A comparative study of some reconstruction methods for linear inverse problems (original) (raw)
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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.
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 iterative Lagrange method for the regularization of discrete ill-posed inverse problems
Computers & Mathematics with Applications, 2010
In this paper, an iterative method is presented for the computation of regularized solutions of discrete ill-posed problems. In the proposed method, the regularization problem is formulated as an equality constrained minimization problem and an iterative Lagrange method is used for its solution. The Lagrange iteration is terminated according to the discrepancy principle. The relationship between the proposed approach and classical Tikhonov regularization is discussed. Results of numerical experiments are presented to illustrate the effectiveness and usefulness of the proposed method.
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, ...
Minimum variance regularization in linear inverse problems
Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 2004
In this paper we propose a minimum variance regularization procedure for the solution of ill-posed linear inverse problems. The method was applied to the deconvolution of Doppler broadening positron-electron annihilation radiation in aluminum. The characteristics of the procedure were tested through simulation.
The main goal of the workshop is to gather young participants (mostly mathematicians and geophysicists) from China and overseas together to discuss how to solve inverse and ill-posed problems using different solving strategies. Eminent specialists from China, Russia (partially sponsored by the Russian Foundation of Basic Research), USA, India and Norway were invited to present their lectures. Other young scientists also present their recent researches during the conference. The book covers many directions in the modern theory of inverse and illposed problems-mathematical physics, optimal inverse design, inverse scattering, inverse vibration, biomedical imaging, oceanography, seismic imaging and remote sensing; methods including standard regularization, parallel computing for multidimensional problems, Nyström method, numerical differentiation, analytic continuation, perturbation regularization, filtering, optimization and sparse solving methods are fully addressed. This issue attempts to bridge the gap between theoretical studies of ill-posed inverse problems and practical applications. Let us continue our efforts for further progress. This book will be helpful to researchers and teachers in developing courses on various inverse and ill-posed problems of mathematical physics, geosciences, designing technology, imaging, high performance computing, inverse scattering and vibration, and so on. It could be also beneficial for senior undergraduate students, graduate and Ph.D. students, recent graduates getting practical experience, engineers and researchers who study inverse and ill-posed problems and solve them in practice.
Survey of Computational Methods for Inverse Problems
Recent Trends in Computational Science and Engineering, 2018
Inverse problems occur in a wide range of scientific applications, such as in the fields of signal processing, medical imaging, or geophysics. This work aims to present to the field practitioners, in an accessible and concise way, several established and newer cutting-edge computational methods used in the field of inverse problems-and when and how these techniques should be employed.
A statistical view of iterative methods for linear inverse problems
TEST, 2007
In this article we study the problem of recovering the unknown solution of a linear ill-posed problem, via iterative regularization methods. We review the problem of projection-regularization from a statistical point of view. A basic purpose of the paper is the consideration of adaptive model selection for determining regularization parameters. This article introduces a new regularized estimator which has the best possible adaptive properties for a wide range of linear functionals. We derive non asymptotic upper bounds for the mean square error of the estimator and give the optimal convergence rates.
Reconstruction Methods for Inverse Problems
2019
The reconstruction in quantitative coupled physics imaging often requires that the solutions of certain PDEs satisfy some non-zero constraints, such as the absence of critical points or nodal points. After a brief review of several methods used to construct such solutions, I will focus on a recent approach that combines the Runge approximation and the Whitney embedding