Underlying Eigenvalue Problem Research Articles

On refinements to QMFD based chirp parameter estimation

Commuting matrix methods furnish a full basis of orthogonal eigenvectors for the discrete Fourier transform or its centered version needed for computing the discrete fractional Fourier transform and multicomponent chirp signal analysis. However, these approaches suffer from ill-conditioning issues at higher matrix sizes, and require a computationally expensive eigenvalue decomposition.In this paper, ill-conditioning issues associated with the QMFD approach developed previously by the authors are addressed via diagonal modification. Further symmetries of the eigenvectors are used to reduce the size of the underlying eigenvalue problem. These modifications are then incorporated into the real-arithmetic implementation of the QMFD approach that is shown to be significantly superior to the conventional implementation and the corresponding MSE of the chirp parameter estimates are shown to approach their Cramer–Rao lower bounds.

The hot spots conjecture can be false: some numerical examples

The hot spots conjecture is only known to be true for special geometries. This paper shows numerically that the hot spots conjecture can fail to be true for easy to construct bounded domains with one hole. The underlying eigenvalue problem for the Laplace equation with Neumann boundary condition is solved with boundary integral equations yielding a non-linear eigenvalue problem. Its discretization via the boundary element collocation method in combination with the algorithm by Beyn yields highly accurate results both for the first non-zero eigenvalue and its corresponding eigenfunction which is due to superconvergence. Additionally, it can be shown numerically that the ratio between the maximal/minimal value inside the domain and its maximal/minimal value on the boundary can be larger than 1 + 10− 3. Finally, numerical examples for easy to construct domains with up to five holes are provided which fail the hot spots conjecture as well.

Thermal instability in a liquid layer with permeable boundaries under the influence of variable gravity

In this paper, we study the thermal instability of a horizontal layer of liquid heated from below and confined between thermally conducting permeable boundaries under the action of a variable gravitational field across the layer. It is established that the principle of the exchange of stabilities holds good in the more general framework of hydrodynamical boundary conditions when the parameter representing the gravity variation increasing vertically upward is less than equal to unity. The underlying eigenvalue problem is solved using the Chandrasekhar’s technique which is then computed numerically. The numerical results for a wide range of the parameters are presented and discussed. It is found that the gravity variation increasing vertically upward has a destabilizing effect and that the permeability of a boundary has a stabilizing effect on the onset of convection. Influence of the gravity variation increasing vertically upward for various particular cases of the boundary conditions is obtained as limiting cases of the permeability parameters of the boundaries and illustrated graphically.

Transparent boundary conditions for discretized non-Hermitian eigenvalue problems

A method is presented for transparent, energy-dependent boundary conditions for open, non-Hermitian systems, and is illustrated on an example of Stark resonances in a single-particle quantum system. The approach provides an alternative to external complex scaling, and is applicable when asymptotic solutions can be characterized at large distances from the origin. Its main benefit consists in a drastic reduction of the dimesnionality of the underlying eigenvalue problem. Besides application to quantum mechanics, the method can be used in other contexts such as in systems involving unstable optical cavities and lossy waveguides.

Computing eigenfunctions and eigenvalues of boundary-value problems with the orthogonal spectral renormalization method

The spectral renormalization method was introduced in 2005 as an effective way to compute ground states of nonlinear Schr\"odinger and Gross-Pitaevskii type equations. In this paper, we introduce an orthogonal spectral renormalization (OSR) method to compute ground and excited states (and their respective eigenvalues) of linear and nonlinear eigenvalue problems. The implementation of the algorithm follows four simple steps: (i) reformulate the underlying eigenvalue problem as a fixed point equation, (ii) introduce a renormalization factor that controls the convergence properties of the iteration, (iii) perform a Gram-Schmidt orthogonalization process in order to prevent the iteration from converging to an unwanted mode; and (iv) compute the solution sought using a fixed-point iteration. The advantages of the OSR scheme over other known methods (such as Newton's and self-consistency) are: (i) it allows the flexibility to choose large varieties of initial guesses without diverging, (ii) easy to implement especially at higher dimensions and (iii) it can easily handle problems with complex and random potentials. The OSR method is implemented on benchmark Hermitian linear and nonlinear eigenvalue problems as well as linear and nonlinear non-Hermitian $\mathcal{PT}$-symmetric models.

Dispersion analysis of composite acousto-elastic waveguides

Abstract Propagation characteristics of guided waves are widely utilized for nondestructive characterization of laminated composites which are often immersed in fluid. These propagation characteristics are quantified through dispersion and attenuation curves, and need to be computed for a large number of estimated structure and material property combinations, over a wide range of frequencies. To solve this central problem, an efficient approach is proposed for dispersion analysis of three types of immersed waveguides: laminated plates; laminated rods and pipes; composite waveguides with generic cross-section. The approach is based on Semi-Analytical Finite Element (SAFE) method enhanced with two novel discretization methods: Complex-length Finite Element Method (CFEM) for the solid domain and Perfectly Matched Discrete Layers (PMDL) for the surrounding fluid. The resulting approach is computationally more efficient than the existing methods in that it keeps the underlying eigenvalue problem linear and substantially smaller. The paper presents the basic ideas of the proposed approach, specific algorithms for determining the discretization parameters, and open-source implementation of the resulting waveguide models. Several numerical examples are presented to illustrate the method's efficiency. Finally, the theoretical predictions from the method are validated using experimental observations for several structural members.

