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We present a componentwise iterated relative entropy regularization method (COMET) where the prior and regularization parameter could be updated in the iterates. Such a reconstruction method could be useful for multicomponent inverse problems, such as the one occurring in electron tomography. The paper also contains a brief introduction to regularization theory with emphasis on variational based regularization methods, and we rigorously prove that the tolerance-based entropy reconstruction method that occurs in the COMET iterates is a regularization method. We conclude by showing examples of COMET applied to electron tomography data.

We examine the method of local regularization for the solution of linear first-kind integral equations on . We provide a theoretical analysis of the method and prove that the regularized solutions converge to the true solution as the level of error in perturbed data goes to zero. We also develop an iterative numerical algorithm based on this theory and describe its implementation. Our testing with a number of examples shows that local regularization tends to perform better than a classical method we call Tik-CG (a method based on a conjugate gradient algorithm with stopping criteria applied to standard Tikhonov regularization) when performance is measured in terms of relative error in solutions and/or in perceived sharpness of images. Unfortunately, this improvement can come at a cost as testing shows that the local regularization algorithm tends to be slower than the Tik-CG approach when applied to 2D images. As we illustrate with our numerical results, however, a compromise can be found by using the converged Tik-CG image as the starting value for the iterative local regularization method.

We will consider the situation of ill-posed problems with regularization methods which demand the choice of interdependent regularization parameters. We show that one can adapt the Lepskij-type balancing principle to this situation.

We discuss a natural linearization approach to the identification of a diffusion coefficient in a quasilinear parabolic equation from observations of its solution during a short period of time. In this context, a natural linearization means a reduction of the original nonlinear inverse problem to two linear ill-posed problems, when the solution of one of them is used as data for the next one. For both these problems no information concerning solution smoothness is usually given. Therefore, an a posteriori regularization parameter choice strategy should be implemented for their regularization. Even if such a strategy allows an order-optimal regularization parameter choice for the first linear problem, it does not provide us with a reliable regularization error estimate. It means that the data perturbation level for the second linear problem is not given exactly. The main accomplishment of the present paper is to propose a new strategy of the regularization parameter choice which allows an order-optimal regularization of the second linear problem under these circumstances. We also estimate the best possible order of accuracy that in principle can be obtained within a natural linearization approach, and discuss its correlation with known results. Numerical experiments on model problems support a theoretical analysis.

Recently a new parameter choice strategy, often called the Lepskii-type strategy, became attractive for regularizing ill-posed problems. Here we present this construction in a simple and unified way. We emphasize that previous modifications can be seen to choose the same parameter, albeit drawing different conclusions from it. Finally we exhibit the application of this a posteriori parameter choice for Tikhonov regularization.

In this paper error estimates for non-quadratic regularization of nonlinear ill-posed problems in Banach spaces are derived. Our analysis is based on a few novel features: in comparison with the classical analysis of regularization methods for inverse and ill-posed problems where a Lipschitz continuity for the Frechet derivative is required, we use a differentiability condition with respect to the Bregman distance. Also, a stability result for the regularized solutions in terms of Bregman distances is proven. Moreover, a source-wise representation of the solution as used in standard theory is interpreted in terms of data enhancement. It is also shown that total variation Bregman distance regularization for image analysis, as developed recently, can be considered as a two-step regularization method consisting of a combination of total variation regularization and additional enhancement. This technique can also be applied for filtering.

For solving linear ill-posed problems regularization methods are required when the available data include some noise. In the present paper regularized approximations are obtained by a general regularization scheme in Hilbert scales which include well-known regularization methods such as the method of Tikhonov regularization and its higher-order forms, spectral methods, asymptotical regularization and iterative regularization methods. For both the cases of high- and low-order regularization, we study a priori and a posteriori rules for choosing the regularization parameter and provide order optimal error bounds that characterize the accuracy of the regularized approximations. These error bounds have been obtained under general smoothing conditions. The results extend earlier results and cover the case of finitely and infinitely smoothing operators. The theory is illustrated by a special ill-posed deconvolution problem arising in geoscience. ? 2005 IOP Publishing Ltd.

