Computation Of Eigenvalues Research Articles

On perturbations for spectrum and singular value decompositions followed by deflation techniques

The calculation of the dominant eigenvalues of a symmetric matrix A together with its eigenvectors, followed by the calculation of the deflation of A1=A−ρUkUkT corresponds to one step of the Wielandt deflation technique, where ρ is a shift and Uk are eigenvectors of A. In this paper, we investigate how the eigenspace of A1 changes when A1 is perturbed to A˜1=A−ρU˜kU˜kT, where U˜k are approximate eigenvectors of A. We establish the bounds for the angle of eigenspaces of A1 and A˜1 based on the Davis-Kahan theorem. Moreover, in the practical implementation for singular value decomposition, once one or several singular triplets converge to a preset accuracy, they should be deflated by B1=B−γWkVkH with γ being a shift, Wk and Vk are singular vectors of B, so that they will not be re-computed. We investigate how the singular subspaces of B1=B−γWkVkH change when B1 is perturbed to B˜1=B−γW˜kV˜kH, W˜k and V˜k are approximate singular vectors of B. We also establish the bounds for the angle of singular subspaces of B1 and B˜1 based on the Wedin theorem.

Parallel finite-element codes for the Bogoliubov-de Gennes stability analysis of Bose-Einstein condensates

We present and distribute a parallel finite-element toolbox written in the free software FreeFEM for computing the Bogoliubov-de Gennes (BdG) spectrum of stationary solutions to one- and two-component Gross-Pitaevskii (GP) equations, in two or three spatial dimensions. The parallelization of the toolbox relies exclusively upon the recent interfacing of FreeFEM with the PETSc library. The latter contains itself a wide palette of state-of-the-art linear algebra libraries, graph partitioners, mesh generation and domain decomposition tools, as well as a suite of eigenvalue solvers that are embodied in the SLEPc library. Within the present toolbox, stationary states of the GP equations are computed by a Newton method. Branches of solutions are constructed using an adaptive step-size continuation algorithm. The combination of mesh adaptivity tools from FreeFEM with the parallelization features from PETSc makes the toolbox efficient and reliable for the computation of stationary states. Their BdG spectrum is computed using the SLEPc eigenvalue solver. We perform extensive tests and validate our programs by comparing the toolbox's results with known theoretical and numerical findings that have been reported in the literature. Program summaryProgram Title: FFEM_BdG_ddm_toolbox.zipCPC Library link to program files:https://doi.org/10.17632/w9hg964wpb.1Licensing provisions: GPLv3Programming language:FreeFEM (v 4.12) free software (www.freefem.org)Nature of problem: Among the plethora of configurations that may exist in Gross-Pitaevskii (GP) equations modeling one or two-component Bose-Einstein condensates, only the ones that are deemed spectrally stable (or even, in some cases, weakly unstable) have high probability to be observed in realistic ultracold atoms experiments. To investigate the spectral stability of solutions requires the numerical study of the linearization of GP equations, the latter commonly known as the Bogoliubov-de Gennes (BdG) spectral problem. The present software offers an efficient and reliable tool for the computation of eigenvalues (or modes) of the BdG problem for a given two- or three-dimensional GP configuration. Then, the spectral stability (or instability) can be inferred from its spectrum, thus predicting (or not) its observability in experiments.Solution method: The present toolbox in FreeFEM consists of the following steps. At first, the GP equations in two (2D) and three (3D) spatial dimensions are discretized by using P2 (piece-wise quadratic) Galerkin triangular (in 2D) or tetrahedral (in 3D) finite elements. For a given configuration of interest, mesh adaptivity in FreeFEM is deployed in order to reduce the size of the problem, thus reducing the toolbox's execution time. Then, stationary states of the GP equations are obtained by a Newton method whose backbone involves the use of a reliable and efficient linear solver judiciously selected from the PETSc1 library. Upon identifying stationary configurations, to trace branches of such solutions a parameter continuation method over the chemical potential in the GP equations (effectively controlling the number of atoms in a BEC) is employed with step-size adaptivity of the continuation parameter. Finally, the computation of the stability of branches of solutions (i.e. the BdG spectrum), is carried out by accurately solving, at each point in the parameter space, the underlying eigenvalue problem by using the SLEPc2 library. Three-dimensional computations are made affordable in the present toolbox by using the domain decomposition method (DDM). In the course of the computation, the toolbox stores not only the solutions but also the eigenvalues and respective eigenvectors emanating from the solution to the BdG problem. We offer examples for computing stationary configurations and their BdG spectrum in one- and two-component GP equations.Running time: From minutes to hours depending on the mesh resolution and space dimension.

