Iterative regularization with minimum-residual methods

Abstract

Abstract.We study the regularization properties of iterative minimum-residual methods applied to discrete ill-posed problems. In these methods, the projection onto the underlying Krylov subspace acts as a regularizer, and the emphasis of this work is on the role played by the basis vectors of these Krylov subspaces. We provide a combination of theory and numerical examples, and our analysis confirms the experience that MIN-RES and MR-II can work as general regularization methods. We also demonstrate theoretically and experimentally that the same is not true, in general, for GMRES and RRGMRES -their success as regularization methods is highly problem dependent. (2000): 65F22, 65F10. AMS subject classification

Solving an Ill-Posed Cauchy Problem for a Two-Dimensional Parabolic PDE with Variable Coefficients Using a Preconditioned GMRES Method

“…Gradually the "ill-posed part" starts to influence the solution, and then the solution explodes [13]. However, unpreconditioned GMRES does not work for solving the SPE problem [23,13].…”

“…Thus in this paper we propose a method for solving numerically a 2D SPE with variable coefficients, discrete in space, using a preconditioned Krylov subspace method, GMRES (generalized minimum residual) [38]. The properties of Krylov methods applied to ill-posed problems have recently been studied in several papers [24,6,7,8,20,23,4]; also preconditioned GMRES has been proposed [20]. It has been reported [23] that without preconditioner GMRES fails to solve even 1D sideways parabolic problems.…”

“…The properties of Krylov methods applied to ill-posed problems have recently been studied in several papers [24,6,7,8,20,23,4]; also preconditioned GMRES has been proposed [20]. It has been reported [23] that without preconditioner GMRES fails to solve even 1D sideways parabolic problems. We here show that it is possible to construct an efficient preconditioner for the 2D case using a nearby equation with constant coefficients.…”

Fast convergence of the Coiflet-Galerkin method for general elliptic BVPs

“…The fast rate of convergence results in very good numerical results, at low levels of approximation, with low degrees of Coiflets, because the size of the corresponding linear system becomes (relatively) small and we can use regularization techniques to solve linear systems (Jensen and Hansen, 2007). These techniques apply (somehow) dense factorizations of the stiffness matrix which is not suitable for large scale systems, in contrast to preconditioning methods, where sparse (usually deflected) factorizations are considered (Saad, 1996).…”

Reconstruction of Single-Grain Orientation Distribution Functions for Crystalline Materials

“…We use iterative regularization methods to solve the large sparse systems of linear equations involved in this approach, and we develop a new robust stopping criterion for these iterative methods. Three methods are considered: ART [7,22], conjugate gradient least squares (CGLS) [3,19], and preconditioned CGLS (PCGLS) [14]. The iterative regularization methods are compared by means of simulations, and their dependence on noise and on the number of projections available is characterized.…”