Fractional Tikhonov regularization methods have been recently proposed to reduce the oversmoothing property of the Tikhonov regularization in standard form, in order to preserve the details of the approximated solution. Their regularization and convergence properties have been previously investigated showing that they are of optimal order. This paper provides saturation and converse results on their convergence rates. Using the same iterative refinement strategy of iterated Tikhonov regularization, new iterated fractional Tikhonov regularization methods are introduced. We show that these iterated methods are of optimal order and overcome the previous saturation results. Furthermore, nonstationary iterated fractional Tikhonov regularization methods are investigated, establishing their convergence rate under general conditions on the iteration parameters. Numerical results confirm the effectiveness of the proposed regularization iterations.

We propose a non-stationary iterated network Tikhonov (iNETT) method for the solution of ill-posed inverse problems. The iNETT employs deep neural networks to build a data-driven regularizer, and it avoids the difficult task of estimating the optimal regularization parameter. To achieve the theoretical convergence of iNETT, we introduce uniformly convex neural networks to build the data-driven regularizer. Rigorous theories and detailed algorithms are proposed for the construction of convex and uniformly convex neural networks. In particular, given a general neural network architecture, we prescribe sufficient conditions to achieve a trained neural network which is component-wise convex or uniformly convex; moreover, we provide concrete examples of realizing convexity and uniform convexity in the modern U-net architecture. With the tools of convex and uniformly convex neural networks, the iNETT algorithm is developed and a rigorous convergence analysis is provided. Lastly, we show applications of the iNETT algorithm in 2D computerized tomography, where numerical examples illustrate the efficacy of the proposed algorithm.

Given a linear self-adjoint differential operator $$\mathscr {L}$$L along with a discretization scheme (like Finite Differences, Finite Elements, Galerkin Isogeometric Analysis, etc.), in many numerical applications it is crucial to understand how good the (relative) approximation of the whole spectrum of the discretized operator $$\mathscr {L}\,^{(n)}$$L ( n ) is, compared to the spectrum of the continuous operator $$\mathscr {L}$$L . The theory of Generalized Locally Toeplitz sequences allows to compute the spectral symbol function $$\omega $$ω associated to the discrete matrix $$\mathscr {L}\,^{(n)}$$L ( n ) . Inspired by a recent work by T. J. R. Hughes and coauthors, we prove that the symbol $$\omega $$ω can measure, asymptotically, the maximum spectral relative error $$\mathscr {E}\ge 0$$E ≥ 0 . It measures how the scheme is far from a good relative approximation of the whole spectrum of $$\mathscr {L}$$L , and it suggests a suitable (possibly non-uniform) grid such that, if coupled to an increasing refinement of the order of accuracy of the scheme, guarantees $$\mathscr {E}=0$$E = 0 .

We are mainly concerned with sequences of graphs having a grid geometry, with a uniform local structure in a bounded domain $${\Omega} {\subset} \mathbb{R}^{d}, d \geq 1$$Ω ⊂ R d , d ≥ 1 . When $$\Omega = [0, 1]$$Ω = [ 0 , 1 ] , such graphs include the standard Toeplitz graphs and, for $$\Omega = [0, 1]^{d}$$Ω = [ 0 , 1 ] d , the considered class includes d-level Toeplitz graphs. In the general case, the underlying sequence of adjacency matrices has a canonical eigenvalue distribution, in the Weyl sense, and we show that we can associate to it a symbol $$\mathfrak{f}$$f . The knowledge of the symbol and of its basic analytical features provides many information on the eigenvalue structure, of localization, spectral gap, clustering, and distribution type.Few generalizations are also considered in connection with the notion of generalized locally Toeplitz sequences and applications are discussed, stemming e.g. from the approximation of differential operators via numerical schemes. Nevertheless, more applications can be taken into account, since the results presented here can be applied as well to study the spectral properties of adjacency matrices and Laplacian operators of general large graphs and networks.