A regularization method is proposed for the function reconstruction from approximate average fluxes. Such problems often occur in environmental science and mathematical statistics. The unique solvability of the method is proved and a number of conditions are given to characterize the solution. The error estimates are established after the introduction of some interpolation operators. A series of numerical examples are provided to illustrate the effectiveness and computational performance of the method. Some ideas for the choice of the regularization parameter are also suggested by the computational experience.

Local regularization methods allow for the application of sequential solution techniques for the solution of Volterra problems, retaining the causal structure of the original Volterra problem and leading to fast solution techniques. Stability and convergence of these methods was shown to hold on a large class of linear Volterra problems, i.e., the class of ν-smoothing problems for ν = 1, 2, ... in Lamm (2005 Inverse Problems 21 785–803). In this paper, we enlarge the family of convergent local regularization methods to include sequential versions of classical regularization methods such as sequential Tikhonov regularization. In fact, sequential Tikhonov regularization was considered earlier by Lamm and Elden (1997 SIAM J. Numer. Anal. 34 1432–50) but there the theory was limited to the class of discretized one-smoothing Volterra problems. An interesting feature of sequential classical regularization methods is that they involve two regularization parameters: the usual local regularization parameter r controls the size of the local problem while a second parameter α controls the amount of regularization to be applied in each subproblem. This approach suggests a wavelet type of regularization method with the parameter r controlling spatial resolution and α controlling frequency resolution. In this paper, we also show how the 'future polynomial regularization' method of Cinzori (2004 Inverse Problems 20 1791–806) can be viewed as a special case of the general framework of Lamm (2005) in the 1-smoothing case. In addition, we extend the results of Lamm (2005) to nonlinear Volterra problems of Hammerstein type and give numerical results to illustrate the effectiveness of the method in this case.

This paper deals with quantitative aspects of regularization for ill-posed linear equations in Banach spaces, when the regularization is done using a general convex penalty functional. The error estimates shown here by means of Bregman distances yield better convergences rates than those already known for maximum entropy regularization, as well as for total variation regularization.

Local regularization methods for ill-posed linear Volterra equations have been shown to be efficient regularization procedures preserving the causal structure of the Volterra problem and allowing for sequential solution methods. However questions posed recently in Ring and Prix (2000 Inverse Problems 16 619–34) raise doubts as to whether such methods are convergent for problems which are more than just mildly ill-posed. In this paper we address these questions by reformulating the local regularization method via the use of signed Borel measures instead of the positive Borel measures used in earlier approaches. The result is a new theory for the local regularization of ν-smoothing Volterra problems for which stability and convergence is assured for all ν = 1, 2, .... In this paper we discuss this new local regularization theory for general finitely smoothing Volterra problems and demonstrate convergence and stability of the resulting method. In addition we indicate why using signed Borel measures instead of positive measures makes sense in the context of the Volterra problem and also has connections to the theory of mollification and approximate inverses (Louis A K 1996 Inverse Problems 12 175–90). Finally we include numerical examples which illustrate the improvement which comes from using signed measures instead of positive measures and which facilitates an examination of the role played by the regularization parameter.

This paper provides some new a priori choice strategy for regularization parameters in order to obtain convergence rates in Tikhonov regularization for solving ill-posed problems Af0 = g0, f0 X, g0 Y, with a linear operator A mapping in Hilbert spaces X and Y. Our choice requires only that the range of the adjoint operator A* includes a member of some variable Hilbert scale and is, in principle, applicable in the case of general f0 without source conditions imposed otherwise in the existing papers. For testing our strategies, we apply them to the determination of a wave source, to the Abel integral equation, to a backward heat equation and to the determination of initial temperature by boundary observation.

Based on the thin plate spline approximation theory, we propose in this paper an effective regularization algorithm for the reconstruction of numerical derivatives from two-dimensional scattered noisy data. The convergence results under two different choice rules for the regularization parameter are given. Numerical results show that the proposed method is effective and stable.

