Convergence analysis of the discretization of continuous-domain inverse problems

Abstract We study continuous-domain linear inverse problems that involve a general data-fidelity term and a regularisation term. We consider a regularisation that is formed by the sparsity-promoting total-variation norm, pre-composed with a differential operator that specifies some underlying dictionary of atoms. It has been previously shown that such problems have sparse spline solutions with adaptive knots. These knots are part of the parameterization of the solution and their estimation is itself a difficult non-convex problem. To alleviate this difficulty, we rely on an exact discretization of the optimization problem, where the spline knots are chosen on a dense regular grid. We then follow a multiresolution strategy to refine this grid. In this work, we investigate the convergence of the discretization to the original continuous-domain problem when the grid goes from coarse to fine. We provide an in-depth study of this convergence, concluding that its strength depends on the regularity of the Green’s function of the differential operator. We show that uniform convergence holds in very general settings. We carry a numerical analysis to illustrate our theoretical results.