Some matrix nearness problems suggested by Tikhonov regularization

The numerical solution of linear discrete ill-posed problems typically requires regularization, i.e., replacement of the available ill-conditioned problem by a nearby better conditioned one. The most popular regularization methods for problems of small to moderate size are Tikhonov regularization and truncated singular value decomposition (TSVD). By considering matrix nearness problems related to Tikhonov regularization, several novel regularization methods are derived. These methods share properties with both Tikhonov regularization and TSVD, and can give approximate solutions of higher quality than either one of these methods

A GCV based Arnoldi-Tikhonov regularization method

For the solution of linear discrete ill-posed problems, in this paper we consider the Arnoldi-Tikhonov method coupled with the Generalized Cross Validation for the computation of the regularization parameter at each iteration. We study the convergence behavior of the Arnoldi method and its properties for the approximation of the (generalized) singular values, under the hypothesis that Picard condition is satisfied. Numerical experiments on classical test problems and on image restoration are presented

Regularization matrices determined by matrix nearness problems

This paper is concerned with the solution of large-scale linear discrete ill-posed problems with error-contaminated data. Tikhonov regularization is a popular approach to determine meaningful approximate solutions of such problems. The choice of regularization matrix in Tikhonov regularization may significantly affect the quality of the computed approximate solution. This matrix should be chosen to promote the recovery of known important features of the desired solution, such as smoothness and monotonicity. We describe a novel approach to determine regularization matrices with desired properties by solving a matrix nearness problem. The constructed regularization matrix is the closest matrix in the Frobenius norm with a prescribed null space to a given matrix. Numerical examples illustrate the performance of the regularization matrices so obtained

Embedded techniques for choosing the parameter in Tikhonov regularization

This paper introduces a new strategy for setting the regularization parameter when solving large-scale discrete ill-posed linear problems by means of the Arnoldi-Tikhonov method. This new rule is essentially based on the discrepancy principle, although no initial knowledge of the norm of the error that affects the right-hand side is assumed; an increasingly more accurate approximation of this quantity is recovered during the Arnoldi algorithm. Some theoretical estimates are derived in order to motivate our approach. Many numerical experiments, performed on classical test problems as well as image deblurring are presented

Regularization matrices for discrete ill-posed problems in several space-dimensions

Many applications in science and engineering require the solution of large linear discrete ill-posed problems that are obtained by the discretization of a Fredholm integral equation of the first kind in several space dimensions. The matrix that defines these problems is very ill conditioned and generally numerically singular, and the right-hand side, which represents measured data, is typically contaminated by measurement error. Straightforward solution of these problems is generally not meaningful due to severe error propagation. Tikhonov regularization seeks to alleviate this difficulty by replacing the given linear discrete ill-posed problem by a penalized least-squares problem, whose solution is less sensitive to the error in the right-hand side and to roundoff errors introduced during the computations. This paper discusses the construction of penalty terms that are determined by solving a matrix nearness problem. These penalty terms allow partial transformation to standard form of Tikhonov regularization problems that stem from the discretization of integral equations on a cube in several space dimensions

Regularized solution of a nonlinear problem in electromagnetic sounding

We propose a regularization method to solve a nonlinear ill-posed problem connected to inversion of data gathered by a ground conductivity meter

Inversion of multiconfiguration complex EMI data with minimum gradient support regularization: A case study

Frequency-domain electromagnetic instruments allow the collection of data in different configurations, that is, varying the intercoil spacing, the frequency, and the height above the ground. Their handy size makes these tools very practical for near-surface characterization in many fields of applications, for example, precision agriculture, pollution assessments, and shallow geological investigations. To this end, the inversion of either the real (in-phase) or the imaginary (quadrature) component of the signal has already been studied. Furthermore, in many situations, a regularization scheme retrieving smooth solutions is blindly applied, without taking into account the prior available knowledge. The present work discusses an algorithm for the inversion of the complex signal in its entirety, as well as a regularization method that promotes the sparsity of the reconstructed electrical conductivity distribution. This regularization strategy incorporates a minimum gradient support stabilizer into a truncated generalized singular value decomposition scheme. The results of the implementation of this sparsity-enhancing regularization at each step of a damped Gauss-Newton inversion algorithm (based on a nonlinear forward model) are compared with the solutions obtained via a standard smooth stabilizer. An approach for estimating the depth of investigation, that is, the maximum depth that can be investigated by a chosen instrument configuration in a particular experimental setting is also discussed. The effectiveness and limitations of the whole inversion algorithm are demonstrated on synthetic and real data sets