This paper is devoted to the regularization of a class of evolution hemivariational inequalities. The operator involved is taken to be non-coercive and the data are assumed to be known approximately. Under the assumption that the evolution hemivariational inequality is solvable, a strongly convergent approximation procedure is designed by means of the so-called Browder–Tikhonov regularization method.

We examine a continuous, local regularization method for 1-smoothing, ill-posed, first-kind Volterra problems. We show that the method converges to the true solution as the noise in the data decreases to zero. Numerical examples are provided to demonstrate the performance of the method in comparison to earlier techniques.

The aim of this paper is to provide quantitative estimates for the minimizers of non-quadratic regularization problems in terms of the regularization parameter, respectively the noise level. As usual for ill-posed inverse problems, these estimates can be obtained only under additional smoothness assumptions on the data, the so-called source conditions, which we identify with the existence of Lagrange multipliers for a limit problem. Under such a source condition, we shall prove a quantitative estimate for the Bregman distance induced by the regularization functional, which turns out to be the natural distance measure to use in this case. We put a special emphasis on the case of total variation regularization, which is probably the most important and prominent example in this class. We discuss the source condition for this case in detail and verify that it still allows discontinuities in the solution, while imposing some regularity on its level sets.

We consider the solution of a distributed parameter estimation problem where the data are contaminated by noise. A common approach to solve such a problem is to use Tikhonov style regularization; however, it is not always clear what type of regularization penalty should be used for a given problem as different regularization operators may yield very different solutions. Here we use supervised learning techniques to determine a regularization functional given a training set of feasible solutions. Our approach leads to a constraint optimization problem that we solve using inexact sequential quadratic programming type methods. We illustrate the methodology with two examples.

We consider the local regularization problem for integral equations of the first kind, generalizing previous work which applied only to problems of Volterra type. Our approach allows for local control of the regularization process, allowing for resolution of fine/sharp features of solutions without having to resort to nondifferentiable optimization techniques. In addition, we present examples illustrating the numerical implementation of one version of the resulting local regularization algorithm and show that, under quite reasonable assumptions, the operation count of the local method compares well with that of standard Tikhonov regularization.

In this paper, we discuss a classical ill-posed problem—numerical differentiation by the Tikhonov regularization. Based on the conditional stability estimate for this ill-posed problem, a new simple method for choosing regularization parameters is proposed. We show that it has an almost optimal convergence rate when the exact solution is in H2. The advantages of our method are (1) we can get similar computational results with much less computation, in comparison with other methods, and (2) we can find the discontinuous points numerically.

The selection of multiple regularization parameters is considered in a generalized L-curve framework. Multiple-dimensional extensions of the L-curve for selecting multiple regularization parameters are introduced, and a minimum distance function (MDF) is developed for approximating the regularization parameters corresponding to the generalized corner of the L-hypersurface. For the single-parameter (i.e. L-curve) case, it is shown through a model that the regularization parameters minimizing the MDF essentially maximize the curvature of the L-curve. Furthermore, for both the single-and multiple-parameter cases the MDF approach leads to a simple fixed-point iterative algorithm for computing regularization parameters. Examples indicate that the algorithm converges rapidly thereby making the problem of computing parameters according to the generalized corner of the L-hypersurface computationally tractable.

In this paper dynamic inverse problems are studied, where the investigated object is allowed to change during the measurement procedure. In order to achieve reasonable results, temporal a priori information will be considered. Here, 'temporal smoothness' is used as a quite general, but for many applications sufficient, a priori information. This is justified in the case of slight movements during an x-ray scan in computerized tomography, or in the field of current density reconstruction, where one wants to conclude from electrical measurements on the surface of the head, the locations of brain activity. First, the notion of a dynamic inverse problem is introduced, then we describe how temporal smoothness can be incorporated in the regularization of the problem, and finally an efficient solver and some regularization properties of this solver are presented. This theory will be exploited in three practically relevant applications in a following paper.

