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 problems are presented. Copyright © 2014 John Wiley & Sons, Ltd.     

Iterated Tikhonov regularization with a general penalty term

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

Two Tikhonov‐type regularization methods for inverse source problem on the Poisson equation

In this paper, we investigate a problem of the identification of an unknown source on Poisson equation from some fixed location. A conditional stability estimate for an inverse heat source problem is proved. We show that such a problem is mildly ill‐posed and further present two Tikhonov‐type regularization methods (a generalized Tikhonov regularization method and a simplified generalized Tikhonov regularization method) to deal with this problem. Convergence estimates are presented under the a priori choice of the regularization parameter. Numerical results are presented to illustrate the accuracy and efficiency of our methods. Copyright © 2012 John Wiley & Sons, Ltd.     

Tikhonov regularization of nonlinear III-posed problems in hilbert scales

In this paper we consider nonlinear ill-posed problems F(x) = y ₀, where x and y ₀ are elements of Hilbert spaces X and Y, respectively. We solve these problems by Tikhonov regularization in a Hilbert scale. This means that the regularizing norm is stronger than the norm in X. Smoothness conditions are given that guarantee convergence rates with respect to the data noise in the original norm in X. We also propose a variant of Tikhonov regularization that yields these rates without needing the knowledge of the smoothness conditions. In this variant F is allowed to be known only approximately and X can be approximated by a finite-dimensional subspace. Finally, we illustrate the required conditions for a simple parameter estimation problem for regularization in Sobolev spaces.     

A new variant of L-curve for Tikhonov regularization

In this paper we introduce a new variant of L-curve to estimate the Tikhonov regularization parameter for the regularization of discrete ill-posed problems. This method uses the solution norm versus the regularization parameter. The numerical efficiency of this new method is also discussed by considering some test problems.     

Adaptive iterative regularization schemes for two‐dimensional integral‐algebraic systems

The main idea of this paper is to utilize the adaptive iterative schemes based on regularization techniques for moderately ill‐posed problems that are obtained by a system of linear two‐dimensional Volterra integral equations with a singular matrix in the leading part. These problems may arise in the modeling of certain heat conduction processes as well as in the dynamic simulation packages such as compressible flow through a plant piping network. Owing to the ill‐posed nature of the first kind Volterra equation that appears in the system, we will focus on the two families of regularization algorithms, ie, the Landweber and Lavrentiev type methods, where we treat both the exact and perturbed data. Our aim is to work directly with the original Volterra equations without any kind of reduction. Two fast iterative algorithms with reasonable computational complexity are developed. Numerical experiments on a few test problems are used to illustrate the validity and efficiency of the proposed iterative methods in comparison with the classical regularization methods.     

An inverse problem for space‐fractional backward diffusion problem

In this paper, an inverse problem for space‐fractional backward diffusion equation, which is highly ill‐posed, is considered. This problem is obtained from the classical diffusion equation by replacing the second‐order space derivative with a Riesz–Feller derivative of order α ∈ (0,2]. We show that such a problem is severely ill‐posed, and further present a simplified Tikhonov regularization method to deal with this problem. Convergence estimate is presented under a priori choice of regularization parameter. Numerical experiments are given to illustrate the accuracy and efficiency of the proposed method. Copyright © 2013 John Wiley & Sons, Ltd.     

A Riesz–Feller space‐fractional backward diffusion problem with a time‐dependent coefficient: regularization and error estimates

In this paper, we consider a Riesz–Feller space‐fractional backward diffusion problem with a time‐dependent coefficient We show that this problem is ill‐posed; therefore, we propose a convolution regularization method to solve it. New error estimates for the regularized solution are given under a priori and a posteriori parameter choice rules, respectively. Copyright © 2016 John Wiley & Sons, Ltd.     

Multi-parameter Tikhonov regularization and model function approach to the damped Morozov principle for choosing regularization parameters

In this paper, we study the multi-parameter Tikhonov regularization method which adds multiple different penalties to exhibit multi-scale features of the solution. An optimal error bound of the regularization solution is obtained by a priori choice of multiple regularization parameters. Some theoretical results of the regularization solution about the dependence on regularization parameters are presented. Then, an a posteriori parameter choice, i.e., the damped Morozov discrepancy principle, is introduced to determine multiple regularization parameters. Five model functions, i.e., two hyperbolic model functions, a linear model function, an exponential model function and a logarithmic model function, are proposed to solve the damped Morozov discrepancy principle. Furthermore, four efficient model function algorithms are developed for finding reasonable multiple regularization parameters, and their convergence properties are also studied. Numerical results of several examples show that the damped discrepancy principle is competitive with the standard one, and the model function algorithms are efficient for choosing regularization parameters.     

The Tikhonov regularization extended to equilibrium problems involving pseudomonotone bifunctions

We extend the Tikhonov regularization method widely used in optimization and monotone variational inequality studies to equilibrium problems. It is shown that the convergence results obtained from the monotone variational inequality remain valid for the monotone equilibrium problem. For pseudomonotone equilibrium problems, the Tikhonov regularized subproblems have a unique solution only in the limit, but any Tikhonov trajectory tends to the solution of the original problem, which is the unique solution of the strongly monotone equilibrium problem defined on the basis of the regularization bifunction.     