A novel approach to crack modeling using extended finite element and substructures methods

In the present work, a combination of Extended Finite Element Method (XFEM) and substructures methods has been utilized for crack modeling. Structures are divided into several substructures, and XFEM is employed to model the presence of cracks within each substructure. In another method, crack modeling has been carried out without using the Extended Finite Element Method. Instead, cracks are modeled by eliminating boundary constraints from two substructures. The Krylov-Schur algorithm is utilized to accelerate the computation of structural modes by preparing stiffness and mass matrices for eigenvalue and eigenvector calculations related to the superstructure. For evaluating this method, four examples have been considered: In the first example, an experimental L-shaped steel structure comprising a beam and a column was examined. The second example involved an experimental steel beam with a box-shaped cross-section. In the third example, a steel beam with simple supports at both ends and a rectangular cross-section was proposed. In the fourth example, a real two-span bridge (Turtle Mountain Bridge) is also presented to evaluate the proposed method. The method was applied for modeling, and the obtained results were validated using experimental and analytical results. In this work, the Structural Similarity Index Method (SSIM) for mode shape similarity assessment has been employed for crack identification. Based on the results obtained from the modeling and crack identification evaluations, it can be concluded that the current method can model multiple cracks in structures with variable cross-sections, real concrete bridges, frame-shaped structures, and diverse structures by assembling substructures. Despite simplifying the problems, it maintains an acceptable level of accuracy in validation. This advantage enables the identification of cracks through a straightforward comparison of mode shapes.

High-efficiency parametric iterative schemes for solving nonlinear equations with and without memory

Many practical problems, such as the Malthusian population growth model, eigenvalue computations for matrices, and solving the Van der Waals' ideal gas equation, inherently involve nonlinearities. This paper initially introduces a two-parameter iterative scheme with a convergence order of two. Building on this, a three-parameter scheme with a convergence order of four is proposed. Then we extend these schemes into higher-order schemes with memory using Newton's interpolation, achieving an upper bound for the efficiency index of 7.8874813≈1.99057. Finally, we validate the new schemes by solving various numerical and practical examples, demonstrating their superior efficiency in terms of computational cost, CPU time, and accuracy compared to existing methods.

Experiment and industrial application of dust concentration measurement method using image transmittance characteristic

Abstract To prevent the harm of respirable dust and dust explosions, it is necessary to detect the concentration of dust generated in the process of bulk materials to help dust suppression. In this investigation, a dust concentration measurement method is proposed based on the eigenvalue calculation of image transmittance. The method is combined with the dark channel prior theory into the computer system and calibrated in the laboratory to calculate the transmittance of dust images. Furthermore, a dust concentration online measurement experimental platform is established to verify the correctness and reliability. Moreover, the industrial application is also carried out. The average relative error of dust concentration is 4.53%, and the relative error is less than 10% of the dust concentration measured by online measurement experimental platforms and industrial applications, which satisfied the measurement requirements. It is concluded that the measurement method has high applicability and can be practically applied to dust concentration measurements.

Spectral properties for discontinuous Dirac system with eigenparameter‐dependent boundary condition

In this paper, Dirac system with interface conditions and spectral parameter dependent boundary conditions is investigated. By introducing a new Hilbert space, the original problem is transformed into an operator problem. Then the continuity and differentiability of the eigenvalues with respect to the parameters in the problem are showed. In particular, the differential expressions of eigenvalues for each parameter are given. These results would provide theoretical support for the calculation of eigenvalues of the corresponding problems.

Accurate bidiagonal decompositions of Cauchy–Vandermonde matrices of any rank

AbstractWe present a new decomposition of a Cauchy–Vandermonde matrix as a product of bidiagonal matrices which, unlike its existing bidiagonal decompositions, is now valid for a matrix of any rank. The new decompositions are insusceptible to the phenomenon known as subtractive cancellation in floating point arithmetic and are thus computable to high relative accuracy. In turn, other accurate matrix computations are also possible with these matrices, such as eigenvalue computation amongst others.

An optimizated additional damping control for suppressing ultra‐low frequency oscillation suppression based on SVC

AbstractThe ultra‐low frequency oscillation (ULFO) imposes an emerging stability challenge to the high proportion of hydropower grids. To suppress ULFO without reducing the primary frequency regulation of the hydro governor, a novel idea to exploit the voltage regulation effect of load is implemented. First, the additional damping controller is designed and configured in static var compensator (SVC), and the frequency analysis model considering the configuration of SVC as well as damping controller is established. Based on the model, the specific influence of SVC and controller on ULFO is analysed through eigenvalue calculation. In addition, the influence of various load models and SVC location on the damping level are further studied. Consequently, a parameter optimization design method for SVC additional damping control is proposed, it is modelled as the optimization problem of damping ratio in ULFO mode under multi‐operation conditions, which is solved by a particle swarm optimization algorithm. Finally, the effectiveness of the designed SVC additional damping controller is verified in the improved four‐machine two‐area power system and the actual power grid in China.