Computing regularization parameters for general-form Tikhonov regularization can be an expensive and difficult task, especially if multiple parameters or many solutions need to be computed in real time. In this work, we assume training data is available and describe an efficient learning approach for computing regularization parameters that can be used for a large set of problems. We consider an empirical Bayes risk minimization framework for finding regularization parameters that minimize average errors for the training data. We first extend methods from Chung et al (2011 SIAM J. Sci. Comput. 33 3132–52) to the general-form Tikhonov problem. Then we develop a learning approach for multi-parameter Tikhonov problems, for the case where all involved matrices are simultaneously diagonalizable. For problems where this is not the case, we describe an approach to compute near-optimal regularization parameters by using operator approximations for the original problem. Finally, we propose a new class of regularizing filters, where solutions correspond to multi-parameter Tikhonov solutions, that requires less data than previously proposed optimal error filters, avoids the generalized SVD, and allows flexibility and novelty in the choice of regularization matrices. Numerical results for 1D and 2D examples using different norms on the errors show the effectiveness of our methods.

We investigate the reconstruction of the domain D from the measured far field pattern for scattering of some incident field ui, e.g. for the inverse two- or three-dimensional acoustic sound-soft or two-dimensional electromagnetic perfect-conductor obstacle scattering problems. Both a perfect Newton scheme, a partially regularized and a fully regularized Newton scheme with stopping rule are investigated. As regularization methods we combine quasi-solutions with either the point-source method, minimum norm solutions, Tikhonov regularization or spectral cut-off. We prove superlinear local convergence of regularized Newton updates towards a regularized solution and the convergence of the regularized solution towards the true solution when the data error tends to zero, both under the condition that the true solution has an analytic boundary and the normal derivative of the total field has no zeros on the boundary. In addition, the strong relation of Newton's method to a version of the point-source method is shown when Newton updates are used to find the unknown domain from the reconstructed scattered field us.

In this paper we apply a new regularization method, regularization for surface representations to two-dimensional parameter estimation problems where the parameter is supposed to have jumps. This method is well suited for ill-posed problems with discontinuous solutions. It is a generalization of the recently developed method, regularization for curve representations, which has been successfully applied to linear and nonlinear one-dimensional ill-posed problems. We prove convergence of the finite-dimensional Tikhonov regularized solutions and present some numerical results.

We are interested in this paper in recovering lacking data on some part of a domain boundary, from the knowledge of Cauchy data on the other part. It is first proved that the desired solution is the unique fixed point of some appropriate operator, which naturally gives rise to an iterative process that is proved to be convergent. Discretization provides an additional regularization: the algorithm reads as a least square fitting of the given data, with a regularization term the effect of which fades as iterations go on. Displayed numerical results highlight its accuracy, as well as its robustness.

In this paper we propose a cubically convergent algorithm. Our basic tool is the Tikhonov regularization and Morozov's and damped Morozov's discrepancy principles. Numerical experiments for integral equations of the first kind are presented to compare the efficiency of the proposed algorithms.

In this paper, we investigate the inverse problem of determining the potential of the Schrodinger equation from finite measurements on the boundary. It is well known that this is an ill posed problem in the sense of Hadamard. The stability estimate is proved under the assumption that the potentials have some a priori constraints. Based on this conditional stability result, we propose one kind of Tikhonov regularization and prove the convergence rate for the regularized solution.

In this paper we investigate the stability and convergence rates of the widely used output least-squares method with Tikhonov regularization for the identification of the conductivity distribution in a heat conduction system. Due to the rather restrictive source conditions and regularity assumptions on the nonlinear parameter-to-solution operator concerned, the existing Tikhonov regularization theory for nonlinear inverse problems is difficult to apply for the convergence rate analysis here. By introducing some new techniques, we are able to relax these regularity requirements and derive a much simpler and easily interpretable source condition but still achieve the same convergence rates as the standard Tikhonov regularization theory does.

In this paper we study the regularization of linear and nonlinear ill-posed operator equations by projection onto finite-dimensional spaces with a posteriori chosen space dimension. We show that this results in a regularization method, i.e. a stable solution method for the ill-posed problem, that converges to an exact solution as the data noise level goes to zero, with optimal rates under additional regularity conditions.

