Generalized Arnoldi--Tikhonov Method for Sparse Reconstruction

Gazzola, Silvia; Nagy, James G.

doi:10.1137/130917673

Cited by 59 publications

(97 citation statements)

References 38 publications

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“…In [16,28,29] a generalized Arnoldi-Tikhonov method (GAT) was introduced that iteratively solves the Tikhonov problem (1) using a Krylov subspace method based on the Arnoldi decomposition of the matrix A. Simultaneously, after each Krylov iteration, the regularization parameter is updated in order to approximate the value for which the discrepancy is equal to ηε.…”

Section: Reference Method: Generalized Bidiagonal-tikhonovmentioning

confidence: 99%

“…where r(z k ) = ||B k+1,k z k − c k+1 || and r(y k ) = ||B k+1,k y k − c k+1 || are the residuals. A brief sketch of this method is given in algorithm 3, but for more information we refer to [16,28,29]. Note that in the original GAT method, the non-regularized iterates z k are equivalent to the GMRES [30] iterations for the solution of Ax = b.…”

Section: Reference Method: Generalized Bidiagonal-tikhonovmentioning

confidence: 99%

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Projected Newton method for noise constrained Tikhonov regularization

2020

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Tikhonov regularization is a popular approach to obtain a meaningful solution for ill-conditioned linear least squares problems. A relatively simple way of choosing a good regularization parameter is given by Morozov's discrepancy principle. However, most approaches require the solution of the Tikhonov problem for many different values of the regularization parameter, which is computationally demanding for large scale problems. We propose a new and efficient algorithm which simultaneously solves the Tikhonov problem and finds the corresponding regularization parameter such that the discrepancy principle is satisfied. We achieve this by formulating the problem as a nonlinear system of equations and solving this system using a line search method. We obtain a good search direction by projecting the problem onto a low dimensional Krylov subspace and computing the Newton direction for the projected problem. This projected Newton direction, which is significantly less computationally expensive to calculate than the true Newton direction, is then combined with a backtracking line search to obtain a globally convergent algorithm, which we refer to as the Projected Newton method. We prove convergence of the algorithm and illustrate the improved performance over current state-of-the-art solvers with some numerical experiments.

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Section: Reference Method: Generalized Bidiagonal-tikhonovmentioning

confidence: 99%

Section: Reference Method: Generalized Bidiagonal-tikhonovmentioning

confidence: 99%

Projected Newton method for noise constrained Tikhonov regularization

2020

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“…In [26], a flexible Arnoldi Tikhonov method was presented for sparse image reconstruction, where a procedure for adapting the regularisation parameter was demonstrated within the solution of a linear least squares problem. However, this procedure differs from the approach proposed in this paper, in the sense that a discrepancy principle was used in [26] with a formulation to recursively change λ k until the data residual fall just above the noise level that is assumed to be known in the data. Once this balance is achieved, no more changes in λ k are observed, which prompted the authors to consider a restarting procedure to be implemented within the algorithm.…”

Section: Adaptive Gauss Newton Methodsmentioning

confidence: 99%

Absolute Imaging of Low Conductivity Material Distributions Using Nonlinear Reconstruction Methods in Magnetic Induction Tomography

Dekdouk¹,

Ktistis²,

Armitage³

et al. 2016

PIER

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Abstract-Magnetic Induction Tomography (MIT) is a newly developing technique of electrical tomography that in principle is able to map the electrical conductivity distribution in the volume of objects. The image reconstruction problem in MIT is similar to electrical impedance tomography (EIT) in the sense that both seek to recover the conductivity map, but differ remarkably in the fact that data being inverted in MIT is derived from induction theory and related sources of noise are different. Progress in MIT image reconstruction is still limited, and so far mainly linear algorithms have been implemented. In difference imaging, step inversion was demonstrated for recovering perturbations within conductive media, but at the cost of producing qualitative images, whilst in absolute imaging, linear iterative algorithms have mostly been employed but mainly offering encouraging results for imaging isolated high conductive targets.In this paper, we investigate the possibility of absolute imaging in 3D MIT of a target within conductive medium for low conductivity application (σ < 5 Sm −1 ). For this class of problems, the MIT image reconstruction exhibits non-linearity and ill-posedness that cannot be treated with linear algorithms. We propose to implement for the first time in MIT two effective inversion methods known in non-linear optimization as Levenberg Marquardt (LMM) and Powell's Dog Leg (PDLM) methods. These methods employ damping and trust region techniques for controlling convergence and improving minimization of the objective function. An adaptive version of Gauss Newton is also presented (AGNM), which implements a damping mechanism to the regularization parameter. Here, the level of penalty is varied during the iterative process. As a comparison between the methods, different criteria are examined from image reconstructions using the LMM, PDLM and AGNM. For test examples, volumetric image reconstruction of a perturbation within homogeneous cylindrical background is considered. For inversion, an independent finite element FEM software package Maxwell by Ansys is employed to generate simulated data using a model of a 16 channel MIT system. Numerical results are employed to show different performance characteristics between the methods based on convergence, stability and sensitivity to the choice of the regularization parameter. To demonstrate the effect of scalability of absolute imaging in MIT for more realistic problems, a human head model with an internal anomaly is used to produce reconstructions for different finer resolutions. AGNM is adopted here and employs the Krylov subspace method to replace the computationally demanding direct inversion of the regularized Hessian.

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“…In addition, most of the theory and methods have been developed for standard‐form Tikhonov and extensions of hybrid iterative methods to the general‐form Tikhonov problem may be non‐trivial, especially if WL does not have full rank . Various researchers have proposed hybrid approaches based on the Arnoldi process , generalized Krylov bases , and other basis extensions . However, a fully automated implementation that works for extremely large‐scale problems is not readily available.…”

Section: Methodsmentioning

confidence: 99%

Computational methods for image reconstruction

Chung

Ruthotto

2016

NMR in Biomedicine

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Reconstructing images from indirect measurements is a central problem in many applications, including the subject of this special issue, quantitative susceptibility mapping (QSM). The process of image reconstruction typically requires solving an inverse problem that is ill-posed and large-scale and thus challenging to solve. Although the research field of inverse problems is thriving and very active with diverse applications, in this part of the special issue we will focus on recent advances in inverse problems that are specific to deconvolution problems, the class of problems to which QSM belongs. We will describe analytic tools that can be used to investigate underlying ill-posedness and apply them to the QSM reconstruction problem and the related extensively studied image deblurring problem. We will discuss state-of-the-art computational tools and methods for image reconstruction, including regularization approaches and regularization parameter selection methods. We finish by outlining some of the current trends and future challenges. Copyright © 2016 John Wiley & Sons, Ltd.

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Generalized Arnoldi--Tikhonov Method for Sparse Reconstruction

Cited by 59 publications

References 38 publications

Projected Newton method for noise constrained Tikhonov regularization

Projected Newton method for noise constrained Tikhonov regularization

Absolute Imaging of Low Conductivity Material Distributions Using Nonlinear Reconstruction Methods in Magnetic Induction Tomography

Computational methods for image reconstruction

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