Israel Koltracht

On the convergence of asynchronous paracontractions with application to tomographic reconstruction from incomplete data

Abstract Convergence of iterative processes in C k of the form x i+r i =α j i x 1+r i -1 +(1-α)P j i x i , where j i ϵ{1,2,…, n }, i = 1,2,…, is analyzed. It is shown that if the matrices P 1 ,…, P n are paracontracting in the same smooth, strictly convex norm and if the sequence { j i } ∞ i = 1 has certain regularity properties, then the above iterates converge. This result implies the convergence of a parallel asynchronous implementation of the algebraic reconstruction technique (ART) algorithm often used in tomographic reconstruction from incomplete data.

Convergence of sequential and asynchronous nonlinear paracontractions

SummaryWe establish the convergence of sequential and asynchronous iteration schemes for nonlinear paracontracting operators acting in finite dimensional spaces. Applications to the solution of linear systems of equations with convex constraints are outlined. A first generalization of one of our convergence results to an infinite pool of asymptotically paracontracting operators is also presented.

Nyström-Clenshaw-Curtis quadrature for integral equations with discontinuous kernels

A new highly accurate numerical approximation scheme based on a Gauss type Clenshaw-Curtis quadrature for Fredholm integral equations of the second kind x(t) + ∫ab k(t, s)x(s)ds = y(t), whose kernel k(t,s) is either discontinuous or not smooth along the main diagonal, is presented. This scheme is of spectral accuracy when k(t,s) is infinitely differentiable away from the diagonal t = s. Relation to the singular value decomposition is indicated. Application to integro-differential Schrodinger equations with nonlocal potentials is given.

Inverse Scattering with Noisy Data

In this paper we analyze error propagation in layer-peeling inversion methods. A bound for the error in recovering the reflection coefficient at a certain depth is given in terms of the estimated reflection coefficients. The error propagation results are then used to discuss some practical inversion algorithms that exploit available prior information on the reflection coefficient sequence.

On accurate computations of the Perron root

This paper establishes a new componentwise perturbation result for the Perron root of a nonnegative and irreducible matrix. The error bound is independent of the angle between left and right Perron eigenvectors. It is shown that a known inverse iteration algorithm with new stopping criteria will have a small componentwise backward error, which is consistent with the perturbation result. Numerical experiments demonstrate that the accuracy of the Perron root computed by the proposed algorithm is, indeed, independent of the angle.

Convergence properties of ART and SOR algorithms

SummaryART algorithms with relaxation parameters are studied for general (consistent or inconsistent) linear algebraic systemsRx=f, and a general convergence theorem is formulated. The advantage of severe underrelaxation is reexamined and clarified. The relationship to solutions obtained by applying SOR methods to the equationRRTy=f is investigated.

Threshold algorithms for the prediction of reflection coefficients in a layered medium

An algorithm is presented for the solution of the inverse problem of reflection seismology in the presence of noise. The algorithm is based on a new representation of reflection coefficients in terms of the recorded seismogram. This representation allows use of matrix perturbation methods for the analysis of error magnification in the recursive reconstruction of a stratified acoustic medium. Our analysis indicates that one of the main reasons for uncontrollable noise magnification is the assignment of significant values to very small reflection coefficients, most of which reflect only noise in the data rather than reflection information. Our analysis also allows one to decide when a small computed reflection coefficient should be set to zero. The strategy of setting small reflection coefficients to zero, which is called thresholding, has a stabilizing effect on inverse scattering algorithms. The threshold algorithm also permits adaptive change of noise barriers, which can be used for more detailed exposur...

A novel method for the solution of the Schrödinger equation in the presence of exchange terms

In the Hartree–Fock approximation the Pauli exclusion principle leads to a Schrodinger equation of an integro-differential form. We describe the extension of a new spectral noniterative method (S-IEM), previously developed for solving the Lippmann–Schwinger integral equation with local potentials, so as to include the exchange nonlocality. We apply it to the restricted case of electron-hydrogen scattering in which the bound electron remains in the ground state and the incident electron has zero angular momentum, and we compare the acuracy and economy of the new method to two other methods. One is a noniterative solution of the integral equation as described by Sams and Kouri in 1969. Another is an iterative method introduced by Kim and Udagawa in 1990 for nuclear physics applications, which makes an expansion of the solution into an especially favorable basis obtained by a method of moments. The S-IEM method turns out to be more accurate than the two comparison methods by many orders of magnitude for the ...

Triangular factors of Cauchy and Vandermonde matrices

Explicit formulas for triangular factors of Cauchy and Vandermonde matrices and their inverses in terms of entries of these matrices are presented.

The structure of some matrices arising in tomography

Linear algebraic systems Rx = y are considered which are generated by typical limited-angle tomographic problems. A detailed examination is made of vectors x for which Rx = 0 and which are responsible for “ghost images” frequently referred to in the literature. In particular, the construction of an orthogonal basis for the subspace of all such vectors x is investigated and is complete in some special cases. It is shown how advantage can be taken of this information.