In this paper, based on the conditional stability estimate for ill-posed inverse problems, we propose a new strategy for a priori choice of regularizing parameters in Tikhonov's regularization and we show that it can be applied to a wide class of inverse problems. The convergence rate of the regularized solutions is also proved.

We analyse the convergence of a class of discrete predictor-corrector methods for the sequential regularization of first-kind Volterra integral equations. In contrast to classical methods such as Tikhonov regularization, this class of methods preserves the Volterra (causal) structure of the original problem. The result is a discretized regularization method for which the number of arithmetic operations is (N 2 ) (where N is the dimension of the approximating space) in contrast to standard Tikhonov regularization which requires (N 3 ) operations. In addition, the method considered here is defined using functional regularization parameters so that the possibility for more or less smoothing at different points in the domain of the solution is allowed. We establish a convergence theory for these methods and present relevant numerical examples, illustrating how one functional regularization parameter may be adaptively selected as part of the sequential regularization process. This work generalizes earlier results by the first author to the case of a penalized predictor-corrector formulation, functional regularization parameters, and nonconvolution Volterra equations.

We discuss a new a posteriori rule for choosing the regularization parameter in regularization methods for solving linear ill-posed problems Ax = y with minimum-norm solution x. Main emphasis is devoted to the methods of ordinary and iterated Tikhonov regularization. We suppose that instead of y there are given noisy data y satisfying |y-y| with known noise level . The approach of the new rule consists of finding the smallest = ME for which it can be proved that the error |x-x| of the regularized solution xisstrictlymonotonicallyincreasingforincreasing-values, that is, there holds {d}{{d\alpha}}|x-x|2>0 for all (ME,). Due to this property we call our rule of choosing the monotone error rule (ME rule). Our ME rule leads to smaller errors than the a posteriori rule of Raus (1985 Acta Comment. Univ. Tartuensis 715 12-20 (in Russian)) and Gfrerer (1987 Math. Comput. 49 507-22). Furthermore, we discuss discretization issues and illustrate some of our theoretical results by numerical experiments.

In this paper we apply the recently developed method regularization for curve representations to nonlinear ill-posed problems, especially to parameter estimation problems where the parameter might have discontinuities. We prove convergence of the finite-dimensional Tikhonov regularized solutions and present some numerical results.

The problem of piecewise-uniform regularization for two-dimensional ill-posed problems with bounded discontinuous solutions is under discussion. The algorithm of modified Tikhonov regularization is applied to solve the problem on the class of functions of two variables with bounded VH-variation as previously proposed by the author. In finite-dimensional form, the algorithm is reduced to mathematical programming with a nonsmooth target function. After smooth approximation, the algorithm is numerically implemented in an effective way to ensure the piecewise-uniform convergence of the approximate solutions to the exact solution of an ill-posed problem. The quality of obtained piecewise-uniform regularization is illustrated by numerical experiments in blurred and noisy image reconstruction.

In this paper we consider the a posteriori parameter choice strategy proposed by Scherzer et al in 1993 for the Tikhonov regularization of nonlinear ill-posed problems and obtain some results on the convergence and convergence rate of Tikhonov regularized solutions under suitable assumptions. Finally we present some illustrative examples.

The aim of this paper is to study regularization methods for linear ill-posed problems. Linear methods are Tikhonov-Phillips methods, iterative methods, truncated singular value decomposition, Backus-Gilbert-type methods and approximate inverse, for example. The first three are generally studied as filter methods where a special filter for the singular value decomposition can be computed. In the other methods mentioned the regularization is achieved by either smoothing the data or the solution. More general is the approximate inverse introduced by Louis (1996 Inverse Problems 12 175-90). Here we show that all these methods can be viewed either as smoothing the pseudo-inverse or equivalently as first smoothing the data and then applying the pseudo-inverse. The smoothing of the data or of the pseudo-inverse has to be at least of the order of the smoothing of the operator in the problem to be solved. Conditions for the order-optimality of the methods are given.