Condition numbers and perturbation analysis for the Tikhonov regularization of discrete ill‐posed problems

One of the most successful methods for solving the least‐squares problem min_x∥Ax?b∥₂ with a highly ill‐conditioned or rank deficient coefficient matrix A is the method of Tikhonov regularization. In this paper, we derive the normwise, mixed and componentwise condition numbers and componentwise perturbation bounds for the Tikhonov regularization. Our results are sharper than the known results. Some numerical examples are given to illustrate our results. Copyright © 2010 John Wiley & Sons, Ltd.     

Fixed-point iterations in determining a Tikhonov regularization parameter in Kirsch’s factorization method

Kirsch’s factorization method is a fast inversion technique for visualizing the profile of a scatterer from measurements of the far-field pattern. The mathematical basis of this method is given by the far-field equation, which is a Fredholm integral equation of the first kind in which the data function is a known analytic function and the integral kernel is the measured (and therefore noisy) far-field pattern. We present a Tikhonov parameter choice approach based on a fast fixed-point iteration method which constructs a regularization parameter associated with the corner of the L-curve in log-log scale. The performance of the method is evaluated by comparing our reconstructions with those obtained via the L-curve and we conclude that our method yields reliable reconstructions at a lower computational cost.     

Asymptotic L-based comparison of Tikhonov regularization and Landweber iteration

The distance between the Tikhonov and Landweber regularized solutions of a linear inverse problem is partly controlled by the L^∞-norm of the difference in their corresponding singular value filters. For large Landweber iteration number, the regularization parameter of the closest Tikhonov filter to a given Landweber filter is determined. This asymptotically computed parameter compares well with the numerically computed value even for moderate sized iteration number. A consequence of the analysis is to determine the range of singular values to which the difference in regularized solutions is most sensitive.     

A reduced basis approach to large‐scale pseudospectra computation

For studying spectral properties of a nonnormal matrix $A\in {\mathbb{C}}^{n\times n}$, information about its spectrum σ(A) alone is usually not enough. Effects of perturbations on σ(A) can be studied by computing ε‐pseudospectra, i.e. the level sets of the resolvent norm function $\mathsf{g}(z)=\Vert {(zI?A)}^{?1}{\Vert }_2$. The computation of ε‐pseudospectra requires determining the smallest singular values ${\sigma }_{\min }(zI?A)$ for all z on a portion of the complex plane. In this work, we propose a reduced basis approach to pseudospectra computation, which provides highly accurate estimates of pseudospectra in the region of interest, in particular, for pseudospectra estimates in isolated parts of the spectrum containing few eigenvalues of A. It incorporates the sampled singular vectors of zI ? A for different values of z, and implicitly exploits their smoothness properties. It provides rigorous upper and lower bounds for the pseudospectra in the region of interest. In addition, we propose a domain splitting technique for tackling numerically more challenging examples. We present a comparison of our algorithms to several existing approaches on a number of numerical examples, showing that our approach provides significant improvement in terms of computational time.     

On a non‐uniqueness in fragmentation models

Equations of fragmentation describing e.g. the polymer degradation, despite their apparent simplicity, have numerous non‐typical features. One of them is the existence of multiple solutions. In this paper, applying classical results of Hille and Phillips, we explain their occurrence and prove that nevertheless the uniqueness holds for a large class of physically reasonable solutions. Copyright © 2002 John Wiley & Sons, Ltd.     

A modified truncated singular value decomposition method for discrete ill‐posed problems

Truncated singular value decomposition is a popular method for solving linear discrete ill‐posed problems with a small to moderately sized matrix A. Regularization is achieved by replacing the matrix A by its best rank‐k approximant, which we denote by A_k. The rank may be determined in a variety of ways, for example, by the discrepancy principle or the L‐curve criterion. This paper describes a novel regularization approach, in which A is replaced by the closest matrix in a unitarily invariant matrix norm with the same spectral condition number as A_k. Computed examples illustrate that this regularization approach often yields approximate solutions of higher quality than the replacement of A by A_k.Copyright © 2014 John Wiley & Sons, Ltd.     

GCV for Tikhonov regularization via global Golub–Kahan decomposition

Generalized cross validation is a popular approach to determining the regularization parameter in Tikhonov regularization. The regularization parameter is chosen by minimizing an expression, which is easy to evaluate for small‐scale problems, but prohibitively expensive to compute for large‐scale ones. This paper describes a novel method, based on Gauss‐type quadrature, for determining upper and lower bounds for the desired expression. These bounds are used to determine the regularization parameter for large‐scale problems. Computed examples illustrate the performance of the proposed method and demonstrate its competitiveness. Copyright © 2016 John Wiley & Sons, Ltd.     

Discrepancy principles for Tikhonov regularization of ill-posed problems leading to optimal convergence rates

We propose a class ofa posteriori parameter choice strategies for Tikhonov regularization (including variants of Morozov's and Arcangeli's methods) that lead to optimal convergence rates toward the minimal-norm, least-squares solution of an ill-posed linear operator equation in the presence of noisy data.     

Modified Tikhonov regularization method for the Cauchy problem of the Helmholtz equation

In this paper, the Cauchy problem for the Helmholtz equation is investigated. By Green’s formulation, the problem can be transformed into a moment problem. Then we propose a modified Tikhonov regularization algorithm for obtaining an approximate solution to the Neumann data on the unspecified boundary. Error estimation and convergence analysis have been given. Finally, we present numerical results for several examples and show the effectiveness of the proposed method.