Interior Eigenvalues Research Articles

A modification of the factorization method for the classical acoustic inverse scattering problems

It is well-known that sampling type methods for solving inverse scattering problems fail if the wave number is an eigenvalue of a corresponding interior eigenvalue problem. By adding the far field patterns corresponding to an artificial ball lying within the obstacle and imposing an impedance boundary condition on the boundary of this ball we propose a modification of the factorization method which provides the characterization of the unknown obstacle for all wave numbers. Some numerical experiments are presented to demonstrate the feasibility and effectiveness of our method.

A parallel implementation of Davidson methods for large-scale eigenvalue problems in SLEPc

In the context of large-scale eigenvalue problems, methods of Davidson type such as Jacobi-Davidson can be competitive with respect to other types of algorithms, especially in some particularly difficult situations such as computing interior eigenvalues or when matrix factorization is prohibitive or highly inefficient. However, these types of methods are not generally available in the form of high-quality parallel implementations, especially for the case of non-Hermitian eigenproblems. We present our implementation of various Davidson-type methods in SLEPc, the Scalable Library for Eigenvalue Problem Computations. The solvers incorporate many algorithmic variants for subspace expansion and extraction, and cover a wide range of eigenproblems including standard and generalized, Hermitian and non-Hermitian, with either real or complex arithmetic. We provide performance results on a large battery of test problems.

Interior Eigenvalues from Density Matrix Expansions in Quantum Mechanical Molecular Dynamics

An accelerated polynomial expansion scheme to construct the density matrix in quantum mechanical molecular dynamics simulations is proposed. The scheme is based on recursive density matrix expansions, e.g., [A. M. N. Niklasson, Phys. Rev. B, 66 (2002), 155115], which are accelerated by a scale-and-fold technique [E. H. Rubensson, J. Chem. Theory Comput., 7 (2011), pp. 1233--1236]. The acceleration scheme requires interior eigenvalue estimates, which may be expensive and cumbersome to come by. Here we show how such eigenvalue estimates can be extracted from the recursive expansion by a simple and robust procedure at a negligible computational cost. Our method is illustrated with density functional tight-binding Born--Oppenheimer molecular dynamics simulations, where the computational effort is dominated by the density matrix construction. In our analysis we identify two different phases of the recursive polynomial expansion, the conditioning and purification phases, and we show that the acceleration repres...

An interior eigenvalue problem from electronic structure calculations

We consider the generalized eigenvalue problem Ax = λBx, where A and B are real symmetric matrices and B is also positive definite. All the eigenvalues of this problem are real, and it is often necessary to compute only a few eigenvalues which are important for applications. In electronic structure calculations of materials, specific interior eigenvalues are of fundamental interest, since they play crucial roles in various industrial applications. In this paper, we propose an approach based on the inertia of the linear matrix pencil of A and B. The eigenvalue problem is restated, and a class of algorithms is presented for separating the target eigenvalues from the others.

The inside–outside duality for scattering problems by inhomogeneous media

This paper investigates the relationship between interior transmission eigenvalues k0 > 0 and the accumulation point 1 of the eigenvalues of the scattering operator when k approaches k0. As is well known, the spectrum of is discrete, the eigenvalues μn(k) lie on the unit circle in and converge to 1 from one side depending on the sign of the contrast. Under certain (implicit) conditions on the contrast it is shown that interior transmission eigenvalues k0 can be characterized by the fact that one eigenvalue of converges to 1 from the opposite side if k tends to k0 from below. The proof uses the Cayley transform, Courant’s maximum–minimum principle, and the factorization of the far field operator. For constant contrasts that are positive and large enough or negative and small enough, we show that the conditions necessary to prove this characterization are satisfied at least for the smallest transmission eigenvalue.

A multi-level method for transmission eigenvalues of anisotropic media

In this paper, we propose a multi-level finite element method for the transmission eigenvalue problem of anisotropic media. The problem is non-standard and non-self-adjoint with important applications in inverse scattering theory. We employ a suitable finite element method to discretize the problem. The resulting generalized matrix eigenvalue problem is large, sparse and non-Hermitian. To compute the smallest real transmission eigenvalue, which is usually an interior eigenvalue, we devise a multi-level method using Arnoldi iteration. At the coarsest mesh, the eigenvalue is obtained using Arnoldi iteration with an adaptive searching technique. This value is used as the initial guess for Arnoldi iteration at the next mesh level. This procedure is then repeated until the finest mesh level. Numerical examples are presented to show the viability of the method.

