Natural products as leads to the synthesis of new anticancer compounds.

missin

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

Natural products as leads to the synthesis of new anticancer compounds.

missin

Flexible Krylov Methods for <i>L</i>_p regularization

In this paper we develop flexible Krylov methods for efficiently computing regularized solutions to large-scale linear inverse problems with an `2 fit-to-data term and an `p penalization term, for p ≥ 1. First we approximate the p-norm penalization term as a sequence of 2-norm penalization terms using adaptive regularization matrices in an iterative reweighted norm fashion, and then we exploit flexible preconditioning techniques to efficiently incorporate the weight updates. To handle general (nonsquare) `p-regularized least-squares problems, we introduce a flexible Golub–Kahan approach and exploit it within a Krylov–Tikhonov hybrid framework. Furthermore, we show that both the flexible Golub–Kahan and the flexible Arnoldi approaches for p = 1 can be used to efficiently compute solutions that are sparse with respect to some transformations. The key benefits of our approach compared to existing optimization methods for `p regularization are that inner-outer iterationschemes are replaced by efficient projection methods on linear subspaces of increasing dimension and that expensive regularization parameter selection techniques can be avoided. Theoretical insights are provided, and numerical results from image deblurring and tomographic reconstruction illustrate the benefits of this approach, compared to well-established methods

Some transpose-free CG-like solvers for nonsymmetric ill-posed problems

This paper introduces and analyzes an original class of Krylov subspace methods that provide an efficient alternative to many well-known conjugate-gradient-like (CG-like) Krylov solvers for square nonsymmetric linear systems arising from discretizations of inverse ill-posed problems. The main idea underlying the new methods is to consider some rank-deficient approximations of the transpose of the system matrix, obtained by running the (transpose-free) Arnoldi algorithm, and then apply some Krylov solvers to a formally right-preconditioned system of equations. Theoretical insight is given, and many numerical tests show that the new solvers outperform classical Arnoldi-based or CG-like methods in a variety of situations

Some transpose-free CG-like solvers for nonsymmetric ill-posed problems

2siThis paper introduces and analyzes an original class of Krylov subspace methods that provide an efficient alternative to many well-known conjugate-gradient-like (CG-like) Krylov solvers for square nonsymmetric linear systems arising from discretizations of inverse ill-posed problems. The main idea underlying the new methods is to consider some rank-deficient approximations of the transpose of the system matrix, obtained by running the (transpose-free) Arnoldi algorithm, and then apply some Krylov solvers to a formally right-preconditioned system of equations. Theoretical insight is given, and many numerical tests show that the new solvers outperform classical Arnoldi-based or CG-like methods in a variety of situations.openembargoed_20210328Gazzola S.; Novati P.Gazzola, S.; Novati, P

Computational methods for large-scale inverse problems:a survey on hybrid projection methods

This paper surveys animportant class of methods that combine iterative projection methods and variational regularization methods for large-scale inverse problems. Iterative methods such as Krylov subspace methods are invaluable in the numerical linear algebra community and have proved important in solving inverse problems due to their inherent regularizing properties and their ability to handle large-scale problems. Variational regularization describes abroad and important class of methods that are used to obtain reliable solutions to inverse problems, whereby one solves a modified problem that incorporates prior knowledge. Hybrid projection methods combine iterative projection methods with variational regularization techniques in a synergistic way, providing researchers with a powerful computational framework for solving very large inverse problems. Although the idea of a hybrid Krylov method for linear inverse problems goes back to the 1980s, several recent advances on new regularization frameworks and methodologies have made this field ripe for extensions, further analyses, and new applications. In this paper, we provide a practical and accessible introduction to hybrid projection methods in the context of solving large (linear) inverse problems