A REFINEMENT AND COARSENING INDICATOR ALGORITHM FOR FINDING SPARSE SOLUTIONS OF INVERSE PROBLEMS

In this paper we extend the idea of adaptive discretization by using refinement and coarsening indicators from papers by Chavent, Bissell, Benameur and Jaffre (cf., e. g., [5], [9]) to a general setting. This allows to make use of the relation between adaptive discretization and sparse paramerization in order to construct an algorithm for finding sparse solutions of inverse problems. We provide some first steps in the analysis of the proposed method and apply it to an inverse problem in systems biology, namely the reconstruction of gene networks in an ordinary differential equation (ODE) model. Here due to the fact that not all genes interact with each other, reconstruction of a sparse connectivity matrix is a key issue.