Switching to hydrogen fuel adoption in developing economies: A case study of Vietnam

ABSTRACT Extensive research has been conducted on the transition to environmentally friendly fuels in developed countries worldwide. However, there has been a significant lack of attention given to this crucial topic in developing countries. The study aims to bridge this gap by conducting a comprehensive analysis of the existing obstacles and prospects associated with transitioning to hydrogen fuel, with a specific focus on Vietnam. To achieve this objective, a rigorous SWOT (Strengths, Weaknesses, Opportunities, and Threats) analysis approach is employed, drawing upon an extensive literature review and the insights of selective experts in the field. Additionally, the Analytic Hierarchy Process (AHP) is utilized to effectively prioritize the identified factors from the SWOT analysis through eigenvalue calculation. As a result, the weaknesses are the highest first-class priority factors, with “Dedicated infrastructure for the transition to hydrogen fuel” and “Technological capacity for converting to hydrogen fuel” taking the next priorities in the second class. The outcomes of this research serve as a valuable resource for Vietnam and other developing countries, guiding policy-making and practical applications in the pursuit of transitioning to hydrogen fuel. Furthermore, this paper establishes a robust framework that can guide future studies in this specific area of research.

High-accuracy measurement, advanced theory and analysis of the evolution of satellite transitions in manganese Kα using XR-HERFD.

Here, the novel technique of extended-range high-energy-resolution fluorescence detection (XR-HERFD) has successfully observed the n = 2 satellite in manganese to a high accuracy. The significance of the satellite signature presented is many hundreds of standard errors and well beyond typical discovery levels of three to six standard errors. This satellite is a sensitive indicator for all manganese-containing materials in condensed matter. The uncertainty in the measurements has been defined, which clearly observes multiple peaks and structure indicative of complex physical quantum-mechanical processes. Theoretical calculations of energy eigenvalues, shake-off probability and Auger rates are also presented, which explain the origin of the satellite from physical n = 2 shake-off processes. The evolution in the intensity of this satellite is measured relative to the full Kα spectrum of manganese to investigate satellite structure, and therefore many-body processes, as a function of incident energy. Results demonstrate that the many-body reduction factor S02 should not be modelled with a constant value as is currently done. This work makes a significant contribution to the challenge of understanding many-body processes and interpreting HERFD or resonant inelastic X-ray scattering spectra in a quantitative manner.

Kinetics Parameters Estimation for the Coupled Neutron-Photon Point Kinetics Model in Monte Carlo Eigenvalue Calculations

The delayed photoneutron effect should be considered in the point kinetics equation (PKE) applied to D2O and beryllium-moderated or -reflected systems because of its importance on reactor kinetics. This paper proposes neutron-photon-coupled PKE with a clear definition of adjoint-weighted kinetics parameters after providing the mathematical derivation from the neutron and photon transport and precursor equations. The Monte Carlo algorithms to estimate the kinetics parameters are induced from their definition and implemented in the eigenvalue calculation module of McCARD. They are verified through homogeneous infinite-medium problems with few-group neutron and photon cross sections by comparing with analytic solutions. The proposed PKE and kinetics parameters estimation methods are applied for the kinetics analyses of the artificial control rod drop transient problem in the RB reactor. Photonuclear physics is also implemented in a time-dependent Monte Carlo simulation to calculate the reference data of neutron amplitude in the core to be compared with the PKE results. As good agreement is shown in the neutron amplitude, it is validated that the developed PKE is capable of predicting the reactor kinetics considering photoneutrons.

Advanced Computational Framework to Analyze the Stability of Non-Newtonian Fluid Flow through a Wedge with Non-Linear Thermal Radiation and Chemical Reactions

