Showing papers in "Inverse Problems in 2012"

Solving the split feasibility problem without prior knowledge of matrix norms

Abstract: The split feasibility problem (SFP) consists in finding a point in a given closed convex subset of a Hilbert space such that its image under a bounded linear operator belongs to a given closed convex subset of another Hilbert space. Iterative methods can be employed to solve the SFP. The most popular iterative method is Byrne’s CQ algorithm. However, to employ Byrne’s CQ algorithm, one needs to know a priori the norm (or at least an estimate of the norm) of the bounded linear operator (matrix in the finite-dimensional framework). It is the purpose of this paper to introduce a way of selecting the stepsizes such that the implementation of the CQ algorithm does not need any prior information about the operator norm. We also practise this way of selecting stepsizes for variants of the CQ algorithm, including a relaxed CQ algorithm where the two closed convex sets are both level sets of convex functions, and a Halpern-type algorithm. Both weak and strong convergence are investigated. Numerical experiments are included to illustrate the applications in signal processing of the CQ algorithm with stepsizes selected in an adaptive way.

Alternating direction methods for classical and ptychographic phase retrieval

Abstract: In this paper, we show how the augmented Lagrangian alternating direction method (ADM) can be used to solve both the classical and ptychographic phase retrieval problems. We point out the connection between ADM and projection algorithms such as the hybrid input–output algorithm, and compare its performance against standard algorithms for phase retrieval on a number of test images. Our computational experiments show that ADM appears to be less sensitive to the choice of relaxation parameters, and it usually outperforms the existing techniques for both the classical and ptychographic phase retrieval problems.

A direct sampling method to an inverse medium scattering problem

Abstract: In this work we present a novel sampling method for time harmonic inverse medium scattering problems. It provides a simple tool to directly estimate the shape of the unknown scatterers (inhomogeneous media), and it is applicable even when the measured data are only available for one or two incident directions. A mathematical derivation is provided for its validation. Two- and three-dimensional numerical simulations are presented, which show that the method is accurate even with a few sets of scattered field data, computationally efficient, and very robust with respect to noises in the data.

An inverse problem for a one-dimensional time-fractional diffusion problem

Abstract: We study an inverse problem of recovering a spatially varying potential term in a one-dimensional time-fractional diffusion equation from the flux measurements taken at a single fixed time corresponding to a given set of input sources. The unique identifiability of the potential is shown for two cases, i.e. the flux at one end and the net flux, provided that the set of input sources forms a complete basis in L2(0, 1). An algorithm of the quasi-Newton type is proposed for the efficient and accurate reconstruction of the coefficient from finite data, and the injectivity of the Jacobian is discussed. Numerical results for both exact and noisy data are presented.

Sparsity regularization for parameter identification problems

Abstract: The investigation of regularization schemes with sparsity promoting penalty terms has been one of the dominant topics in the field of inverse problems over the last years, and Tikhonov functionals with lp-penalty terms for 1 ⩽ p ⩽ 2 have been studied extensively. The first investigations focused on regularization properties of the minimizers of such functionals with linear operators and on iteration schemes for approximating the minimizers. These results were quickly transferred to nonlinear operator equations, including nonsmooth operators and more general function space settings. The latest results on regularization properties additionally assume a sparse representation of the true solution as well as generalized source conditions, which yield some surprising and optimal convergence rates. The regularization theory with lp sparsity constraints is relatively complete in this setting; see the first part of this review. In contrast, the development of efficient numerical schemes for approximating minimizers of Tikhonov functionals with sparsity constraints for nonlinear operators is still ongoing. The basic iterated soft shrinkage approach has been extended in several directions and semi-smooth Newton methods are becoming applicable in this field. In particular, the extension to more general non-convex, non-differentiable functionals by variational principles leads to a variety of generalized iteration schemes. We focus on such iteration schemes in the second part of this review. A major part of this survey is devoted to applying sparsity constrained regularization techniques to parameter identification problems for partial differential equations, which we regard as the prototypical setting for nonlinear inverse problems. Parameter identification problems exhibit different levels of complexity and we aim at characterizing a hierarchy of such problems. The operator defining these inverse problems is the parameter-to-state mapping. We first summarize some general analytic properties derived from the weak formulation of the underlying differential equation, and then analyze several concrete parameter identification problems in detail. Naturally, it is not possible to cover all interesting parameter identification problems. In particular we do not include problems related to inverse scattering or nonlinear tomographic problems such as optical, thermo-acoustic or opto-acoustic imaging. Also we do not review the extensive literature on the closely related field of control problems for partial differential equations. However, we include one example which highlights the differences and similarities between control theory and the inverse problems approach in this context.

