New POCS algorithms for regularization of inverse problems (original) (raw)
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Analysis of Some Optimization Techniques for Regularization of Inverse Problems
2016
The main objective in inverse problems is to approximate some unknown parameters or attributes of interest, given some measurements that are only indirectly related to these parameters. This type of problem appears in many areas of science, engineering and industry. Examples can be found in medical computerized tomography, groundwater flow modeling, etc. In the process of solving these problems often appears an instability phenomenon known as ill-posedness which requires regularization. Ill-posedness is related to the fact that the presence of even a small amount of noise in the data can lead to enormous errors in the approximated solution. Different regularization techniques have been proposed in the literature. In this thesis our focus is put on Total Variation regularization. We study the total variation regularization for both image denoising and image deblurring problems. Three algorithms for total variation regularization will be analysed, namely the split Bregman algorithms, ...
Inverse Problems, Regularization and Applications
ArXiv, 2019
Inverse problems arise in a wide spectrum of applications in fields ranging from engineering to scientific computation. Connected with the rise of interest in inverse problems is the development and analysis of regularization methods, such as truncated singular value decomposition (TSVD), Tikhonov regularization or iterative regularization methods (like Landerweb), which are a necessity in most inverse problems due to their ill-posedness. In this thesis we propose a new iterative regularization technique to solve inverse problems, without any dependence on external parameters and thus avoiding all the difficulties associated with their involvement. To boost the convergence rate of the iterative method different descent directions are provided, depending on the source conditions, which are based on some specific a-priori knowledge about the solution. We show that this method is very robust to the presence of (extreme) errors in the data. In addition, we also provide a very efficient ...
arXiv: Numerical Analysis, 2020
In this paper we extend a recent idea of formulating and regularizing inverse problems as minimization problems, so without using a forward operator, thus avoiding explicit evaluation of a parameter-to-state map. We do so by rephrasing three application examples in this minimization form, namely (a) electrical impedance tomography with the complete electrode model (b) identification of a nonlinear magnetic permeability from magnetic flux measurements (c) localization of sound sources from microphone array measurements. To establish convergence of the proposed regularization approach for these problems, we first of all extend the existing theory. In particular, we take advantage of the fact that observations are finite dimensional here, so that inversion of the noisy data can to some extent be done separately, using a right inverse of the observation operator. This new approach is actually applicable to a wide range of real world problems.
An Iteratively Regularized Projection Method for Nonlinear Ill-posed Problems
An iterative regularization method in the setting of a finite dimen-sional subspace X h of the real Hilbert space X has been considered for obtaining stable approximate solution to nonlinear ill-posed oper-ator equations F (x) = y where F : D(F) ⊆ X −→ X is a nonlinear monotone operator on X. We assume that only a noisy data y δ with y − y δ ≤ δ are available. Under the assumption that the Fréchet derivative F of F is Lipschitz continuous, a choice of the regularization parameter using an adaptive selection of the parameter and a stopping rule for the iteration index using a majorizing sequence are presented. We prove that under a general source condition on x 0 x, the error x h,δ n,α x between the regularized approximation x h,δ n,α , (x h,δ 0,α := P h x 0 where P h is an orthogonal projection on to X h) and the solutio x is of optimal order. The results of computational experiments are provided which shows the reliability of our method.
Regularization by projection: Approximation theoretic aspects and distance functions
Journal of Inverse and Ill-posed Problems, 2000
The authors study regularization of non-linear inverse problems by projection methods. Non-linearity is controlled by some range invariance assumption. Emphasis is on approximation theoretic properties of the discretization which determine the convergence rates. Instead of using source conditions for the true solution to represent the error, the authors show how distance functions with respect to some benchmark smoothness are able to replace this. Some examples indicate how the results can be applied.
IEEE Transactions on Signal Processing
We consider one-dimensional (1-D) linear inverse problems that are formulated in the continuous domain. The object of recovery is a function that is assumed to minimize a convex objective functional. The solutions are constrained by imposing a continuous-domain regularization. We derive the parametric form of the solution (representer theorems) for Tikhonov (quadratic) and generalized total-variation (gTV) regularizations. We show that, in both cases, the solutions are splines that are intimately related to the regularization operator. In the Tikhonov case, the solution is smooth and constrained to live in a fixed subspace that depends on the measurement operator. By contrast, the gTV regularization results in a sparse solution composed of only a few dictionary elements that are upper-bounded by the number of measurements and independent of the measurement operator. Our findings for the gTV regularization resonates with the minimization of the 1 norm, which is its discrete counterpart and also produces sparse solutions. Finally, we find the experimental solutions for some measurement models in one dimension. We discuss the special case when the gTV regularization results in multiple solutions and devise an algorithm to find an extreme point of the solution set which is guaranteed to be sparse.
Regularization of Inverse Problems
The problem of determining the thermophysical properties by means of a discrete set of observations on the temperatures of the test object given with measurement errors is examined. The investigation of complex processes by using inverse problems has attracted considerable attention lately. Their solution is associated with certain singularities, particularly the influence of errors in the initial data on the desired solution. As is known from [i], in such cases it is necessary to limit the domain of the allowable solutions and to match the measurement errors. Since a number of stabilizing functionals with the same problem can be set in correspondence and different norms for the deviation from the quantities observed can be selected, then it is interesting to determine those among them which will permit, for sufficiently general assumptions about the desired quantities, obtaining the most exact solutions under conditions of unimprovable observations for a broad range of measurement errors. In addition, the question of selecting the method of matching the observations occurs in the solution of applied ill-posed problems. One condition that establishes a relation between the accuracy of the solution and the measurement error [2] is used in the widespread problem, in practice, of restoring the thermal flux. This condition expresses the total error in all observations for measurements executed at several points. However, one condition can turn out to be inadequate to determine several parameters of a model that is characteristic for the inverse coefficient problems, while taking total account of the errors results in a loss in accuracy of the solution of the inverse problem [3]. This paper is devoted to investigating the properties of the regularized solution of an inverse coefficient problem for the nonlinear heat-conduction equation as a function of the degree of limitation of the domain of admissible solutions, the form of the observation error estimate, and the methods of matching them. In the domain Q = {~, t):O < x < i, 0 < t < T} we examine the one-dimensional heat-conduction equation a~ O~-ox a~-~x + I (x, 0 (1) for which the initial and boundary conditions assuring uniqueness and stability in the determination of the function u(x, t) for given values of the specific heat a~(u) and the heat conductivity an(U) and any T > 0 are assumed known. Let us also assume that at m points of space, and for each of n times of the domain Q observation results are given u~l=u(x,, t~)+efy, i= 1, m, ]= 1, n, (2) with a known magnitude of the deviation norm 62=j-(l~y-uij)", i = i, ~, i=I (3) Balashikhinskoe NPO Kriogenmash.
Regularization of inverse problems by an approximate matrix-function technique
Numerical Algorithms
In this work, we introduce and investigate a class of matrix-free regularization techniques for discrete linear ill-posed problems based on the approximate computation of a special matrix-function. In order to produce a regularized solution, the proposed strategy employs a regular approximation of the Heavyside step function computed into a small Krylov subspace. This particular feature allows our proposal to be independent from the structure of the underlying matrix. If on the one hand, the use of the Heavyside step function prevents the amplification of the noise by suitably filtering the responsible components of the spectrum of the discretization matrix, on the other hand, it permits the correct reconstruction of the signal inverting the remaining part of the spectrum. Numerical tests on a gallery of standard benchmark problems are included to prove the efficacy of our approach even for problems affected by a high level of noise.
International Journal of Mathematical Analysis
An iteratively regularized projection method, which converges quadrat-ically, has been considered for obtaining stable approximate solution to nonlinear ill-posed operator equations F (x) = y where F : D(F) ⊆ X −→ X is a nonlinear monotone operator defined on the real Hilbert space X. We assume that only a noisy data y δ with − y δ ≤ δ are available. Under the assumption that the Fréchet derivative F of F is Lipschitz continuous, a choice of the regularization parameter using an adaptive selection of the parameter and a stopping rule for the iteration index using a majorizing sequence are presented. We prove that under a general source condition on x 0 x, the error h,δ n,α x between the regularized approximation x h,δ n,α , (x h,δ 0,α := P h x 0 where P h is an orthog-onal projection on to a finite dimensional subspace X h of X) and the solutio x is of optimal order.