The Anderson model of localization: a challenge for modern eigenvalue methods

We present a comparative study of the application of modern eigenvalue algorithms to an eigenvalue problem arising in quantum physics, namely, the computation of a few interior eigenvalues and their associated eigenvectors for the large, sparse, real, symmetric, and indefinite matrices of the Anderson model of localization. We compare the Lanczos algorithm in the 1987 implementation of Cullum and Willoughby with the implicitly restarted Arnoldi method coupled with polynomial and several shift-and-invert convergence accelerators as well as with a sparse hybrid tridiagonalization method. We demonstrate that for our problem the Lanczos implementation is faster and more memory efficient than the other approaches. This seemingly innocuous problem presents a major challenge for all modern eigenvalue algorithms.Comment: 16 LaTeX pages with 3 figures include

A Wiener-Laguerre model of VIV forces given recent cylinder velocities

Slender structures immersed in a cross flow can experience vibrations induced by vortex shedding (VIV), which cause fatigue damage and other problems. VIV models in engineering use today tend to operate in the frequency domain. A time domain model would allow to capture the chaotic nature of VIV and to model interactions with other loads and non-linearities. Such a model was developed in the present work: for each cross section, recent velocity history is compressed using Laguerre polynomials. The compressed information is used to enter an interpolation function to predict the instantaneous force, allowing to step the dynamic analysis. An offshore riser was modeled in this way: Some analyses provided an unusually fine level of realism, while in other analyses, the riser fell into an unphysical pattern of vibration. It is concluded that the concept is promissing, yet that more work is needed to understand orbit stability and related issues, in order to further progress towards an engineering tool

Fast hybrid numerical-asymptotic boundary element methods for high frequency screen and aperture problems based on least-squares collocation

We present a hybrid numerical-asymptotic (HNA) boundary element method (BEM) for high frequency scattering by two-dimensional screens and apertures, whose computational cost to achieve any prescribed accuracy remains bounded with increasing frequency. Our method is a collocation implementation of the high order hp HNA approximation space of Hewett et al. (IMA J Numer Anal 35:1698–1728, 2015), where a Galerkin implementation was studied. An advantage of the current collocation scheme is that the one-dimensional highly oscillatory singular integrals appearing in the BEM matrix entries are significantly easier to evaluate than the two-dimensional integrals appearing in the Galerkin case, which leads to much faster computation times. Here we compute the required integrals at frequency-independent cost using the numerical method of steepest descent, which involves complex contour deformation. The change from Galerkin to collocation is nontrivial because naive collocation implementations based on square linear systems suffer from severe numerical instabilities associated with the numerical redundancy of the HNA basis, which produces highly ill-conditioned BEM matrices. In this paper we show how these instabilities can be removed by oversampling, and solving the resulting overdetermined collocation system in a weighted least-squares sense using a truncated singular value decomposition. On the basis of our numerical experiments, the amount of oversampling required to stabilise the method is modest (around 25% typically suffices), and independent of frequency. As an application of our method we present numerical results for high frequency scattering by prefractal approximations to the middle-third Cantor set

Hybrid chaotic map with L-shaped fractal Tromino for image encryption and decryption

Insecure communication in digital image security and image storing are considered as important challenges. Moreover, the existing approaches face problems related to improper security at the time of image encryption and decryption. In this research work, a wavelet environment is obtained by transforming the cover image utilizing integer wavelet transform (IWT) and hybrid discrete cosine transform (DCT) to completely prevent false errors. Then the proposed hybrid chaotic map with L-shaped fractal Tromino offers better security to maintain image secrecy by means of encryption and decryption. The proposed work uses fractal encryption with the combination of L-shaped Tromino theorem for enhancement of information hiding. The regions of L-shaped fractal Tromino are sensitive to variations, thus are embedded in the watermark based on a visual watermarking technique known as reversible watermarking. The experimental results showed that the proposed method obtained peak signal-to-noise ratio (PSNR) value of 56.82dB which is comparatively higher than the existing methods that are, Beddington, free, and Lawton (BFL) map with PSNR value of 8.10 dB, permutation substitution, and Boolean operation with PSNR value of 21.19 dB and deoxyribonucleic acid (DNA) level permutation-based logistic map with PSNR value of 21.27 dB

Passivity Enforcement via Perturbation of Hamiltonian Matrices

This paper presents a new technique for the passivity enforcement of linear time-invariant multiport systems in statespace form. This technique is based on a study of the spectral properties of related Hamiltonian matrices. The formulation is applicable in case the system input-output transfer function is in admittance, impedance, hybrid, or scattering form. A standard test for passivity is first performed by checking the existence of imaginary eigenvalues of the associated Hamiltonian matrix. In the presence of imaginary eigenvalues the system is not passive. In such a case, a new result based on first-order perturbation theory is presented for the precise characterization of the frequency bands where passivity violations occur. This characterization is then used for the design of an iterative perturbation scheme of the state matrices, aimed at the displacement of the imaginary eigenvalues of the Hamiltonian matrix. The result is an effective algorithm leading to the compensation of the passivity violations. This procedure is very efficient when the passivity violations are small, so that first-order perturbation is applicable. Several examples illustrate and validate the procedure