# CBC Talk on Sparse Regularisation for Inverse Problems - May 7, 2015

Total number of participants: 9

Total number of guests outside of CBC: 1

Number of different nationalities represented: 4

Total number of speakers: 1

Total number of talks: 1

**Title: **Sparse regularisation for inverse problems

**Abstract:**

There exist a large number of applications where one tries to reconstruct from given noisy measurements an object that is characterized by its sparsity properties. Examples include, for instance, natural images, which tend to be sparse with respect to a wavelet basis, or also medical images, which may consist of uniform regions separated by well defined edges; in this case it is the gradients that can said to be sparse. While classical, quadratic regularisation methods are often well capable of producing fairly good results, they tend to perform badly in the reconstruction of sparsity patterns, because coefficients close to zero are penalised only in a very weak form. As a remedy, one can replace the quadratic penalization by a sub-quadratic penalization. Apart from enhancing the sparsity, this has the additional effect that the accuracy of the reconstructions is, potentially, vastly improved. In the case of ill-posed problems, there are theoretical results that show that the worst case error obtained with quadratical regularisation decreases in all non-trivial situations much slower than the measurement error. That is, the accuracy of the reconstruction can be several orders of magnitude worse than the accuracy of the measurements. In contrast, non-smooth regularisation methods allow for reconstructions with errors of the same order as the noise level. In this talk, I will give a short overview about the possibilities but also shortcomings of sparsity enhancing regularisation methods.

Participants:

Martin Alnæs

Gabriel Balaban

Henrik Finsberg

Simon Funke

Markus Grasmair

Valeriya Naumova

Magne Nordaas

Marie Rognes

Kristian Valen-Sendstad

Center for Biomedical Computing (CBC) aims to develop and apply novel simulation technologies to reach new understanding of complex physical processes affecting human health. We target selected medical problems where insight from mathematical modeling can contribute to changing clinical practice.