1,446 research outputs found
A DEIM Induced CUR Factorization
We derive a CUR matrix factorization based on the Discrete Empirical
Interpolation Method (DEIM). For a given matrix , such a factorization
provides a low rank approximate decomposition of the form ,
where and are subsets of the columns and rows of , and is
constructed to make a good approximation. Given a low-rank singular value
decomposition , the DEIM procedure uses and to
select the columns and rows of that form and . Through an error
analysis applicable to a general class of CUR factorizations, we show that the
accuracy tracks the optimal approximation error within a factor that depends on
the conditioning of submatrices of and . For large-scale problems,
and can be approximated using an incremental QR algorithm that makes one
pass through . Numerical examples illustrate the favorable performance of
the DEIM-CUR method, compared to CUR approximations based on leverage scores
Green's functions for multiply connected domains via conformal mapping
A method is described for the computation of the Green's function in the complex plane corresponding to a set of K symmetrically placed polygons along the real axis. An important special case is a set of K real intervals. The method is based on a Schwarz-Christoffel conformal map of the part of the upper half-plane exterior to the problem domain onto a semi-infinite strip whose end contains K-1 slits. From the Green's function one can obtain a great deal of information about polynomial approximations, with applications in digital filters and matrix iteration. By making the end of the strip jagged, the method can be generalised to weighted Green's functions and weighted approximations
Spectral Properties of Schr\"odinger Operators Arising in the Study of Quasicrystals
We survey results that have been obtained for self-adjoint operators, and
especially Schr\"odinger operators, associated with mathematical models of
quasicrystals. After presenting general results that hold in arbitrary
dimensions, we focus our attention on the one-dimensional case, and in
particular on several key examples. The most prominent of these is the
Fibonacci Hamiltonian, for which much is known by now and to which an entire
section is devoted here. Other examples that are discussed in detail are given
by the more general class of Schr\"odinger operators with Sturmian potentials.
We put some emphasis on the methods that have been introduced quite recently in
the study of these operators, many of them coming from hyperbolic dynamics. We
conclude with a multitude of numerical calculations that illustrate the
validity of the known rigorous results and suggest conjectures for further
exploration.Comment: 56 page
Spectral Approximation for Quasiperiodic Jacobi Operators
Quasiperiodic Jacobi operators arise as mathematical models of quasicrystals
and in more general studies of structures exhibiting aperiodic order. The
spectra of these self-adjoint operators can be quite exotic, such as Cantor
sets, and their fine properties yield insight into associated dynamical
systems. Quasiperiodic operators can be approximated by periodic ones, the
spectra of which can be computed via two finite dimensional eigenvalue
problems. Since long periods are necessary to get detailed approximations, both
computational efficiency and numerical accuracy become a concern. We describe a
simple method for numerically computing the spectrum of a period- Jacobi
operator in operations, and use it to investigate the spectra of
Schr\"odinger operators with Fibonacci, period doubling, and Thue-Morse
potentials
The Spectra of Large Toeplitz Band Matrices with a Randomly Perturbed Entry
This report is concerned with the union of all possible spectra that may emerge when perturbing a large Toeplitz band matrix in the site by a number randomly chosen from some set . The main results give descriptive bounds and, in several interesting situations, even provide complete identifications of the limit of as . Also discussed are the cases of small and large sets as well as the "discontinuity of the infinite volume case", which means that in general does not converge to something close to as , where is the corresponding infinite Toeplitz matrix. Illustrations are provided for tridiagonal Toeplitz matrices, a notable special case. \ud
\ud
The second author was supported by UK Enginering and Physical Sciences Research Council Grant GR/M1241
Generational Phenomenology
Information from a book (by Lancaster and Stillman, 2002 β see later) about generational gaps and conflicts in American companies is used to show that there is a generational dimension to the socio-cultural lifeworld. In relation to that, some indications are offered about how attitudes toward oneβs own as well as other generations can be reflectively analyzed. Other societies probably have similar differences between generations.Indo-Pacific Journal of Phenomenology, Volume 3, Edition 1, November 200
How descriptive are GMRES convergence bounds?
Eigenvalues with the eigenvector condition number, the field of values, and pseudospectra have all been suggested as the basis for convergence bounds for minimum residual Krylov subspace methods applied to non-normal coefficient matrices. This paper analyzes and compares these bounds, illustrating with six examples the success and failure of each one. Refined bounds based on eigenvalues and the field of values are suggested to handle low-dimensional non-normality. It is observed that pseudospectral bounds can capture multiple convergence stages. Unfortunately, computation of pseudospectra can be rather expensive. This motivates an adaptive technique for estimating GMRES convergence based on approximate pseudospectra taken from the Arnoldi process that is the basis for GMRES
The Tortoise and the Hare restart GMRES
When solving large nonsymmetric systems of linear equations with the restarted GMRES algorithm, one is inclined to select a relatively large restart parameter in the hope of mimicking the full GMRES process. Surprisingly, cases exist where small values of the restart parameter yield convergence in fewer iterations than larger values. Here, two simple examples are presented where GMRES(1) converges exactly in three iterations, while GMRES(2) stagnates. One of these examples reveals that GMRES(1) convergence can be extremely sensitive to small changes in the initial residual
- β¦