Sparse deterministic approximation of Bayesian inverse problems

Abstract: We present a parametric deterministic formulation of Bayesian inverse problems with an input parameter from infinite-dimensional, separable Banach spaces. In this formulation, the forward problems are parametric, deterministic elliptic partial differential equations, and the inverse problem is to determine the unknown, parametric deterministic coefficients from noisy observations comprising linear functionals of the solution. We prove a generalized polynomial chaos representation of the posterior density with respect to the prior measure, given noisy observational data. We analyze the sparsity of the posterior density in terms of the summability of the input data's coefficient sequence. The first step in this process is to estimate the fluctuations in the prior. We exhibit sufficient conditions on the prior model in order for approximations of the posterior density to converge at a given algebraic rate, in terms of the number N of unknowns appearing in the parametric representation of the prior measure. Similar sparsity and approximation results are also exhibited for the solution and covariance of the elliptic partial differential equation under the posterior. These results then form the basis for efficient uncertainty quantification, in the presence of data with noise.

Reconstructing absorption and scattering distributions in quantitative photoacoustic tomography

Abstract: Quantitative photoacoustic tomography is a novel hybrid imaging technique aiming at estimating optical parameters inside tissues. The method combines (functional) optical information and accurate anatomical information obtained using ultrasound techniques. The optical inverse problem of quantitative photoacoustic tomography is to estimate the optical parameters within tissue when absorbed optical energy density is given. In this paper we consider reconstruction of absorption and scattering distributions in quantitative photoacoustic tomography. The radiative transport equation and diffusion approximation are used as light transport models and solutions in different size domains are investigated. The simulations show that scaling of the data, for example by using logarithmic data, can be expected to significantly improve the convergence of the minimization algorithm. Furthermore, both the radiative transport equation and diffusion approximation can give good estimates for absorption. However, depending on the optical properties and the size of the domain, the diffusion approximation may not produce as good estimates for scattering as the radiative transport equation.

Local analysis of inverse problems: Hölder stability and iterative reconstruction

Abstract: We consider a class of inverse problems defined by a nonlinear mapping from parameter or model functions to the data, where the inverse mapping is Holder continuous with respect to appropriate Banach spaces. We analyze a nonlinear Landweber iteration and prove local convergence and convergence rates with respect to an appropriate distance measure. Opposed to the standard analysis of the nonlinear Landweber iteration, we do not assume source and nonlinearity conditions, but this analysis is based solely on the Holder continuity of the inverse mapping.

A uniform reconstruction formula in integral geometry

Abstract: A new method for analytic inversion of Radon type integral transforms is proposed. Key words: Regular hypersurface family, Funk-Radon transform, Principal value integral, Reconstruction, Hyperbolic algebraic curve MSC 53C65 44A12 65R10 1 Introduction We present a uniform reconstruction method for a class of geometric integral transforms for submanifolds of codimension 1. The reconstruction does not include summation of an in…nite series and looks like a standard inversion of the Radon transform. We specify this method for classical and new acquisition geometries. The condition of regularity is necessary for an inversion operator to be bounded in a Sobolev space scale, but it is not su¢ cient. Existence of an exact reconstruction formula depends on vanishing of some singular integrals of rational forms on a sphere. In §8 we discuss reconstruction for families of spheres. This subject is in focus of recent research, see surveys of related results in [13],[12],[15]. 2 Geometry and integrals Let X and be smooth n dimensional manifolds where n > 1; let Z be a smooth closed hypersurface in X and p : Z ! X; : Z ! be natural projections. We suppose that there exists a real smooth function in X (called generating function) such that Z = f(x; ) ; (x; ) = 0g and dx 6= 0 on Z. Suppose that (i) The map has rank n and the mapping P : N (Z)! T (X) is a local di¤eomorphism. Here, N (Z) denotes the conormal bundle of Z and P (x; ; x; ) = (x; x) 2 T (X) : It follows that the set Z ( ) = 1 ( ) = fx; (x; ) = 0g is for any 2 a smooth hypersurface in X; and for any point x 2 X and for any tangent hyperplane h Tx (X) there is a locally unique hypersurface Z ( ) through x tangent to h: 1 Proposition 2.1 For an arbitrary generating function property (i) is equivalent to the condition det (dx;td ; ) 6= 0 where (x; t; ; ) = t (x; ) ; t; 2 R; t > 0 for any local coordinate system x1; :::; xn in X and any local coordinate system 1; :::; n in : For a proof see [18], Proposition 1.1. De…nition. We call a generating function regular if it satis…es conditions (i) and (ii) there are no conjugate points, that is the equations (x; ) = (y; ) and d (x; ) = d (y; ) are ful…lled for no x 6= y 2 X; 2 . We assume further that X is an open set in an Euclidean space E; let dV be the volume form and dS be a hypersurface element in E. Consider the integral M f ( ) = Z ( (x; )) fdV = c = Z

