Daniela Calvetti

Tikhonov regularization and the L-curve for large discrete ill-posed problems

Discretization of linear inverse problems generally gives rise to very ill-conditioned linear systems of algebraic equations. Typically, the linear systems obtained have to be regularized to make the computation of a meaningful approximate solution possible. Tikhonov regularization is one of the most popular regularization methods. A regularization parameter specifies the amount of regularization and, in general, an appropriate value of this parameter is not known a priori. We review available iterative methods, and present new ones, for the determination of a suitable value of the regularization parameter by the L-curve criterion and the solution of regularized systems of algebraic equations.

Application of ADI Iterative Methods to the Restoration of Noisy Images

The restoration of two-dimensional images in the presence of noise by Wieners minimum mean square error filter requires the solution of large linear systems of equations. When the noise is white and Gaussian, and under suitable assumptions on the image, these equations can be written as a Sylvesters equation \[ T_1^{-1}\hat{F}+\hat{F}T_2=C \] for the matrix

Estimation of the L-Curve via Lanczos Bidiagonalization

\hat{F}

TIKHONOV REGULARIZATION OF LARGE LINEAR PROBLEMS

representing the restored image. The matrices

Computation of Gauss-Kronrod of quadrature rules

T_1

Noninvasive Electrocardiographic Imaging (ECGI): Application of the Generalized Minimal Residual (GMRes) Method

and

GMRES, L-Curves, and Discrete Ill-Posed Problems

T_2

IRBL: An Implicitly Restarted Block-Lanczos Method for Large-Scale Hermitian Eigenproblems

are symmetric positive definite Toeplitz matrices. We show that the ADI iterative method is well suited for the solution of these Sylvesters equations, and illustrate this with computed examples for the case when the image is described by a separable first-order Markov process. We also consider generalizations of the ADI iterative method, propose new algorithms for the generation of iteration parameters, and illustrate the competitiveness of these schemes.

Astrocytes as the Glucose Shunt for Glutamatergic Neurons at High Activity: An In Silico Study

The L-curve criterion is often applied to determine a suitable value of the regularization parameter when solving ill-conditioned linear systems of equations with a right-hand side contaminated by errors of unknown norm. However, the computation of the L-curve is quite costly for large problems; the determination of a point on the L-curve requires that both the norm of the regularized approximate solution and the norm of the corresponding residual vector be available. Therefore, usually only a few points on the L-curve are computed and these values, rather than the L-curve, are used to determine a value of the regularization parameter. We propose a new approach to determine a value of the regularization parameter based on computing an L-ribbon that contains the L-curve in its interior. An L-ribbon can be computed fairly inexpensively by partial Lanczos bidiagonalization of the matrix of the given linear system of equations. A suitable value of the regularization parameter is then determined from the L-ribbon, and we show that an associated approximate solution of the linear system can be computed with little additional work.

Non-negativity and iterative methods for ill-posed problems

Many numerical methods for the solution of linear ill-posed problems apply Tikhonov regularization. This paper presents a new numerical method, based on Lanczos bidiagonalization and Gauss quadrature, for Tikhonov regularization of large-scale problems. An estimate of the norm of the error in the data is assumed to be available. This allows the value of the regularization parameter to be determined by the discrepancy principle.