Restarted Arnoldi Method Research Articles

Discrete variable method for non-integrable quantum systems

The states of an atom in external electric fields become quasi-bound since the electron can ionize by tunneling through the potential barrier into the continuum. Due to the external electric field the ionization threshold of the atom is lowered from the field-free value. This process becomes important for states close to the classical ionization energy or above. These resonance states can be studied using the complex coordinate method. In this method the Hamiltonian of the system is continued into the complex plane by a complex dilatation, therefore the Hamiltonian is no longer Hermitian and can support complex eigenenergies associated with decaying states. Resonances are uncovered by the rotated continuum spectra with complex eigenvalue and square-integrable (complex rotated) eigenfunctions. The basic idea is to combine this complex coordinate rotation method with the finite element method, and the discrete variable technique. These two methods have been successfully used to compute atomic data for the hydrogen atom in external magnetic and electric fields. We obtain a complex symmetric Hamiltonian matrix, which we solve using the implicitly restarted Arnoldi method (ARPACK). These methods have been extended to alkali atoms in external strong magnetic and electric fields by including model potentials and have also been successfully used in studying various effective one-particle problems.

Using Implicitly Restarted Arnoldi Method for Eigenanalysis of Power Systems

In this paper the implicitly restarted Arnoldi method is applied for the partial eigenanalysis of large power systems. The commonly used complex shift-invert and Cayley transformation are proved to be equivalent for implicitly restarted Arnoldi method under certain conditions. New locking technique is exploited to compute eigenvalue clusters in real large power systems and extensions are made to apply for complex matrix. Comparisons are also made with two other variants of restarted Arnoldi method. The tests show that the implicitly restarted Arnoldi method is fast, robust, and reliable.

Finding waveguide modes via implicit operator projection in the Krylov subspace

This work presents an approach toward solving electromagnetic operator eigenproblems based on implicit operator projection in the Krylov subspace. It is shown that this scheme allows reduction in both memory and computational cost of finding modes in dielectric guides for a proposed iterative solver based on the implicitly restarted Arnoldi method. Still, the generality of the presented method provides the means to use it with a number of algorithms which apply the concept of Krylov subspaces.

The implicit restarted Arnoldi method, an efficient alternative to solve the neutron diffusion equation

To calculate the neutronic steady state of a nuclear power reactor core and its subcritical modes, it is necessary to solve a partial eigenvalue problem. In this paper, an implicit restarted Arnoldi method is presented as an advantageous alternative to classical methods as the Power Iteration method and the Subspace Iteration method. The efficiency of these methods, has been compared calculating the dominant Lambda modes of several configurations of the Three Mile Island reactor core.

Analyzing the spectrum of general, non-hermitian Dirac operators

We discuss the computational problems when analyzing general, non-hermitian matrices and in particular the un-modified Wilson lattice Dirac operator. We report on our experiences with the Implicitly Restarted Arnoldi Method. The eigenstates of the Wilson-Dirac operator which have real eigenvalues and correspond to zero modes in the continuum are analyzed by correlating the size of the eigenvalues with the chirality of the eigenstates.

Combined tank and applicator design of radio frequency heating systems

A finite-element eigenmode characterises the combined tank and applicator cavities of radio frequency heating systems. The aim is to enable a designer to specify the geometrical configurations of the combined systems for operation at the ISM (industrial, scientific or medical) frequency bands without having to build many expensive prototypes. The use of mathematical modelling results in a linear eigenvalue problem which is solved by the implicitly restarted Arnoldi method followed by spectral transformation to enhance convergence to a desired portion of the spectrum. After considering first the tank circuit in isolation, the combined tank and applicator circuits are linked and a similar eigenmode analysis shows its effectiveness in examining the various geometrical configurations in terms of tuning and the establishment of other resonances. This is the first step in tackling the problem that such industrial systems may pose as far as electromagnetic compatibility is concerned.

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.

Dynamic Thick Restarting of the Davidson, and the Implicitly Restarted Arnoldi Methods

The Davidson method is a popular preconditioned variant of the Arnoldi method for solving large eigenvalue problems. For theoretical as well as practical reasons the two methods are often used with restarting. Frequently, information is saved through approximated eigenvectors to compensate for the convergence impairment caused by restarting. We call this scheme of retaining more eigenvectors than needed and prove that thick restarted, nonpreconditioned Davidson is equivalent to the implicitly restarted Arnoldi. We also establish a relation between thick restarted Davidson and a Davidson method applied on a deflated system. The theory is used to address the question of which and how many eigenvectors to retain and motivates the development of a dynamic thick restarting scheme for the symmetric case, which can be used in both Davidson and implicit restarted Arnoldi. Several experiments demonstrate the efficiency and robustness of the scheme.