Parameter choice in Banach space regularization under variational inequalities

Abstract: The authors study parameter choice strategies for the Tikhonov regularization of nonlinear ill-posed problems in Banach spaces. The effectiveness of any parameter choice for obtaining convergence rates depends on the interplay of the solution smoothness and the nonlinearity structure, and it can be expressed concisely in terms of variational inequalities. Such inequalities are link conditions between the penalty term, the norm misfit and the corresponding error measure. The parameter choices under consideration include an a priori choice, the discrepancy principle as well as the Lepskii principle. For the convenience of the reader, the authors review in an appendix a few instances where the validity of a variational inequality can be established.

Absolute uniqueness of phase retrieval with random illumination

Abstract: Random illumination is proposed to enforce absolute uniqueness and resolve all types of ambiguity, trivial or nontrivial, in phase retrieval. Almost sure irreducibility is proved for any complex-valued object whose support set has rank 2. While the new irreducibility result can be viewed as a probabilistic version of the classical result by Bruck, Sodin and Hayes, it provides a novel perspective and an effective method for phase retrieval. In particular, almost sure uniqueness, up to a global phase, is proved for complex-valued objects under general two-point conditions. Under a tight sector constraint absolute uniqueness is proved to hold with probability exponentially close to unity as the object sparsity increases. Under a magnitude constraint with random amplitude illumination, uniqueness modulo global phase is proved to hold with probability exponentially close to unity as object sparsity increases. For general complex-valued objects without any constraint, almost sure uniqueness up to global phase is established with two sets of Fourier magnitude data under two independent illuminations. Numerical experiments suggest that random illumination essentially alleviates most, if not all, numerical problems commonly associated with the standard phasing algorithms. (Some figures may appear in colour only in the online journal)

On multi-spectral quantitative photoacoustic tomography in diffusive regime

Abstract: The objective of quantitative photoacoustic tomography (qPAT) is to reconstruct the diffusion, absorption and Gruneisen thermodynamic coefficients of heterogeneous media from knowledge of the interior absorbed radiation. It has been shown in Bal and Ren (2011 Inverse Problems 27 075003), based on diffusion theory, that with data acquired at one given wavelength, all three coefficients cannot be reconstructed uniquely. In this work, we study the multi-spectral qPAT problem and show that when multiple wavelength data are available, all coefficients can be reconstructed simultaneously under minor prior assumptions. Moreover, the reconstructions are shown to be very stable. We present some numerical simulations that support the theoretical results.

Blind backscattering experimental data collected in the field and an approximately globally convergent inverse algorithm

Abstract: An approximately globally convergent numerical method for a 1D coefficient inverse problem for a hyperbolic PDE is applied to image dielectric constants of targets from blind experimental data. The data were collected in the field by the Forward Looking Radar of the US Army Research Laboratory. A posteriori analysis has revealed that computed and tabulated values of dielectric constants are in good agreement. Convergence analysis is presented.

Conditional stability in determining a zeroth-order coefficient in a half-order fractional diffusion equation by a Carleman estimate

Abstract: An inverse problem of determining a zeroth-order coefficient in a one-dimensional fractional diffusion equation of half-order in time is investigated. Under some assumptions on the regularity of the solutions and coefficients, we prove a conditional stability estimate by some additional data. The key is a Carleman estimate, but since we have no Carleman estimates for the fractional diffusion equation, we further take the t-derivative of half-order to obtain the equation where the principal term is . The nonhomogeneous term is coupled with derivatives of the difference between two coefficients, and so we need an additional Carleman estimate, which is different from usual coefficient inverse problems for partial differential equations.

