The convergence behavior of a number of algorithms based on minimizing residual norms over Krylov subspaces, is not well understood. Residual or error bounds currently available are either too loose or depend on unknown constants which can be very large. In this paper we take another look at traditional as well as alternative ways of obtaining upper bounds on residual norms. In particular, we d...

Iterative solution of the Levermore-Pomraning equations for transport in binary statistical mixtures can be extremely slow in certain limits. We propose an iterative method that improves convergence by utilizing a combination of inner iterations, synthetic acceleration schemes and Krylov iterative methods. Spectral analysis and numerical results show that our new scheme outperforms simpler iter...

This paper is concerned with the solution of boundary-element models based on substructuring. Structured matrix-vector products and the matrix-copy option are proposed to increase the efficiency of algorithms based on Krylov solvers. The former technique was designed to avoid the excessive number of conditional tests during solver iterations, and the latter one, to avoid the repeated calculatio...

In this paper, a strategy is proposed for alternative computations of the residual vectors in Krylov subspace methods, which improves the agreement of the computed residuals and the true residuals to the level of O(u)‖A‖‖x‖. Building on earlier ideas on residual replacement and on insights in the finite precision behavior of the Krylov subspace methods, computable error bounds are derived for i...

In various applications, for instance in the detection of a Hopf bifurcation or in solving separable boundary value problems using the two-parameter eigenvalue problem, one has to solve a generalized eigenvalue problem of the form (B1 ⊗A2 −A1 ⊗B2)z = μ(B1 ⊗ C2 − C1 ⊗B2)z, where matrices are 2 × 2 operator determinants. We present efficient methods that can be used to compute a small subset of t...

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 p-norm, p ≥ 1. The regularizat...

This paper takes another look at the convergence of the Arnoldi procedure forsolving nonsymmetric eigenvalue problems. Three different viewpoints are considered. The first usesa bound on the distance from the eigenvector to the Krylov subspace from the smallest singularvalue of matrix consisting of the Krylov basis. A second approach relies on the Schur factorization.Finally, a ...

This paper proposes accelerated subspace optimization methods in the context of image restoration. Subspace optimization methods belong to the class of iterative descent algorithms for unconstrained optimization. At each iteration of such methods, a stepsize vector allowing the best combination of several search directions is computed through a multidimensional search. It is usually obtained by...

In this paper we propose a general approach by which eigenvalues with a special property of a given matrix A can be obtained. In this approach we first determine a scalar function $ : @ + @ whose modulus is maximized by the eigenvalues that have the special property. Next, we compute the generalized power iterations U,+l =if!I(A)Uj, j=O,l,..., where u. is an arbitrary initial vector. Finally, w...

We consider certain speed estimates for Krylov subspace methods (such as GMRES) when applied upon systems consisting of a compact operator K with small unstructured perturbation B. Information about the decay of singular values of K is also assumed. Our main result is that the Krylov method will perform initially at superlinear speed when applied upon such pre-conditioned system. However, with ...