In this paper we deal with Morozov's discrepancy principle as an aposteriori parameter choice rule for Tikhonov regularization with general convex penalty terms Ψ for non-linear inverse problems. It is shown that a regularization parameter α fulfilling the discprepancy principle exists, whenever the operator F satisfies some basic conditions, and that for this parameter choice rule holds α → 0, δ q /α → 0 as the noise level δ goes to 0. It is illustrated that for suitable penalty terms this yields convergence of the regularized solutions to the true solution in the topology induced by Ψ. Finally, we establish convergence rates with respect to the generalized Bregman distance and a numerical example is presented.
We derive convergence rates for Tikhonov-type regularization with convex penalty terms, where the regularization parameter is chosen according to Morozov's discrepancy principle and variational inequalities are used to generalize classical source and nonlinearity conditions. Rates are obtained first with respect to the Bregman distance and a Taylor-type distance and those results are combined to derive rates in norm and the penalty term topology.For the special case of the sparsity promoting weighted ℓp-norms as penalty terms and for a searched-for solution, which is known to be sparse, the above results give convergence rates of up to linear order.
The convergence rates results in ℓ 1 -regularization when the sparsity assumption is narrowly missed, presented by Burger et al. (2013 Inverse Problems 29 025013), are based on a crucial condition which requires that all basis elements belong to the range of the adjoint of the forward operator. Partly it was conjectured that such a condition is very restrictive. In this context, we study sparsity-promoting varieties of Tikhonov regularization for linear ill-posed problems with respect to an orthonormal basis in a separable Hilbert space using ℓ 1 and sublinear penalty terms. In particular, we show that the corresponding range condition is always satisfied for all basis elements if the problems are well-posed in a certain weaker topology and the basis elements are chosen appropriately related to an associated Gelfand triple. The Radon transform, Symm's integral equation and linear integral operators of Volterra type are examples for such behaviour, which allows us to apply convergence rates results for non-sparse solutions, and we further extend these results also to the case of non-convex ℓ q -regularization with 0 < q < 1.2010 Mathematics Subject Classification. 65J20, 47A52, 44A12, 49J40.
Abstract. The SD-SPIDER method for the characterization of ultrashort laser pulses requires the solution of a nonlinear integral equation of autoconvolution type with a device-based kernel function. Taking into account the analytical background of a variational regularization approach for solving the corresponding ill-posed operator equation formulated in complex-valued L 2 -spaces over finite real intervals, we suggest and evaluate numerical procedures using NURBS and the TIGRA method for calculating the regularized solutions in a stable manner. In this context, besides the complex deautoconvolution problem with noisy but full data, a phase retrieval problem is introduced which adapts to the experimental state of the art in laser optics. For the treatment of this problem facet, which is formulated as a tensor product operator equation, we derive wellposedness of variational regularization methods. Case studies with synthetic and real optical data show the capability of the implemented approach as well as its limitation due to measurement deficits. MSC2010 subject classification: 47A52, 47J06, 78A60, 65R32, 45Q05, 65J15
We formulate the problem of designing gradient-index optical coatings as the task of solving a system of operator equations. We use iterative numerical procedures known from the theory of inverse problems to solve it with respect to the coating refractive index profile and thickness. The mathematical derivations necessary for the application of the procedures are presented, and different numerical methods (Landweber, Newton, and Gauss-Newton methods, Tikhonov minimization with surrogate functionals) are implemented. Procedures for the transformation of the gradient coating designs into quasi-gradient ones (i.e., multilayer stacks of homogeneous layers with different refractive indices) are also developed. The design algorithms work with physically available coating materials that could be produced with the modern coating technologies.
Abstract. In this paper we present a globally convergent algorithm for the computation of a minimizer of the Tikhonov functional with sparsity promoting penalty term for nonlinear forward operators in Banach space. The dual TIGRA method uses a gradient descent iteration in the dual space at decreasing values of the regularization parameter α j , where the approximation obtained with α j serves as the starting value for the dual iteration with parameter α j+1 . With the discrepancy principle as a global stopping rule the method further yields an automatic parameter choice. We prove convergence of the algorithm under suitable step-size selection and stopping rules and illustrate our theoretic results with numerical experiments for the nonlinear autoconvolution problem.
Abstract. Magnetic Resonance Imaging with parallel data acquisition requires algorithms for reconstructing the patient's image from a small number of measured lines of the Fourier domain (k-space).In contrast to well-known algorithms like SENSE and GRAPPA and its flavors we consider the problem as a non-linear inverse problem. However, in order to avoid cost intensive derivatives we will use Landweber-Kaczmarz iteration and in order to improve the overall results some additional sparsity constraints.
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