Stabilizing inverse problems by internal data

Abstract: Several newly developing hybrid imaging methods (e.g., those combining electrical impedance or optical imaging with acoustics) enable one to obtain some auxiliary interior information (usually some combination of the electrical conductivity and the current) about the parameters of the tissues. This information, in turn, happens to stabilize the exponentially unstable and thus low-resolution optical and electrical impedance tomography. Various known instances of this effect have been studied individually. We show that there is a simple general technique (covering all known cases) that shows what kinds of interior data stabilize the reconstruction, and why. Namely, we show when the linearized problem becomes an elliptic pseudo-differential one, and thus stable. Stability here is meant as the problem being Fredholm, so the local uniqueness is not shown and probably does not hold in such generality.

Sparsity-promoting Bayesian inversion

Abstract: A computational Bayesian inversion model is demonstrated. It is discretization invariant, describes prior information using function spaces with a wavelet basis and promotes reconstructions that are sparse in the wavelet transform domain. The method makes use of the Besov space prior with p = 1, q = 1 and s = 1, which is related to the total variation prior. Numerical evidence is presented in the context of a one-dimensional deconvolution task, suggesting that edge-preserving and noise-robust reconstructions can be achieved consistently at various resolutions.

Hybrid tomography for conductivity imaging

Abstract: Hybrid imaging techniques utilize couplings of physical modalities—they are called hybrid, because typically, the excitation and measurement quantities belong to different modalities. Recently there has been an enormous research interest in this area because these methods promise very high resolution. In this paper, we give a review on hybrid tomography methods for electrical conductivity imaging. The reviewed imaging methods utilize couplings between electric, magnetic and ultrasound modalities. By this it is possible to perform high-resolution electrical impedance imaging and to overcome the low-resolution problem of electric impedance tomography.

Joint inversion approaches for geophysical electromagnetic and elastic full-waveform data

Abstract: We present joint inversion approaches for integrating controlled source electromagnetic data and seismic full-waveform data for geophysical applications. The first approach is the joint petrophysical inversion carried out by reconstructing petrophysical parameters such as porosity and saturations instead of the usual geophysical parameters such as resistivity, seismic velocities and mass density. This approach utilizes the strong correlation between the electromagnetic and seismic parameters through the petrophysical relationships. Another approach that does not require the a priori petrophysical correlation is the joint structural inversion method. In this approach, the inversion is carried out by employing a regularization function for enforcing the structural similarity between the resistivity and the seismic velocities and the mass density. In this method, we employ the cross-gradient function, which has been shown on many occasions to be quite effective. By using a time-lapse reservoir monitoring example, we show that both joint inversion approaches produce results that are superior to those obtained by disjointed inversions. Hence, these joint inversions have great potential to be the next-generation tools for reservoir characterization and monitoring.

Complex transmission eigenvalues for spherically stratified media

Abstract: We investigate the existence of complex transmission eigenvalues for spherically stratified media in and . In with the index of refraction being constant, we show that there exists an infinite number of complex eigenvalues. In and constant index of refraction, we show that if the index of refraction is an integer there are no complex eigenvalues, whereas if the index of refraction is a rational number, complex eigenvalues can exist. Under appropriate assumptions we also show that complex eigenvalues can exist for a spherically stratified variable index of refraction.

A primal–dual interior-point framework for using the L1 or L2 norm on the data and regularization terms of inverse problems

Abstract: Maximum a posteriori estimates in inverse problems are often based on quadratic formulations, corresponding to a least-squares fitting of the data and to the use of the L2 norm on the regularization term. While the implementation of this estimation is straightforward and usually based on the Gauss–Newton method, resulting estimates are sensitive to outliers and result in spatial distributions of the estimates that are smooth. As an alternative, the use of the L1 norm on the data term renders the estimation robust to outliers, and the use of the L1 norm on the regularization term allows the reconstruction of sharp spatial profiles. The ability therefore to use the L1 norm either on the data term, on the regularization term, or on both is desirable, though the use of this norm results in non-smooth objective functions which require more sophisticated implementations compared to quadratic algorithms. Methods for L1-norm minimization have been studied in a number of contexts, including in the recently popular total variation regularization. Different approaches have been used and methods based on primal–dual interior-point methods (PD-IPMs) have been shown to be particularly efficient. In this paper we derive a PD-IPM framework for using the L1 norm indifferently on the two terms of an inverse problem. We use electrical impedance tomography as an example inverse problem to demonstrate the implementation of the algorithms we derive, and the effect of choosing the L2 or the L1 norm on the two terms of the inverse problem. Pseudo-codes for the algorithms and a public domain implementation are provided.