The main idea of this investigation is to introduce an integrated intelligence approach that investigates the chemically reacting flow of non-Newtonian fluid with a backpropagation neural network (LMS-BPNN). The AI-based LMS-BPNN approach is utilized to obtain the optimal solution of an MHD flow of Eyring–Powell over a porous shrinking wedge with a heat source and nonlinear thermal radiation (Rd). The partial differential equations (PDEs) that define flow problems are transformed into a system of ordinary differential equations (ODEs) through efficient similarity variables. The reference solution is obtained with the bvp4c function by changing parameters as displayed in Scenarios 1–7. The label data are divided into three portions, i.e., 80% for training, 10% for testing, and 10% for validation. The label data are used to obtain the approximate solution using the activation function in LMS-BPNN within the MATLAB built-in command ‘nftool’. The consistency and uniformity of LMS-BPNN are supported by fitness curves based on the MSE, correlation index (R), regression analysis, and function fit. The best validation performance of LMS-BPNN is obtained at 462, 369, 642, 542, 215, 209, and 286 epochs with MSE values of 8.67 × 10−10, 1.64 × 10−9, 1.03 × 10−9, 302 9.35 × 10−10, 8.56 × 10−10, 1.08 × 10−9, and 6.97 × 10−10, respectively. It is noted that f′(η), θ(η), and ϕ(η) satisfy the boundary conditions asymptotically for Scenarios 1–7 with LMS-BPNN. The dual solutions for flow performance outcomes (Cfx, Nux, and Shx) are investigated with LMS-BPNN. It is concluded that when the magnetohydrodynamics increase (M=0.01, 0.05, 0.1), then the solution bifurcates at different critical values, i.e., λc=−1.06329,−1.097,−1.17694. The stability analysis is conducted using an LMS-BPNN approximation, involving the computation of eigenvalues for the flow problem. The deduction drawn is that the upper (first) branch solution remains stable, while the lower branch solution causes a disturbance in the flow and leads to instability. It is observed that the boundary layer thickness for the lower branch (second) solution is greater than the first solution. A comparison of numerical results and predicted solutions with LMS-BPNN is provided and they are found to be in good agreement.

Computation of Laplacian eigenvalues of two-dimensional shapes with dihedral symmetry

We numerically compute the lowest Laplacian eigenvalues of several two-dimensional shapes with dihedral symmetry at arbitrary precision arithmetic. Our approach is based on the method of particular solutions with domain decomposition. We are particularly interested in asymptotic expansions of the eigenvalues λ(n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda (n)$$\\end{document} of shapes with n edges that are of the form λ(n)∼x∑k=0∞Ck(x)nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lambda (n) \\sim x\\sum _{k=0}^{\\infty } \\frac{C_k(x)}{n^k}$$\\end{document} where x is the limiting eigenvalue for n→∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n\\rightarrow \\infty $$\\end{document}. Expansions of this form have previously only been known for regular polygons with Dirichlet boundary conditions and (quite surprisingly) involve Riemann zeta values and single-valued multiple zeta values, which makes them interesting to study. We provide numerical evidence for closed-form expressions of higher order Ck(x)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_k(x)$$\\end{document} and give more examples of shapes for which such expansions are possible (including regular polygons with Neumann boundary condition, regular star polygons, and star shapes with sinusoidal boundary).

Synchronizability in randomized weighted simplicial complexes.

We present a formula for determining synchronizability in large, randomized, and weighted simplicial complexes. This formula leverages eigenratios and costs to assess complete synchronizability under diverse network topologies and intensity distributions. We systematically vary coupling strengths (pairwise and three body), degree, and intensity distributions to identify the synchronizability of these simplicial complexes of the identical oscillators with natural coupling. We focus on randomized weighted connections with diffusive couplings and check synchronizability for different cases. For all these scenarios, eigenratios and costs reliably gauge synchronizability, eliminating the need for explicit connectivity matrices and eigenvalue calculations. This efficient approach offers a general formula for manipulating synchronizability in diffusively coupled identical systems with higher-order interactions simply by manipulating degrees, weights, and coupling strengths. We validate our findings with simplicial complexes of Rössler oscillators and confirm that the results are independent of the number of oscillators, connectivity components, and distributions of degrees and intensities. Finally, we validate the theory by considering a real-world connection topology using chaotic Rössler oscillators.

Fission source convergence diagnosis in Monte Carlo eigenvalue calculations by skewness and kurtosis estimation methods

In this study, a skewness estimation method (SEM) and kurtosis estimation method (KEM) are introduced to determine the number of inactive cycles in Monte Carlo eigenvalue calculations. The SEM and KEM can determine the number of inactive cycles on the basis that fully converged fission source distributions may follow normal distributions without asymmetry or outliers. Two convergence criteria values and a minimum cycle length for the SEM and KEM were determined from skewness and kurtosis analyses of the AGN-201K benchmark and 1D slab problems. The SEM and KEM were then applied to two OECD/NEA slow convergence benchmark problems to evaluate the performance and reliability of the developed methods. Results confirmed that the SEM and KEM provide appropriate and effective convergence cycles when compared to other methods and fission source density fraction trends. Also, the determined criterion value of 0.5 for both ε1 and ε2 was concluded to be reasonable. The SEM and KEM can be utilized as a new approach for determining the number of inactive cycles and judging whether Monte Carlo tally values are fully converged. In the near future, the methods will be applied to various practical problems to further examine their performance and reliability, and optimization will be performed for the convergence criteria and other parameters as well as for improvement of the methodology for practical usage.

