Iterative methods are often used for applications in science and engineering to solve very large scale linear systems. Efficiency of an iterative method depends on the amount of computation needed per iteration, as well as on the number of iterations needed to reconstruct the desired approximate solution. Convergence speed can be accelerated using a technique called preconditioning. Although pr...

Several numerical methods for the solution of large linear ill-posed problems combine Tikhonov regularization with an iterative method based on partial Lanczos bidiagonalization of the operator. This paper discusses the determination of the regularization parameter and the dimension of the Krylov subspace for this kind of methods. A method that requires a Krylov subspace of minimal dimension is...

This paper reports studies on the influence of the regularization parameter and the first estimate on the performance of iterative image restoration algorithms. We discuss regularization parameter estimation methods that have been developed for the linear Tikhonov-Miller filter to restore images distorted by additive Gaussian noise. We have performed experiments on synthetic data to show that t...

This paper is concerned with the solution of large-scale linear discrete ill-posed problems. The determination of a meaningful approximate solution of these problems requires regularization. We discuss regularization by the Tikhonov method and by truncated iteration. The choice of regularization matrix in Tikhonov regularization may significantly affect the quality of the computed approximate s...

We shall be concerned with the construction of Tikhonov–based iteration schemes for solving nonlinear operator equations. In particular, we are interested in algorithms for the computation of a minimizer of the Tikhonov functional. To this end, we introduce a replacement functional, that has much better properties than the classical Tikhonov functional with nonlinear operator. Namely, the repla...

We report on a new iterative method for regularizing a nonlinear operator equation in Hilbert spaces. The proposed algorithm is a combination of Tikhonov regularization and a fixed point algorithm for the minimization of the Tikhonov–functional. Under the assumptions that the operator F is twice continuous Fréchet–differentiable with Lipschitz– continuous first derivative and that the solution ...

A main problem in adaptive optics is to reconstruct the phase spectrum given noisy phase differences. We present an efficient approach to solve the least-squares minimization problem resulting from this reconstruction, using either a truncated singular value decomposition (TSVD)-type or a Tikhonov-type regularization. Both of these approaches make use of Kronecker products and the generalized s...

This paper considers an efficient iterative approach to solve separable nonlinear least squares problems that arise in large scale inverse problems. A variable projection GaussNewton method is used to solve the nonlinear least squares problem, and Tikhonov regularization is incorporated using an iterative Lanczos hybrid scheme. Regularization parameters are chosen automatically using a weighted...

SENSE reconstruction suffers from an ill-conditioning problem, which increasingly lowers the signal-to-noise ratio (SNR) as the reduction factor increases. Ill-conditioning also degrades the convergence behavior of iterative conjugate gradient reconstructions for arbitrary trajectories. Regularization techniques are often used to alleviate the ill-conditioning problem. Based on maximum a poster...