Preconditioned Alternating Projection Algorithms for Maximum a Posteriori ECT Reconstruction

Abstract: We propose a preconditioned alternating projection algorithm (PAPA) for solving the maximum a posteriori (MAP) emission computed tomography (ECT) reconstruction problem. Specifically, we formulate the reconstruction problem as a constrained convex optimization problem with the total variation (TV) regularization. We then characterize the solution of the constrained convex optimization problem and show that it satisfies a system of fixed-point equations defined in terms of two proximity operators raised from the convex functions that define the TV-norm and the constrain involved in the problem. The characterization (of the solution) via the proximity operators that define two projection operators naturally leads to an alternating projection algorithm for finding the solution. For efficient numerical computation, we introduce to the alternating projection algorithm a preconditioning matrix (the EM-preconditioner) for the dense system matrix involved in the optimization problem. We prove theoretically convergence of the preconditioned alternating projection algorithm. In numerical experiments, performance of our algorithms, with an appropriately selected preconditioning matrix, is compared with performance of the conventional MAP expectation-maximization (MAP-EM) algorithm with TV regularizer (EM-TV) and that of the recently developed nested EM-TV algorithm for ECT reconstruction. Based on the numerical experiments performed in this work, we observe that the alternating projection algorithm with the EM-preconditioner outperforms significantly the EM-TV in all aspects including the convergence speed, the noise in the reconstructed images and the image quality. It also outperforms the nested EM-TV in the convergence speed while providing comparable image quality.

Estimating nuisance parameters in inverse problems

Abstract: Many inverse problems include nuisance parameters which, while not of direct interest, are required to recover primary parameters. The structure of these problems allows efficient optimization strategies—a well-known example is variable projection, where nonlinear least-squares problems which are linear in some parameters can be very efficiently optimized. In this paper, we extend the idea of projecting out a subset over the variables to a broad class of maximum likelihood and maximum a posteriori likelihood problems with nuisance parameters, such as variance or degrees of freedom (d.o.f.). As a result, we are able to incorporate nuisance parameter estimation into large-scale constrained and unconstrained inverse problem formulations. We apply the approach to a variety of problems, including estimation of unknown variance parameters in the Gaussian model, d.o.f. parameter estimation in the context of robust inverse problems, and automatic calibration. Using numerical examples, we demonstrate improvement in recovery of primary parameters for several large-scale inverse problems. The proposed approach is compatible with a wide variety of algorithms and formulations, and its implementation requires only minor modifications to existing algorithms.

Convergence rates in expectation for Tikhonov-type regularization of inverse problems with Poisson data

Abstract: In this paper, we study a Tikhonov-type method for ill-posed nonlinear operator equations g† = F(u†), where g† is an integrable, non-negative function. We assume that data are drawn from a Poisson process with density tg†, where t > 0 may be interpreted as an exposure time. Such problems occur in many photonic imaging applications including positron emission tomography, confocal fluorescence microscopy, astronomic observations and phase retrieval problems in optics. Our approach uses a Kullback–Leibler-type data fidelity functional and allows for general convex penalty terms. We prove convergence rates of the expectation of the reconstruction error under a variational source condition as t → ∞ both for an a priori and for a Lepskii-type parameter choice rule.

Accelerated perturbation-resilient block-iterative projection methods with application to image reconstruction

Abstract: We study the convergence of a class of accelerated perturbation-resilient block-iterative projection methods for solving systems of linear equations. We prove convergence to a fixed point of an operator even in the presence of summable perturbations of the iterates, irrespective of the consistency of the linear system. For a consistent system, the limit point is a solution of the system. In the inconsistent case, the symmetric version of our method converges to a weighted least squares solution. Perturbation resilience is utilized to approximate the minimum of a convex functional subject to the equations. A main contribution, as compared to previously published approaches to achieving similar aims, is a more than an order of magnitude speed-up, as demonstrated by applying the methods to problems of image reconstruction from projections. In addition, the accelerated algorithms are illustrated to be better, in a strict sense provided by the method of statistical hypothesis testing, than their unaccelerated versions for the task of detecting small tumors in the brain from X-ray CT projection data.