Spectral methods on the hydrogen atom

This study investigates the computational implementation of spectral theory and its applications in analyzing complex mathematical problems. It explores the use of modern programming languages and scientific libraries for implementing and visualizing complex mathematical concepts, particularly focusing on spectral theory within quantum physics. The research employs computational methods to address the challenges in interpreting spectral properties, utilizing the Lanczos spectral method for eigenvalue calculation in large, sparse matrices. The results illustrate the effectiveness of these computational techniques in visualizing quantum states, demonstrating the potential of advanced programming in understanding and solving intricate problems in quantum physics and spectral graph theory. The study's findings are significant in bridging computational methods with theoretical spectral analysis, offering a new perspective on the application of computational techniques in scientific research.

MaCaw: Domain-Decomposed Monte Carlo Neutral Particle Transport on Unstructured Mesh in MOOSE

MaCaw is a Multiphysics Object-Oriented Simulation Environment (MOOSE)–based application that enables domain-decomposed neutral particle transport calculations in MOOSE. It leverages MOOSE’s ray-tracing module for unstructured mesh particle tracking and OpenMC for collision physics. Additionally, the OpenMC implementation of several calculation steps (e.g. initialization and normalization) needed in a Monte Carlo particle transport eigenvalue calculation were adapted for domain decomposition. This paper reports on MaCaw’s implementation and several limitations, a single verification case, and early single-node scaling studies. This paper also serves as an announcement of the public release of MaCaw on the Idaho National Laboratory GitHub at https://github.com/idaholab/macaw.

The Application of the Bidiagonal Factorization of Totally Positive Matrices in Numerical Linear Algebra

The approach to solving linear systems with structured matrices by means of the bidiagonal factorization of the inverse of the coefficient matrix is first considered in this review article, the starting point being the classical Björck–Pereyra algorithms for Vandermonde systems, published in 1970 and carefully analyzed by Higham in 1987. The work of Higham briefly considered the role of total positivity in obtaining accurate results, which led to the generalization of this approach to totally positive Cauchy, Cauchy–Vandermonde and generalized Vandermonde matrices. Then, the solution of other linear algebra problems (eigenvalue and singular value computation, least squares problems) is addressed, a fundamental tool being the bidiagonal decomposition of the corresponding matrices. This bidiagonal decomposition is related to the theory of Neville elimination, although for achieving high relative accuracy the algorithm of Neville elimination is not used. Numerical experiments showing the good behavior of these algorithms when compared with algorithms that ignore the matrix structure are also included.

Stability Analysis in a Direct-Current Shipboard Power System with Parallel Permanent Magnet Synchronous Generators and Supercapacitor Integration

This paper investigates the small-signal stability of a DC shipboard power system (SPS) with the integration of a supercapacitor. As an efficient energy storage solution, supercapacitors can not only provide rapid energy response to sudden power demand spikes, effectively mitigating load fluctuations, but also enhance the system’s resilience to disturbances. In the context of the parallel operation of two Permanent Magnet Synchronous Generators (PMSGs), the inclusion of supercapacitors may alter the system’s dynamic behaviors, thereby affecting its small-signal stability. This paper develops the small-signal model of SPS and explores the small-signal model under various power distribution strategies in the parallel operation of diesel generator sets. Through the calculation of eigenvalues and influence factors, the system’s oscillation modes are analyzed, and key parameters affecting the stability of the DC distribution system are identified. Furthermore, this paper meticulously examines the specific impacts of electrical and control parameter variations on the system’s small-signal stability. Simulation experiments validate the accuracy of the small-signal stability analysis after supercapacitor integration into SPS.

Three-Dimensional Full-Core BEAVRS Using OpenMOC with Transport Equivalence

Using an optimized implementation of the three-dimensional (3D) method of characteristics for neutron transport, along with a novel equivalence method for transport calculations that was designed to correct self-shielding errors from neglecting the angular dependence of resonant group absorption, a 3D full-core light water reactor hybrid stochastic-deterministic eigenvalue calculation was achieved. This paper presents the optimizations developed and compares the transport solutions obtained. For the statepoint, run times near 10 000 CPU hours are achieved—improving on previous works by an order of magnitude—with near 1% error on pin fission to 238U capture ratios and a few dozen pcms on the eigenvalue.