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Iteratively regularized Newtontype methods for general data fidelity functionals and applications to Poisson data
 Nummer. Math
"... We study Newton type methods for inverse problems described by nonlinear operator equations F (u) = g in Banach spaces where the Newton equations F ′(un;un+1−un) = g−F (un) are regularized variationally using a general data misfit functional and a convex regularization term. This generalizes the ..."
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Cited by 7 (3 self)
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We study Newton type methods for inverse problems described by nonlinear operator equations F (u) = g in Banach spaces where the Newton equations F ′(un;un+1−un) = g−F (un) are regularized variationally using a general data misfit functional and a convex regularization term. This generalizes the wellknown iteratively regularized GaussNewton method (IRGNM). We prove convergence and convergence rates as the noise level tends to 0 both for an a priori stopping rule and for a Lepskĭıtype a posteriori stopping rule. Our analysis includes previous order optimal convergence rate results for the IRGNM as special cases. The main focus of this paper is on inverse problems with Poisson data where the natural data misfit functional is given by the KullbackLeibler divergence. Two examples of such problems are discussed in detail: an inverse obstacle scattering problem with amplitude data of the farfield pattern and a phase retrieval problem. The performence of the proposed method for these problems is illustrated in numerical examples. 1
Ground States and Singular Vectors of Convex Variational Regularization Methods
, 2012
"... Singular value decomposition is the key tool in the analysis and understanding of linear regularization methods in Hilbert spaces. Besides simplifying computations it allows to provide a good understanding of properties of the forward problem compared to the prior information introduced by the regul ..."
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Cited by 5 (2 self)
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Singular value decomposition is the key tool in the analysis and understanding of linear regularization methods in Hilbert spaces. Besides simplifying computations it allows to provide a good understanding of properties of the forward problem compared to the prior information introduced by the regularization methods. In the last decade nonlinear variational approaches such as ℓ 1 or total variation regularizations became quite prominent regularization techniques with certain properties being superior to standard methods. In the analysis of those, singular values and vectors did not play any role so far, for the obvious reason that these problems are nonlinear, together with the issue of defining singular values and singular vectors in the first place. In this paper however we want to start a study of singular values and vectors for nonlinear variational regularization of linear inverse problems, with particular focus on singular onehomogeneous regularization functionals. A major role is played by the smallest singular value, which we define as the ground state of an appropriate functional combining the (semi)norm introduced by the forward operator and the regularization functional. The optimality condition for the ground state further yields a natural generalization to higher singular values
Convergence rates in expectation for Tikhonovtype regularization of Inverse Problems with Poisson data
, 2014
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Convergence rates in ℓ 1 regularization if the sparsity assumption fails
 Inverse Problems
"... Variational sparsity regularization based on ℓ 1norms and other nonlinear functionals has gained enormous attention recently, both with respect to its applications and its mathematical analysis. A focus in regularization theory has been to develop error estimation in terms of regularization paramet ..."
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Cited by 3 (3 self)
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Variational sparsity regularization based on ℓ 1norms and other nonlinear functionals has gained enormous attention recently, both with respect to its applications and its mathematical analysis. A focus in regularization theory has been to develop error estimation in terms of regularization parameter and noise strength. For this sake specific error measures such as Bregman distances and specific conditions on the solution such as source conditions or variational inequalities have been developed and used. In this paper we provide, for a certain class of illposed linear operator equations, a convergence analysis that works for solutions that are not completely sparse, but have a fast decaying nonzero part. This case is not covered by standard source conditions, but surprisingly can be treated with an appropriate variational inequality. As a consequence the paper also provides the first examples where the variational inequality approach, which was often believed to be equivalent to appropriate source conditions, can indeed go farther than the latter.
A few remarks on variational models for denoising
 Commun. Math. Sci
"... Abstract Variational models for image and signal denoising are based on the minimization of energy functionals consisting of a fidelity term together with higherorder regularization. In addition to the choices of function spaces to measure fidelity and impose regularization, different scaling expo ..."
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Cited by 1 (0 self)
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Abstract Variational models for image and signal denoising are based on the minimization of energy functionals consisting of a fidelity term together with higherorder regularization. In addition to the choices of function spaces to measure fidelity and impose regularization, different scaling exponents appear. In this note we present a few simple, yet novel, remarks on (i) the stability with respect to deterministic noise perturbations, captured via oscillatory sequences converging weakly to zero, and (ii) exact reconstruction.
Total Variation Regularisation in Measurement and Image space for
"... Abstract. The aim of this paper is to test and analyze a novel technique for image reconstruction in positron emission tomography, which is based on (total variation) regularization on both the image space and the projection space. We formulate our variational problem considering both total variatio ..."
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Abstract. The aim of this paper is to test and analyze a novel technique for image reconstruction in positron emission tomography, which is based on (total variation) regularization on both the image space and the projection space. We formulate our variational problem considering both total variation penalty terms on the image and on an idealized sinogram to be reconstructed from a given Poisson distributed noisy sinogram. We prove existence, uniqueness and stability results for the proposed model and provide some analytical insight into the structures favoured by joint regularization. For the numerical solution of the corresponding discretized problem we employ the split Bregman algorithm and extensively test the approach in comparison to standard total variation regularization on the image. The numerical results show that an additional penalty on the sinogram performs better on reconstructing images with thin structures. 1.
Institut für Numerische und Angewandte Mathematik Iteratively regularized Newtontype methods with general data misfit functionals and applications to Poisson data
, 2012
"... Iteratively regularized Newtontype methods for general data misfit functionals and applications to Poisson data ..."
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Iteratively regularized Newtontype methods for general data misfit functionals and applications to Poisson data