Nested Lanczos: implicitly restarting an unsymmetric Lanczos algorithm

In this text, we present a generalization of the idea of the Implicitly Restarted Arnoldi method to the unsymmetric Lanczos algorithm, using the two-sided Gram-Schmidt process or using a full Lanczos tridiagonalization. The resulting implicitly restarted Lanczos method is called Nested Lanczos. Nested Lanczos can be combined with an implicit filter. It can also be used in case of breakdown and offers an alternative for look-ahead.

The implicit application of a rational filter in the RKS method

The implicitly restarted Arnoldi method implicitly applies a polynomial filter to the Arnoldi vectors by use of orthogonal transformations. In this paper, an implicit filtering by rational functions is proposed for the rational Krylov method. This filtering is performed in an efficient way. Two applications are considered. The first one is the filtering of unwanted eigenvalues using exact shifts. This approach is related to the use of exact shifts in the implicitly restarted Arnoldi method. Second, eigenvalue problems can have an infinite eigenvalue without physical relevance. This infinite eigenvalue can corrupt the eigensolution. An implicit filtering is proposed for avoiding such corruptions.

The Restarted Arnoldi Method Applied to Iterative Linear System Solvers for the Computation of Rightmost Eigenvalues

For the computation of a few eigenvalues of $Ax=\mu Bx$, the restarted Arnoldi method is often applied to transformations, e.g., the shift-invert transformation. Such transformations typically require the solution of linear systems. This paper presents an analysis of the application of the transformation $(M_A-\alpha M_B)^{-1}(A-\lambda B)$ to Arnoldi's method where $\alpha$ and $\lambda$ are parameters and $M_A-\alpha M_B$ is some approximation to $A-\alpha B$. In fact, $(M_A-\alpha M_B)^{-1}$ corresponds to an iterative linear system solver for the system $(A-\alpha B)x=b$. The transformation is an alternative to the shift-invert transformation $(A-\alpha B)^{-1}B$ when direct system solvers are not available or not feasible. The restarted Arnoldi method is analyzed in the case of detection of the rightmost eigenvalues of real nonsymmetric matrices. The method is compared to Davidson's method by use of numerical examples.

Implicitly restarted Arnoldi with purification for the shift-invert transformation

The need to determine a few eigenvalues of a large sparse generalised eigenvalue problem A x = λ B x Ax=\lambda Bx with positive semidefinite B B arises in many physical situations, for example, in a stability analysis of the discretised Navier-Stokes equation. A common technique is to apply Arnoldi’s method to the shift-invert transformation, but this can suffer from numerical instabilities as is illustrated by a numerical example. In this paper, a new method that avoids instabilities is presented which is based on applying the implicitly restarted Arnoldi method with the B B semi-inner product and a purification step. The paper contains a rounding error analysis and ends with brief comments on some extensions.

Micrometer U-tube manometers for medium vacuum measurements: A M Thomas and J L Cross,J vac sci technol, 4 (1) Jan–Feb 1967, 1–5

The states of an atom in external electric fields become quasi-bound since the electron can ionize by tunneling through the potential barrier into the continuum. Due to the external electric field the ionization threshold of the atom is lowered from the field-free value. This process becomes important for states close to the classical ionization energy or above. These resonance states can be studied using the complex coordinate method. In this method the Hamiltonian of the system is continued into the complex plane by a complex dilatation, therefore the Hamiltonian is no longer Hermitian and can support complex eigenenergies associated with decaying states. Resonances are uncovered by the rotated continuum spectra with complex eigenvalue and square-integrable (complex rotated) eigenfunctions. The basic idea is to combine this complex coordinate rotation method with the finite element method, and the discrete variable technique. These two methods have been successfully used to compute atomic data for the hydrogen atom in external magnetic and electric fields. We obtain a complex symmetric Hamiltonian matrix, which we solve using the implicitly restarted Arnoldi method (ARPACK). These methods have been extended to alkali atoms in external strong magnetic and electric fields by including model potentials and have also been successfully used in studying various effective one-particle problems.