Many problems arising in (geo)physics and technology can be formulated as compact operator equations of the first kind A F = G. Due to the ill-posedness of the equation, a variety of regularization methods are being considered for an approximate solution, where particular emphasis must be put on balancing the data and the approximation error. In doing so one is interested in optimal parameter choice strategies. In this paper our interest lies in an efficient algorithmic realization of a special class of regularization methods. More precisely, we implement regularization methods based on filtered singular-value decomposition as a wavelet analysis. This enables us to perform, for example, Tikhonov-Philips regularization as multiresolution. In other words, we are able to pass over from one regularized solution to another by adding or subtracting so-called detail information in terms of wavelets. It is shown that regularization wavelets as proposed here are efficiently applicable to a future problem in satellite geodesy, viz satellite gravity gradiometry.

In this paper we consider a combination of Newton's method with linear Tikhonov regularization, linear Landweber iteration and truncated SVD, for regularizing an abstract, nonlinear, ill-posed operator equation. We show that under certain smoothness conditions on the nonlinear operator, these methods converge locally. For perturbed data we propose an a priori stopping rule, that guarantees convergence of the iterates to a solution, as the noise level goes to zero. Under appropriate closeness and smoothness assumptions on the starting value and the solution, we obtain convergence rates.

We present a `predictor - corrector' type of regularization method for inverse problems modelled by first-kind Volterra integral equations and extend the convergence/regularization theory developed previously to the case where the integral kernel satisfies general -smoothing conditions. The theoretical basis for this method comes from replacing the original first-kind equation by a related second-kind equation which is constructed using `future values' of the original kernel and the data on a small interval of length . In practical implementation this method takes the form of a sequential regularization scheme in which one first predicts a rigid (regularized) solution over a small interval and then, before moving forward in the sequential process, one makes a correction of the solution in order to avoid over-regularization and to improve accuracy. In addition to the convergence theory developed for noise-free data, we show how selection of the regularization parameter as a function of the level of error present in the data serves to facilitate convergence in the case of noisy data. Finally, to further examine the extent to which improves stability, we show how an increase in serves to decrease the condition number of the matrices associated with a discretization of the original problem.

The L-curve method was developed for the selection of regularization parameters in the solution of discrete systems obtained from ill-posed problems. An analysis of this method is given for selecting a parameter for Tikhonov regularization. This analysis, which is carried out in a semi-discrete, semi-stochastic setting, shows that the L-curve approach yields regularized solutions which fail to converge for a certain class of problems. A numerical example is also presented which indicates that this lack of convergence can arise in practical applications.

In this paper we present a method for solving problems such as Af = g by constructing an approximate inverse which maps the data g to a regularized solution of this equation of the first kind. No discretization for f is needed. The solution operator can be precomputed independently of the data. This works for linear problems and for nonlinear problems with a special structure. The regularization is achieved by computing mollified versions of the (minimum-norm) solution. It is shown that this class of regularization operators contains, as special cases, the classical methods such as Tikhonov - Phillips, iteration methods and also discretization methods. In the case where the operator has some invariance properties the storage needs are dramatically reduced.

This paper is devoted to identifying a time-dependent source term in a multi-dimensional time-fractional diffusion equation from boundary Cauchy data. The existence and uniqueness of a strong solution for the corresponding direct problem with homogeneous Neumann boundary condition are firstly proved. We provide the uniqueness and a stability estimate for the inverse time-dependent source problem. Then we use the Tikhonov regularization method to solve the inverse source problem and propose a conjugate gradient algorithm to find a good approximation to the minimizer of the Tikhonov regularization functional. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed method.

We use the singular value decomposition (SVD) to analyse the linear systems of equations that are the heart of the Backus-Gilbert method (1968). We show that the method has a regularizing effect, and our analysis also leads to an alternative implementation, based on the Lanczos algorithm, which is faster than the standard implementation when few SVD components are sufficient in the regularized solution.

