaviour of ants? Read more about the fanscinating work being carri… https://t.co/iCODvvxqE6 View on Twitter