In this work we study the minimization of a linear functional defined on a set of approximate solutions of a discrete ill-posed problem. The primary application of interest is the computation of confidence intervals for components of the solution of such a problem. We exploit the technique introduced by Eldén in 1990, utilizing a parametric programming reformulation involving the solution of a ...

Tikhonov regularization for large-scale linear ill-posed problems is commonly implemented by determining a partial Lanczos bidiagonalization of the matrix of the given system of equations. This paper explores the possibility of instead computing a partial Arnoldi decomposition of the given matrix. Computed examples illustrate that this approach may require fewer matrix-vector product evaluation...

We present a fast 5D (frequency and 4 spatial axes) reconstruction method that uses Multichannel Singular Spectrum Analysis / Cazdow algorithm. Rather than embedding the 4D spatial volume in a Hankel matrix, we propose to embed the data into a block Toeplitz form. Rank reduction is carried out via Lanczos bidiagonalization with fast block Toeplitz matrix-times-vector multiplications via 4D Fast...

This paper presents a preconditioning method based on a recursive multilevel lowrank approximation approach. The basic idea is to recursively divide the problem into two and apply a low-rank approximation to a matrix obtained from the Sherman-Morrison formula. The low-rank approximation may be computed by the partial Singular Value Decomposition (SVD) or it can be approximated by the Lanczos bi...

Generalized matrix functions (GMFs) extend the concept of a matrix function to rectangular matrices via the singular value decomposition. Several applications involving directed graphs, Hamiltonian dynamical systems, and optimization problems with low-rank constraints require the action of a GMF of a large, sparse matrix on a vector. We present a new method for applying GMFs to vectors based on...

We introduce a randomized algorithm for computing the minimal-norm solution to an underdetermined system of linear equations. Given an arbitrary full-rank matrix Am×n with m < n, any vector bm×1, and any positive real number ε less than 1, the procedure computes a vector xn×1 approximating to relative precision ε or better the vector pn×1 of minimal Euclidean norm satisfying Am×n pn×1 = bm×1. T...

In theory, the Lanczos algorithm generates an orthogonal basis of corresponding Krylov subspace. However, in finite precision arithmetic orthogonality and linear independence computed vectors is usually lost quickly. this paper we study a class matrices starting having special nonzero structure that guarantees exact computations whenever floating point satisfying IEEE 754 standard used. Analogo...

it is well known that if the coefficient matrix in a linear system is large and sparse or sometimes not readily available, then iterative solvers may become the only choice. the block solvers are an attractive class of iterative solvers for solving linear systems with multiple right-hand sides. in general, the block solvers are more suitable for dense systems with preconditioner. in this paper,...