Discrepancy principle for statistical inverse problems with application to conjugate gradient iteration

Abstract: The authors discuss the use of the discrepancy principle for statistical inverse problems, when the underlying operator is of trace class. Under this assumption the discrepancy principle is well defined, however a plain use of it may occasionally fail and it will yield sub-optimal rates. Therefore, a modification of the discrepancy is introduced, which corrects both of the above deficiencies. For a variety of linear regularization schemes as well as for conjugate gradient iteration it is shown to yield order optimal a priori error bounds under general smoothness assumptions. A posteriori error control is also possible, however at a sub-optimal rate, in general. This study uses and complements previous results for bounded deterministic noise.

A wave-equation-based Kirchhoff operator

Abstract: In this paper, I will study a Kirchhoff-type integral, which can be seen as a linear operator mapping angle–azimuth-dependent reflection coefficients along a reflector into reflection data for the acoustic wave equation. I will show that a minor adaptation of a construction of angle–azimuth-dependent images as proposed by Sava and Fomel leads to a left inverse of this operator, which maps primary reflection data to angle–azimuth-dependent reflection coefficients. The new construction naturally leads to a reformulation of the Kirchhoff operator, acting on space-shift-extended images, which can be implemented completely in terms of the fundamental solutions of the wave equation. I will study the composition of this new wave-equation-based Kirchhoff operator with an operator forming space-shift-extended images from data. I will show that these operators are partial inverses of each other, with their compositions being pseudo-differential operators that reconstruct suitably microlocalized versions of primary reflection data and extended images focused at space-shift zero.

On the multi-frequency inverse source problem in heterogeneous media

Abstract: The inverse source problem where an unknown source is to be identified from knowledge of its radiated wave is studied. The focus is placed on the effect that multi-frequency data have on establishing uniqueness. In particular, it is shown that data obtained from finitely many frequencies are not sufficient. On the other hand, if the frequency varies within a set with an accumulation point, then the source is determined uniquely, even in the presence of highly heterogeneous media. In addition, an algorithm for the reconstruction of the source using multi-frequency data is proposed. The algorithm, based on a subspace projection method, approximates the minimum-norm solution given the available multi-frequency measurements. A few numerical examples are presented.

L ∞ fitting for inverse problems with uniform noise

Abstract: For inverse problems where the data are corrupted by uniform noise such as arising from quantization errors, the L∞ norm is a more robust data-fitting term than the standard L2 norm. Well-posedness and regularization properties for linear inverse problems with L∞ data fitting are shown, and the automatic choice of the regularization parameter is discussed. After introducing an equivalent reformulation of the problem and a Moreau–Yosida approximation, a superlinearly convergent semi-smooth Newton method becomes applicable for the numerical solution of L∞ fitting problems. Numerical examples illustrate the performance of the proposed approach as well as the qualitative behavior of L∞ fitting.

A mathematical model and inversion procedure for magneto-acousto-electric tomography

Abstract: Magneto-acousto-electric tomography (MAET), also known as the Lorentz force or Hall effect tomography, is a novel hybrid modality designed to be a high-resolution alternative to the unstable electrical impedance tomography. In this paper, we analyze the existing mathematical models of this method, and propose a general procedure for solving the inverse problem associated with the MAET. It consists in applying to the data one of the algorithms of thermo-acoustic tomography, followed by solving the Neumann problem for the Laplace equation and the Poisson equation. For the particular case when the region of interest is a cube, we present an explicit series solution resulting in a fast reconstruction algorithm. As we show, both analytically and numerically, the MAET is a stable technique yielding high-resolution images even in the presence of significant noise in the data.

Iterative regularization with a general penalty term—theory and application to L 1 and TV regularization

Abstract: In this paper, we consider an iterative regularization scheme for linear ill-posed equations in Banach spaces. As opposed to other iterative approaches, we deal with a general penalty functional from Tikhonov regularization and take advantage of the properties of the regularized solutions which where supported by the choice of the specific penalty term. We present convergence and stability results for the presented algorithm. Additionally, we demonstrate how these theoretical results can be applied to L1- and TV-regularization approaches and close the paper with a short numerical example.