In this paper we study some regularization methods to reconstruct the solution q* of the non-linear ill-posed problem A (q)=z from noisy data zdelta in Z where A:D(A) contained in/implied by Q to Z is a non-linear operator between Hilbert spaces Q and Z. The considered regularization methods are (1) the method of Tikhonov regularization in Hilbert scales, (2) the method of regularization in state space which recently has been proposed by Chavent and Kunisch (1992) and (3) some new regularization methods that are based on data smoothing. Assuming certain conditions concerning the non-linear operator A and the smoothness of the solution q* we discuss a priori as well as a posteriori parameter choice strategies that yield regularized solutions with order optimal error bounds.

In this paper convergence and rate of convergence results for nonlinear constrained ill-posed inverse problems formulated as regularized least squares problems are given.

The problem of solving numerically a Fredholm integral equation of the first kind when it is known that f(x)>or=0, by a regularization method based on minimization is considered. The convergence of solutions of this minimization problem, given conditions on the data and the regularization parameter lambda , is demonstrated to show that the procedure leads to a correct regularization method. A generalized cross validation strategy for the selection of the regularization parameter is introduced. A number of numerical experiments are made and comparisons are made with Tikhonov regularization schemes based on differential operators.

The two-dimensional layered volume conductor forward and inverse problems are solved by modelling the medium as an equivalent two-dimensional filter relating the source and measured potentials. This extends previous results to the multi-dimensional multi-layered case. In principle, the inverse problem can be solved as easily as the forward problem, using 2D FFT algorithms. In fact, it is ill-conditioned and must be regularized. The authors show that Tikhonov regularization, constrained least-squares regularization, and stochastic regularization using a Wiener filter all lead to the same spatial low-pass regularizing filter. New contributions include: (1) extension of the medium filter concept from one to two dimensions; (2) a simple layer-recursive formula for computing the medium filter for a layered medium; and (3) application of various regularization techniques, in the form of regularizing filters, to this problem.

The Mumford–Shah model is a very powerful variational approach for edge preserving regularization of image reconstruction processes. However, it is algorithmically challenging because one has to deal with a non-smooth and non-convex functional. In this paper, we propose a new efficient algorithmic framework for Mumford–Shah regularization of inverse problems in imaging. It is based on a splitting into specific subproblems that can be solved exactly. We derive fast solvers for the subproblems which are key for an efficient overall algorithm. Our method neither requires a priori knowledge of the gray or color levels nor of the shape of the discontinuity set. We demonstrate the wide applicability of the method for different modalities. In particular, we consider the reconstruction from Radon data, inpainting, and deconvolution. Our method can be easily adapted to many further imaging setups. The relevant condition is that the proximal mapping of the data fidelity can be evaluated a within reasonable time. In other words, it can be used whenever classical Tikhonov regularization is possible.

In this paper we investigate convergence properties of Tikhonov regularization for linear ill-posed problems under a stochastic error model. Namely, we assume that we are given a finite amount of measurements, each contaminated by Gaussian noise with zero mean and known finite variance. Using Besov-space penalty terms to promote sparse solutions with respect to a preassigned wavelet basis, the Ky Fan metric allows us to lift deterministic convergence results into the stochastic setting. In particular, we formulate a general convergence theorem and propose a formula to directly calculate a suitable regularization parameter. This immediately leads to convergence rates. Numerical examples are presented to verify the theoretical results.

In this paper, we propose an algorithm for solving the large-scale discrete ill-conditioned linear problems arising from the discretization of linear or nonlinear inverse problems. The algorithm combines some existing regularization techniques and regularization parameter choice rules with a randomized singular value decomposition (SVD), so that only much smaller scale systems are needed to solve, instead of the original large-scale regularized system. The algorithm can directly apply to some existing regularization methods, such as the Tikhonov and truncated SVD methods, with some popular regularization parameter choice rules such as the L-curve, GCV function, quasi-optimality and discrepancy principle. The error of the approximate regularized solution is analyzed and the efficiency of the method is well demonstrated by the numerical examples.