Analyzing modal behavior of guided waves using high order eigenvalue derivatives

This paper presents a mode-tracing approach for elastic guided waves based on analytically computed derivatives and includes a study of interesting phenomena in the dispersion curve representation. Numerical simulation is done by means of the Scaled Boundary Finite Element Method (SBFEM). Two approaches are used to identify the characteristics of the resulting wave modes: Taylor approximation and Padé approximation. Higher order differentials of the underlying eigenvalue problem are the basis for these approaches. Remarkable phenomena in potentially critical frequency regions are identified and the tracing approach is adapted to these regions. Additionally, a stabilization of the solution process is suggested.

A Finite Element Method for a Curlcurl-Graddiv Eigenvalue Interface Problem

In this paper we propose and study a finite element method for a curlcurl-graddiv eigenvalue interface problem. Its solution may be of piecewise non-$H^1$. We would like to approximate such a solution in an $H^1$-conforming finite element space. With the discretizations of both curl and div operators of the underlying eigenvalue problem in two finite element spaces, the proposed method is essentially a standard $H^1$-conforming element method, up to element bubbles which can be statically eliminated at element levels. We first analyze the proposed method for the related source interface problem by establishing the stability and the error bounds. We then analyze the underlying eigenvalue interface problem, and we obtain the error bounds $\mathcal{O}(h^{2r_0})$ for eigenvalues which correspond to eigenfunctions in $\prod_{j=1}^J (H^r(\Omega_j))^3\hookrightarrow (H^{r_0}(\Omega))^3$ space, where the piecewise regularity $r$ and the global regularity $r_0$ may belong to the most interesting interval $[0,1]$.

A novel approach to mode‐tracing in SBFEM‐simulations: Higher‐order Taylor‐ and Padé approximation

AbstractThis contribution presents the study of strange phenomena in wave mode representations of waveguides. For this study the waveguides are computed by means of the Scaled Boundary Finite Element Method (SBFEM). Different approaches of mode tracing are used to identify the characteristics of the resulting wave modes. Higher order differentials of the underlying eigenvalue problem are the basis for these approaches.The main idea behind this mode tracing approach is to reduce the cubic computation time to solve the eigenvalue problem for each frequency of interest. This study identifies potentially critical frequency regions and attempts to formulate a solution process. The fascinating effects at critical frequencies are displayed and a suggestion for a stabilization for the solution process is made. This study bases its conclusion on a numerical viewpoint. Main aspects in this study include high order differentials of the eigenvalue problem and the corresponding Taylor and Padé approximations for the eigenvalue problem as a whole. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

An Efficient Calculation of Photonic Crystal Band Structures Using Taylor Expansions

AbstractIn this paper we present an efficient algorithm for the calculation of photonic crystal band structures and band structures of photonic crystal waveguides. Our method relies on the fact that the dispersion curves of the band structure are smooth functions of the quasi-momentum in the one-dimensional Brillouin zone. We show the derivation and computation of the group velocity, the group velocity dispersion, and any higher derivative of the dispersion curves. These derivatives are then employed in a Taylor expansion of the dispersion curves. We control the error of the Taylor expansion with the help of a residual estimate and introduce an adaptive scheme for the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. These derivatives are then employed in a Taylor expansion of the dispersion curves. We control the error of the Taylor expansion with the help of a residual estimate and introduce an adaptive scheme for the selection of nodes in the one-dimensional Brillouin zone at which we solve the underlying eigenvalue problem and compute the derivatives of the dispersion curves. The proposed algorithm is not only advantageous as it decreases the computational effort to compute the band structure but also because it allows for the identification of crossings and anti-crossings of dispersion curves, respectively. This identification is not possible with the standard approach of solving the underlying eigenvalue problem at a discrete set of values of the quasi-momentum without taking the mode parity into account.

Some recent results on MDGKN‐systems

The linearized equations of motion of finite dimensional autonomous mechanical systems are normally written as a second order system and are of the MDGKN type, where the different n × n matrices have certain characteristic properties. These matrix properties have consequences for the underlying eigenvalue problem. Engineers have developed a good intuitive understanding of such systems, particularly for systems without gyroscopic terms (G‐matrix) and circulatory terms (N‐matrix, which may lead to self‐excited vibrations). A number of important engineering problems in the linearized form are described by this type of equations. It has been known for a long time, that damping (D‐matrix) in such systems may either stabilize or destabilize the system depending on the structure of the matrices. Here we present some new results (using a variety of methods of proof) on the influence of the damping terms, which are quite general. Starting from a number of conjectures, they were jointly developed by the authors during recent months.