A refined variant of the inverse-free Krylov subspace method for symmetric generalized eigenvalue problems

A refined variant of the inverse-free Krylov subspace method is proposed in this paper. The new method retains the original Ritz value, and replaces the Ritz vector by a refined Ritz vector in each cycle of the iteration. Each refined Ritz vector is chosen in such a way that the norm of the residual vector formed with the Ritz value is minimized over the subspace involved, and it can be computed cheaply by solving a small sized SVD problem. The refined variant can overcome the irregular convergence behavior of the Ritz vectors which may happen in the inverse-free Krylov subspace method. An a priori error estimate for the refined Ritz vector is given, which shows that the refined Ritz vector converges once the deviation of the eigenvector from the trial Krylov subspace converges to zero. By using spectral transformation, this new method can be applied to compute an interior eigenvalue pair. Numerical experiments are given to show the efficiency of the new methods.

On optimizing Jacobi–Davidson method for calculating eigenvalues in low dimensional structures using eight band k · p model

The paper presents two ways of improving the Jacobi–Davidson method for calculating the eigenvalues and eigenvectors described by eight-band k·p model for quantum dots and other low dimensional structures. First, the method is extended by the application of time reversal symmetry operator. This extension allows efficient calculations of the twofold degeneracy present in the multiband k·p model and other interior eigenvalues. Second, the preconditioner for the indefinite matrix which comes from the discretization of the eight band k·p Hamiltonian is presented. The construction of this preconditioner is based on physical considerations about energy band structure in the k·p model. On the basis of two real examples, it is shown that the preconditioner can significantly shorten the time needed to calculate the interior eigenvalues, despite the fact that the memory usage of the preconditioner and Hamiltionian is comparable. Finally, some technical details for implementing the eight band k·p Hamiltonian and the eigensolver are provided.

Computing Extremal Eigenvalues for Three-Dimensional Photonic Crystals with Wave Vectors Near the Brillouin Zone Center

The band structures of three-dimensional photonic crystals can be determined numerically by solving a sequence of generalized eigenvalue problems. However, not all of the spectral structures of these eigenvalue problems are well-understood, and not all of these eigenvalue problems can be solved efficiently. This article focuses on the eigenvalue problems corresponding to wave vectors that are close to the center of the Brillouin zone of a three dimensional, simple cubic photonic crystal. For these eigenvalue problems, there are (i) many zero eigenvalues, (ii) a couple of near-zero eigenvalues, and (iii) several larger eigenvalues. As the desired eigenvalues are the smallest positive eigenvalues, these particular spectral structures prevent regular eigenvalue solvers from efficiently computing the desired eigenvalues. We study these eigenvalue problems from the perspective of both theory and computation. On the theoretical side, the structures of the null spaces are analyzed to explicitly determine the number of zero eigenvalues of the target eigenvalue problems. On the computational side, the Krylov-Schur and Jacobi-Davidson methods are used to compute the smallest, positive, interior eigenvalues that are of interest. Intensive numerical experiments disclose how the shift values, conditioning numbers, and initial vectors affect the performance of the tested eigenvalue solvers and suggest the most efficient eigenvalue solvers.

Interior transmission eigenvalue problem with refractive index having C2-transition to the background medium

The interior transmission eigenvalue problem for scalar acoustics is studied for a new class of refractive index. Existence of an infinite discrete set of transmission eigenvalues in the case that the acoustic properties of a domain D ⊂ ℝ n are allowed to have a C 2-transition to the homogeneous background medium is established. It is shown that the transmission problem has a weak formulation on certain weighted Sobolev spaces for this class of refractive index. The weak formulation and the discreteness of the spectrum is justified by using the Hardy inequality to prove compact imbedding theorems. Existence of transmission eigenvalues is demonstrated by investigating a generalized eigenvalue problem associated with the weak formulation.

Performance comparison of parallel eigensolvers based on a contour integral method and a Lanczos method

Abstract We study the performance of a parallel nonlinear eigensolver SSEig which is based on a contour integral method. We focus on symmetric generalized eigenvalue problems (GEPs) of computing interior eigenvalues. We chose to focus on GEPs because we can then compare the performance of SSEig with that of a publicly-available software package TRLan, which is based on a thick restart Lanczos method. To solve this type of problems, SSEig requires the solution of independent linear systems with different shifts, while TRLan solves a sequence of linear systems with a single shift. Therefore, while SSEig typically has a computational cost greater than that of TRLan, it also has greater parallel scalability. To compare the performance of these two solvers, in this paper, we develop performance models and present numerical results of solving large-scale eigenvalue problems arising from simulations of modeling accelerator cavities. In particular, we identify the crossover point, where SSEig becomes faster than TRLan. The parallel performance of SSEig solving nonlinear eigenvalue problems is also studied.

