Computing a small number of singular values is required in many practical applications and it is therefore desirable to have efficient and robust methods that can generate such truncated singular value decompositions. A new method based on the Lanczos bidiagonalization and the Krylov-Schur method is presented. It is shown how deflation strategies can be easily implemented in this method and pos...

Given a square matrix A, the inverse subspace problem is concerned with determining a closest matrix to A with a prescribed invariant subspace. When A is Hermitian, the closest matrix may be required to be Hermitian. We measure distance in the Frobenius norm and discuss applications to Krylov subspace methods for the solution of large-scale linear systems of equations and eigenvalue problems as...

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...

The L-curve criterion is often applied to determine a suitable value of the regularization parameter when solving ill-conditioned linear systems of equations with a right-hand side contaminated by errors of unknown norm. However, the computation of the L-curve is quite costly for large problems; the determination of a point on the L-curve requires that both the norm of the regularized approxima...

one of the most remarkable basis of the gravity data inversion is the recognition of sharp boundaries between an ore body and its host rocks during the interpretation step. therefore, in this work, it is attempted to develop an inversion approach to determine a 3d density distribution that produces a given gravity anomaly. the subsurface model consists of a 3d rectangular prisms of known sizes ...

The reduction of a large-scale symmetric linear discrete ill-posed problem with multiple right-hand sides to smaller block tridiagonal matrix can easily be carried out by the application small number steps Lanczos method. We show that subdiagonal blocks reduced converge zero fairly rapidly increasing number. This quick convergence indicates there is little advantage in expressing solutions prob...