Pseudospectra and structured pseudospectra are important tools for the analysis of matrices. Their computation, however, can be very demanding for all but small matrices. A new approach to compute approximations of pseudospectra and structured pseudospectra, based on determining the spectra of many suitably chosen rank-one or projected rank-one perturbations of the given matrix is proposed. The...

A kinematically correct choice of velocity focuses subsurface offset image gathers at zero offset. Infinitesimal warping from the current image towards its focus can be approximated by a horizontal contraction. The image residual can be then taken as the difference between the warped and the original image to account for the velocity error. Least squares fitting of the effect of a velocity pert...

We propose a quasi-Newton line-search method that uses negative curvature directions for solving unconstrained optimization problems. In this method, the symmetric rank-one (SR1) rule is used to update the Hessian approximation. The SR1 update rule is known to have a good numerical performance; however, it does not guarantee positive definiteness of the updated matrix. We first discuss the deta...

This paper studies the H1 Sobolev seminorm of quadratic functions. The research is motivated by the least-norm interpolation that is widely used in derivative-free optimization. We express the H1 seminorm of a quadratic function explicitly in terms of the Hessian and the gradient when the underlying domain is a ball. The seminorm gives new insights into least-norm interpolation. It clarifies th...

A new quasi-Newton scheme for updating a low rank positive semi-definite Hessian approximation is described, primarily for use in sequential quadratic programming methods for nonlinear programming. Where possible the symmetric rank one update formula is used, but when this is not possible a new rank two update is used, which is not in the Broyden family, although invariance under linear transfo...

A new efficient algorithm is presented for joint diagonalization of several matrices. The algorithm is based on the Frobenius-norm formulation of the joint diagonalization problem, and addresses diagonalization with a general, non-orthogonal transformation. The iterative scheme of the algorithm is based on a multiplicative update which ensures the invertibility of the diagonalizer. The algorith...

We consider Hessian approximation schemes for large-scale unconstrained optimization in the context of discretized problems. The considered Hessians typically present a nontrivial sparsity and partial separability structure. This allows iterative quasi-Newton methods to solve them despite of their size. Structured finitedifference methods and updating schemes based on the secant equation are pr...

This note focuses on developing quasi-Newton methods that combine m+ 1 multistep and single-step updates on a single iteration for the sake of constructing the new approximation to the Hessian matrix to be used on the next iteration in computing the search direction. The approach considered here exploits the merits of the multistepmethods and those of El-Baali (1999) to create a hybrid techniqu...

The simple update (SU) and full (FU) are the two paradigmatic time evolution algorithms for a tensor network known as infinite projected entangled pair state (iPEPS). They differ by an error measure that is either, respectively, local or takes into account environment. In this paper we test intermediate neighborhood (NTU) accounting nearest neighbor This small environment can be contracted exac...