Interior transmission eigenvalue problem for Maxwell's equations: the T-coercivity as an alternative approach

In this paper, we examine the interior transmission problem for Maxwell’s equations in the case where both ε and μ, the physical parameters of the scattering medium, differ from ε0 and μ0 modelling the background medium. Using the T-coercivity method, we propose an alternative approach to the classical techniques to prove that this problem is of Fredholm type and that the so-called transmission eigenvalues form at most a discrete set. The T-coercivity approach allows us to deal with cases where ε − ε0 and μ − μ0 can change sign. We also provide results of localization and Faber–Krahn-type inequalities for the transmission eigenvalues.

Fast iterative interior eigensolver for millions of atoms

We show that a combination of the Generalized Davidson method and harmonic Ritz values (called harmonic Davidson) is well-suited for solving large interior eigenvalue problems using a plane wave basis. The algorithm enables us to calculate impurity and band edge states for systems of 100,000 atoms in about one day on 32 cores. We demonstrate the capabilities of the method by calculating the electronic states of a large GaAs quantum dot embedded in an AlAs matrix.

Computing All or Some Eigenvalues of Symmetric $\mathcal{H}_{\ell}$-Matrices

We use a bisection method [B. Parlett, The Symmetric Eigenvalue Problem, Prentice-Hall, Englewood Cliffs, NJ, 1980, p. 51] to compute the eigenvalues of a symmetric $\mathcal{H}_{\ell}$-matrix M. The number of negative eigenvalues of $M-\mu I$ is computed via the LDL$^T$ factorization of $M-\mu I$. For dense matrices, the LDL$^T$ factorization is too expensive to yield an efficient eigenvalue algorithm in general, but not so for $\mathcal{H}_{\ell}$-matrices. In the special structure of $\mathcal{H}_{\ell}$-matrices there is an LDL$^T$ factorization with linear-polylogarithmic complexity. The bisection method requires only matrix-size independent many iterations to find an eigenvalue up to the desired accuracy, so that an eigenvalue can be found in linear-polylogarithmic time. For all n eigenvalues, $\mathcal{O}(n^{2}(\log n )^{4}\log( \Vert {M}\Vert_{2}/\epsilon_{\text{ev}}))$ flops are needed to compute all eigenvalues with an accuracy $\epsilon_{\text{ev}}$. It is also possible to compute only eigenvalues in a specific interval or the jth smallest one. Numerical experiments demonstrate the efficiency of the algorithm, in particular for the case when some interior eigenvalues are required. (A corrected PDF is attached to this article.)

A Filtered Lanczos Procedure for Extreme and Interior Eigenvalue Problems

When combined with Krylov projection methods, polynomial filtering can provide a powerful method for extracting extreme or interior eigenvalues of large sparse matrices. This general approach can be quite efficient in the situation when a large number of eigenvalues is sought. However, its competitiveness depends critically on a good implementation. This paper presents a technique based on such a combination to compute a group of extreme or interior eigenvalues of a real symmetric (or complex Hermitian) matrix. The technique harnesses the effectiveness of the Lanczos algorithm with partial reorthogonalization and the power of polynomial filtering. Numerical experiments indicate that the method can be far superior to competing algorithms when a large number of eigenvalues and eigenvectors is to be computed.

Two‐sided harmonic subspace extractions for the generalized eigenvalue problem

AbstractOne crucial step of the solution of large‐scale generalized eigenvalue problems with iterative subspace methods, e.g. Arnoldi, Jacobi‐Davidson, is a projection of the original large‐scale problem onto a low dimensional subspaces. Here we investigate two‐sided methods, where approximate eigenvalues together with their right and left eigenvectors of the full‐size problem are extracted from the resulting small eigenproblem. The two‐sided Ritz‐Galerkin projection can be seen as the most basic form of this approach. It usually provides a good convergence towards the extremal eigenvalues of the spectrum. For improving the convergence towards interior eigenvalues, we investigate two approaches based on harmonic subspace extractions for the generalized eigenvalue problem. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Computing subdominant unstable modes of turbulent plasma with a parallel Jacobi–Davidson eigensolver

SUMMARYIn the numerical solution of large‐scale eigenvalue problems, Davidson‐type methods are an increasingly popular alternative to Krylov eigensolvers. The main motivation is to avoid the expensive factorizations that are often needed by Krylov solvers when the problem is generalized or interior eigenvalues are desired. In Davidson‐type methods, the factorization is replaced by iterative linear solvers that can be accelerated by a smart preconditioner. Jacobi–Davidson is one of the most effective variants. However, parallel implementations of this method are not widely available, particularly for non‐symmetric problems. We present a parallel implementation that has been included in SLEPc, the Scalable Library for Eigenvalue Problem Computations, and test it in the context of a highly scalable plasma turbulence simulation code. We analyze its parallel efficiency and compare it with a Krylov–Schur eigensolver. Copyright © 2011 John Wiley & Sons, Ltd.

Approximate Quantum and Acoustic Cloaking

For any E\ge 0 , we construct a sequence of bounded potentials V^E_{n},\, n\in\Bbb N , supported in {an annular region B_{out}\setminus B_{inn}\subset\Bbb R ^3 ,} which act as approximate cloaks for solutions of Schrödinger's equation at energy E : For any potential V_0\in L^\infty(B_{inn}) {such that E is not a Neumann eigenvalue of -\Delta+V_0 in B_{inn} }, the scattering amplitudes a_{V_0+V_n^E}(E,\theta,\omega)\to 0 as n\to\infty . The V^E_{ n} thus not only form a family of approximately transparent potentials, but also function as approximate invisibility cloaks in quantum mechanics. {On the other hand, for E close to interior eigenvalues, resonances develop and there exist almost trapped states concentrated in B_{inn} .} We derive the V_n^E from singular, anisotropic transformation optics-based cloaks by a de-anisotropization procedure, which we call \emph{isotropic transformation optics}. This technique uses truncation, inverse homogenization and spectral theory to produce nonsingular, isotropic approximate cloaks. As an intermediate step, we also obtain approximate cloaking for a general class of equations including the acoustic equation.

Valence Excited States in Large Molecules via Local Multireference Singles and Doubles Configuration Interaction.

We demonstrate that valence excited states in large molecules can be treated using local multireference singles and doubles configuration interaction (LMRSDCI). The interior eigenvalues corresponding to the excited states of interest are transformed and shifted to the extrema of the spectrum by way of oblique projections and a matrix shift within a modified Davidson diagonalization scheme. In this way, the approximate wave function associated with the excited state of interest can be isolated independently of the lower lying roots, and residual minimization is used for final convergence to the target eigenstate. We find that vertical excitation energies calculated using LMRSDCI are mostly within 0.2 eV of nonlocal MRSDCI values.

On the multi-frequency obstacle reconstruction via the linear sampling method

This paper investigates the possibility of multi-frequency reconstruction of sound-soft and penetrable obstacles via the linear sampling method involving either far-field or near-field observations of the scattered field. On establishing a suitable approximate solution to the linear sampling equation and making an assumption of continuous frequency sweep, two possible choices for a cumulative multi-frequency indicator function of the scatterer's support are proposed. The first alternative, termed the ‘serial’ indicator, is taken as a natural extension of its monochromatic companion in the sense that its computation entails space-frequency (as opposed to space) L2-norm of a solution to the linear sampling equation. Under a set of assumptions that include experimental observations down to zero frequency and compact frequency support of the wavelet used to illuminate the obstacle, this indicator function is further related to its time-domain counterpart. As a second possibility, the so-called parallel indicator is alternatively proposed as an L2-norm, in the frequency domain, of the monochromatic indicator function. On the basis of a perturbation analysis which demonstrates that the monochromatic solution of the linear sampling equation behaves as O(|k2 − k2∗|−m), m ⩾ 1 in the neighborhood of an isolated eigenvalue, k2∗, of the associated interior (Dirichlet or transmission) problem, it is found that the ‘serial’ indicator is unable to distinguish the interior from the exterior of a scatterer in situations when the prescribed frequency band traverses at least one such eigenvalue. In contrast the ‘parallel’ indicator is, due to its particular structure, shown to be insensitive to the presence of pertinent interior eigenvalues (unknown beforehand and typically belonging to a countable set), and thus to be robust in a generic scattering configuration. A set of numerical results, including both ‘fine’ and ‘coarse’ frequency sampling, is included to illustrate the performance of the competing (multi-frequency) indicator functions, demonstrating behavior that is consistent with the theoretical results.