On $\ell^1$-regularization under continuity of the forward operator in weaker topologies
For researchers in inverse problems and sparse regularization, this provides a theoretical extension to non-reflexive image spaces, though the contribution is incremental as it generalizes existing weak*-to-weak results.
This paper extends ℓ1-regularization theory for linear inverse problems from weak*-to-weak continuous forward operators to weak*-to-weak* continuous operators, proving existence, stability, and convergence of regularized solutions. Convergence rates are derived via variational source conditions, showing linear rates only in borderline cases of proper sparsity or well-posedness.
Our focus is on the stable approximate solution of linear operator equations based on noisy data by using $\ell^1$-regularization as a sparsity-enforcing version of Tikhonov regularization. We summarize recent results on situations where the sparsity of the solution slightly fails. In particular, we show how the recently established theory for weak*-to-weak continuous linear forward operators can be extended to the case of weak*-to-weak* continuity. This might be of interest when the image space is non-reflexive. We discuss existence, stability and convergence of regularized solutions. For injective operators, we will formulate convergence rates by exploiting variational source conditions. The typical rate function obtained under an ill-posed operator is strictly concave and the degree of failure of the solution sparsity has an impact on its behavior. Linear convergence rates just occur in the two borderline cases of proper sparsity, where the solutions belong to $\ell^0$, and of well-posedness. For an exemplary operator, we demonstrate that the technical properties used in our theory can be verified in practice. In the last section, we briefly mention the difficult case of oversmoothing regularization where $x^†$ does not belong to $\ell^1$.