Excitation spectra of many-body systems by linear response: General theory and applications to trapped condensates

We derive a general linear-response many-body theory capable of computing excitation spectra of trapped interacting bosonic systems, e.g., depleted and fragmented Bose-Einstein condensates (BECs). To obtain the linear-response equations we linearize the multiconfigurational time-dependent Hartree for bosons (MCTDHB) method, which provides a self-consistent description of many-boson systems in terms of orbitals and a state vector (configurations), and is in principle numerically-exact. The derived linear-response many-body theory, which we term LR-MCTDHB, is applicable to systems with interaction potentials of general form. From the numerical implementation of the LR-MCTDHB equations and solution of the underlying eigenvalue problem, we obtain excitations beyond available theories of excitation spectra, such as the Bogoliubov-de Gennes (BdG) equations. The derived theory is first applied to study BECs in a one-dimensional harmonic potential. The LR-MCTDHB method contains the BdG excitations and, also, predicts a plethora of additional many-body excitations which are out of the realm of standard linear response. In particular, our theory describes the exact energy of the higher harmonic of the first (dipole) excitation not contained in the BdG theory. We next study a BEC in a very shallow one-dimensional double-well potential. We find with LR-MCTDHB low-lying excitations which are not accounted for by BdG, even though the BEC has only little fragmentation and, hence, the BdG theory is expected to be valid. The convergence of the LR-MCTDHB theory is assessed by systematically comparing the excitation spectra computed at several different levels of theory.

Operator-splitting methods respecting eigenvalue problems for shallow shelf equations with basal drag

We present different numerical methods for solving the shallow shelf equations with basal drag (SSAB). An alternative approach of splitting the SSAB equation into a Laplacian and diagonal shift operator is discussed with respect to the underlying eigenvalue problem. First, we solve the equations using standard methods. Then, the coupled equations are decomposed into operators for membranes stresses, basal shear stress and driving stress. Applying reasonable parameter values, we demonstrate that the operator of the membrane stresses is much stiffer than the operator of the basal shear stress. Here, we could apply a new splitting method, which alternates between the iteration on the membrane-stress operator and the basal-shear operator, with a more frequent iteration on the operator of the membrane stresses. We show that this splitting accelerates and stabilize the computational performance of the numerical method, although an appropriate choice of the standard method used to solve for all operators in one step speeds up the scheme as well.

Convergence analysis for hyperbolic evolution problems in mixed form

SUMMARYIn this paper, we present a convergence analysis for the space discretization of hyperbolic evolution problems in mixed form. The results of previous work on parabolic equations are extended to this situation, showing the relationships between the approximation of the underlying eigenvalue problem and the space discretization of the evolution problem. The theory is applied to the finite element approximation of the wave equation in mixed form and to the Maxwell equations. Some numerical results confirm the theory and make clear how a scheme that does not provide a spectrally correct discretization can perform badly when applied to the approximation of the evolution problem. Copyright © 2012 John Wiley & Sons, Ltd.

On Ideal Stability of Cylindrical Localized Interchange Modes

AbstractStability of cylindrical localized ideal pressure‐driven interchange plasma modes is revisited. Converting the underlying eigenvalue problem into the form of the Schrödinger equation gives a new simple way of deriving the Suydam stability criterion and calculating the growth rates of unstable modes. Near the marginal stability limit the growth rate is exponentially small and the mode has a double‐peak structure. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the stability of N-solitons in integrable systems

The dynamical stability of reflectionless N-solitons for a large class of integrable systems is considered. The underlying eigenvalue problem is the Zakharov–Shabat problem on for any r ≥ 1. Physical examples of interest include the vector nonlinear Schrödinger equation and the integrable (r + 1)-wave interaction problem. It is shown herein that under appropriate assumptions that these solitons are realized as a local minimum of an appropriate Lyapunov function, and are hence dynamically stable.

Computing asymptotic p-values for EDF tests

In this paper the problem of computingp-values for the asymptotic distribution of certain goodness-of-fit test statistics based on the empirical distribution is approached via quadrature. Through examples it is shown that this approach can lead to considerable time savings over the standard practice of discretizing the underlying eigenvalue problem.

Asymptotic stability of a catalyst particle

The catalyst asymptotic stability problem is studied by means of several new methods that allow accurate solutions to be calculated where other methods have given qualitatively erroneous results. The underlying eigenvalue problem is considered in three limiting situations Le = ∞, 1 and 0. These are solved first to give expansion functions for the full eigenproblem at an arbitrary value of Le. A modified Galerkin method based on the two sets of eigenfunctions for the Le = 1 problem even in the lowest order approximation able to solve the full problem for all Le. Perturbation methods developed from the solutions at the limiting Le-values may if properly handled be extremely useful, e.g. for calculating the stability limit. Finally approximate values of the higher eigenvalues are found by simple formulas based on the steady state solution. Thus the whole eigenvalue spectrum can be studied in detail and certain properties of the spectrum that should appear also with other rate expressions are described.