This paper is concerned with the development of adaptive numerical methods for elliptic operator equations. We are especially interested in discretization schemes based on frames. The central objective is to derive an adaptive frame algorithm which is guaranteed to converge for a wide range of cases. As a core ingredient we use the concept of Gelfand frames which induces equivalences between smoothness norms and weighted sequence norms of frame coefficients. It turns out that this Gelfand characteristic of frames is closely related to their localization properties. We also give constructive examples of Gelfand wavelet frames on bounded domains. Finally, an application to the efficient adaptive computation of canonical dual frames is presented.

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We are concerned with Tikhonov regularization of linear ill-posed problems with 1 coefficient penalties. In [Inverse Probl. 24 (2008) 035007], Griesse and Lorenz proposed a semismooth Newton method for the efficient minimization of the corresponding Tikhonov functionals. While the convergence of semismooth Newton methods is locally superlinear in general, their application to 1 Tikhonov regularization is particularly attractive because here, one obtains the exact Tikhonov minimizer after a finite number of iterations, given a sufficiently good initial guess. In this work, we discuss the efficient globalization of B(ouligand)-semismooth Newton methods for 1 Tikhonov regularization by means of damping strategies and suitable descent with respect to an associated merit functional. Numerical examples are provided which show that our method compares well with existing iterative, globally convergent approaches.

Abstract. We are concerned with the efficient numerical solution of minimization problems in Hilbert spaces involving sparsity constraints. These optimizations arise, e.g., in the context of inverse problems. In this work we analyze an efficient variant of the well-known iterative soft-shrinkage algorithm for large or even infinite dimensional problems. This algorithm is modified in the following way. Instead of prescribing a fixed thresholding parameter, we use a decreasing thresholding strategy. Moreover, we use suitable variants of the adaptive schemes derived by Cohen, Dahmen and DeVore for the approximation of the infinite matrix-vector products. We derive a block multiscale preconditioning technique which allows for local well-conditioning of the underlying matrices and for extending the concept of restricted isometry property to infinitely labelled matrices. The combination of these ingredients gives rise to a numerical scheme that is guaranteed to converge with exponential rate, and which allows for a controlled inflation of the support size of the iterations. We also present numerical experiments that confirm the applicability of our approach which extends concepts from compressed sensing to large scale simulation.

This paper is concerned with the numerical treatment of inverse heat conduction problems. In particular, we combine recent results on the regularization of ill-posed problems by iterated soft shrinkage with adaptive wavelet algorithms for the forward problem. The analysis is applied to an inverse parabolic problem that stems from the industrial process of melting Communicated by guest editors: