Tikhonov regularization is one of the most popular approaches to solving linear discrete ill-posed problems. The choice matrix may significantly affect quality computed solution. When identity, iterated can yield approximate solutions higher than (standard) regularization. This paper provides an analysis with a different from identity. Computed examples illustrate performance this method.

We consider the problem of computed tomography (CT). This ill-posed inverse arises when one wishes to investigate internal structure an object with a non-invasive and non-destructive technique. is severely ill-conditioned, meaning it has infinite solutions extremely sensitive perturbations in collected data. sensitivity produces well-known semi-convergence phenomenon if iterative methods are used solve it. In this work, we propose multigrid approach mitigate instability produce fast,...

Fractional Tikhonov regularization methods have been recently proposed to reduce the oversmoothing property of in standard form, order preserve details approximated solution. Their and convergence properties previously investigated showing that they are optimal order. This paper provides saturation converse results on their rates. Using same iterative refinement strategy iterated regularization, new fractional introduced. We show these overcome previous results. Furthermore, nonstationary...

The need to solve discrete ill-posed problems arises in many areas of science and engineering. Solutions these problems, if they exist, are very sensitive perturbations the available data. Regularization replaces original problem by a nearby regularized problem, whose solution is less error contains fidelity term regularization term. Recently, use p-norm measure q-norm has received considerable attention. balance between terms determined parameter. In applications, such as image restoration,...

Abstract Discrete ill-posed inverse problems arise in various areas of science and engineering. The presence noise the data often makes it difficult to compute an accurate approximate solution. To reduce sensitivity computed solution noise, one replaces original problem by a nearby well-posed minimization problem, whose is less sensitive than problem. This replacement known as regularization. We consider situation when consists fidelity term, that defined terms p -norm, regularization q...

Abstract Regularization of certain linear discrete ill-posed problems, as well regression can be formulated large-scale, possibly nonconvex, minimization whose objective function is the sum p th power ℓ -norm a fidelity term and q regularization term, with 0 < , ≤ 2. We describe new restarted iterative solution methods that require less computer storage execution time than described by Huang et al. (BIT Numer. Math. 57 ,351–378, 14). The reduction in achieved periodic restarts method....

Abstract Bregman-type iterative methods have received considerable attention in recent years due to their ease of implementation and the high quality computed solutions they deliver. However, these may require a large number iterations this reduces usefulness. This paper develops computationally attractive linearized Bregman algorithm by projecting problem be solved into an appropriately chosen low-dimensional Krylov subspace. The projection computational effort required for each iteration....

This paper discusses and develops new methods for fitting trigonometric curves, such as circles, ellipses, dumbbells, to data points in the plane. Available circles or ellipses are very sensitive outliers data, time consuming when number of is large. The present focuses on curve that attractive use We propose a direct method two iterative dumbbell curves based polynomials. These efficiently minimize sum squared geometric distances between given fitted curves. In particular, we interested...

In many inverse problems the operator to be inverted is not known precisely, but only a noisy version of it available; we refer this kind problem as semiblind. article, propose functional which involves variables both solution and itself. We first prove that functional, even if nonconvex, admits global minimum its minimization naturally leads regularization method. Later, using popular alternating direction multiplier method (ADMM), describe an algorithm identify stationary point functional....

Iterative soft thresholding algorithms combine one step of a Landweber method (or accelerated variants) with the wavelet (framelet) coefficients. In this paper, we improve these methods by using framelet multilevel decomposition for defining multigrid deconvolution grid transfer operators given low-pass filter frame. Assuming that an estimate noise level is available, recently proposed iterative ℓ2-regularization linear denoising soft-thresholding. This combination allows fast frequency...

Image deblurring is a relevant problem in many fields of science and engineering. To solve this problem, different approaches have been proposed, and, among the various methods, variational ones are extremely popular. These substitute original with minimization where functional composed two terms, data fidelity term regularization term. In paper we propose, classical non-negative constrained $\ell^2$-$\ell^1$ framework, use graph Laplacian as operator. Firstly, describe how to construct from...

Linear systems of equations with a matrix whose singular values decay to zero increasing index number, and without significant gap, are commonly referred as linear discrete ill-posed problems. Such arise, e.g., when discretizing Fredholm integral equation the first kind. The right-hand side vectors problems that arise in science engineering often represent an experimental measurement is contaminated by error. solution these typically very sensitive this Previous works have shown error...

Abstract We consider linear operator equations of the form where $K:{\cal X}\to{\cal Y}$ is a compact between Hilbert spaces ${\cal X} \hbox{ and } {\cal Y}.$ assume y to be attainable, i.e., that problem (1) has solution x † = K minimal norm. Here denotes (Moore‐Penrose) generalized inverse , which unbounded when compact, with infinite dimensional range. Hence ill‐posed regularized in order compute numerical solution. want approximate equation only an approximation δ available ‖ − ≤ δ,...

In many real world problems it is of interest to ascertain which factors are most relevant for determining a given outcome. This the so-called variable selection problem. The present paper proposes new regression model its solution. We show that proposed satisfies continuity, sparsity, and unbiasedness properties. A generalized Krylov subspace method practical solution minimization problem involved described. can be used both small-scale large-scale problems. Several computed examples...

Reconstructing the structure of soil using non-invasive techniques is a very relevant problem in many scientific fields, like geophysics and archaeology. This can be done, for instance, with aid Frequency Domain Electromagnetic (FDEM) induction devices. Inverting FDEM data challenging inverse problem, as extremely ill-posed, i.e., sensible to presence noise measured data, non-linear. Regularization methods substitute original ill-posed well-posed one whose solution an accurate approximation...