Silvia Gazzola

GENERALIZED ARNOLDI-TIKHONOV METHOD FOR SPARSE RECONSTRUCTION ∗

This paper introduces two new algorithms, belonging to the class of Arnoldi--Tikhonov regularization methods, which are particularly appropriate for sparse reconstruction. The main idea is to consider suitable adaptively defined regularization matrices that allow the usual 2-norm regularization term to approximate a more general regularization term expressed in the

Automatic parameter setting for Arnoldi-Tikhonov methods

p

Embedded techniques for choosing the parameter in tikhonov regularization

-norm,

On the Lanczos and Golub–Kahan reduction methods applied to discrete ill‐posed problems

p\geq 1

Image reconstruction and restoration using the simplified topological ε-algorithm

. The regularization matrix can be updated both at each step and after some iterations have been performed, leading to two different approaches: the first one is based on the idea of the iteratively reweighted least squares method and can be obtained considering flexible Krylov subspaces; the second one is based on restarting the Arnoldi algorithm. Numerical examples are given in order to show the effectiveness of these new methods, and comparisons with some other already existing algorithms are made.

Fast Nonnegative Least Squares Through Flexible Krylov Subspaces

In the framework of iterative regularization techniques for large-scale linear ill-posed problems, this paper introduces a novel algorithm for the choice of the regularization parameter when performing the Arnoldi-Tikhonov method. Assuming that we can apply the discrepancy principle, this new strategy can work without restrictions on the choice of the regularization matrix. Moreover, this method is also employed as a procedure to detect the noise level whenever it is just overestimated. Numerical experiments arising from the discretization of integral equations and image restoration are presented.

On Krylov projection methods and Tikhonov regularization

SUMMARY 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

Multi-parameter Arnoldi-Tikhonov methods

The symmetric Lanczos method is commonly applied to reduce large-scale symmetric linear discrete ill-posed problems to small ones with a symmetric tridiagonal matrix. We investigate how quickly the nonnegative subdiagonal entries of this matrix decay to zero. Their fast decay to zero suggests that there is little benefit in expressing the solution of the discrete ill-posed problems in terms of the eigenvectors of the matrix compared with using a basis of Lanczos vectors, which are cheaper to compute. Similarly, we show that the solution subspace determined by the LSQR method when applied to the solution of linear discrete ill-posed problems with a nonsymmetric matrix often can be used instead of the solution subspace determined by the singular value decomposition without significant, if any, reduction of the quality of the computed solution.

Energetic boundary element method analysis of wave propagation in 2D multilayered media

In order to compute meaningful approximations of the solutions of large-scale linear inverse ill-posed problems, some form of regularization should be employed. Cimmino and Landweber methods are well-known iterative regularization methods that can be quite successfully applied for tomographic reconstruction and image restoration problems, despite their usually slow convergence. The goal of this paper is to explore the performance of a recent extrapolation algorithm when applied to accelerate the convergence of these iterative regularization methods. In particular, we provide insight and algorithmic details about the simplified topological e-algorithm applied to slow-converging iterative regularization methods. The results of many numerical experiments and comparisons with other methods are also displayed.

Inheritance of the discrete Picard condition in Krylov subspace methods

Constrained least squares problems arise in a variety of applications, and many iterative methods are already available to compute their solutions. This paper proposes a new efficient approach to solve nonnegative linear least squares problems. The associated KKT conditions are leveraged to form an adaptively preconditioned linear system, which is then solved by a flexible Krylov subspace method. The new method can be easily applied to image reconstruction problems affected by both Gaussian and Poisson noise, where the components of the solution represent nonnegative intensities. {Theoretical insight is given, and} numerical experiments and comparisons are displayed in order to validate the new method, which delivers results of equal or better quality than many state-of-the-art methods for nonnegative least squares solvers, with a significant speedup.