Generalized Eigenvalue Problem Research Articles

Porosity effects on the dynamic response of arbitrary restrained FG nanobeam based on the MCST

Abstract In this study, two different general eigenvalue problems for nanobeams made of functionally graded material with pores in their sections according to Rayleigh beam theory using modified couple stress theory are established. Fourier sine series and Stokes transformation are used for the solution. First, the partial differential equation of motion of the problem is discretized into an ordinary differential equation. Then, the Fourier sine series of infinite series is substituted into this ordinary differential equation to determine the Fourier coefficient. Using the force boundary conditions of the system, Stokes’ transformation is performed at both ends to include elastic spring parameters. The unknown displacement terms are discretized to form two eigenvalue problems. By solving these eigenvalue problems, vibration frequencies for different boundary conditions can be found analytically. The variations of some parameters are discussed in a series of graphs.

Stochastic free vibration analysis of FG-CNTRC plates based on a new stochastic computational scheme

Stochastic analysis can provide more accurate results compared to deterministic analysis. In this study, the spatial variability of material parameters is introduced into the free vibration analysis of functionally graded carbon nanotube reinforced composite (FG-CNTRC) plates to achieve a higher degree of accuracy, and a new stochastic computational scheme is proposed to handle the low-level uncertainties. First-order shear deformation theory (FSDT) is employed to establish the displacement field of the plates. The Young's moduli of carbon nanotube (CNT) and matrix are regarded as one-dimensional (1D) and two-dimensional (2D) random fields, respectively, which is discretized by Karhunen-Loève (K-L) expansion. The obtained random variables are substituted into modified perturbation stochastic method (MPSM) and radial point interpolation method (RPIM) to calculate the first two order estimates of the stochastic non-dimensional natural frequencies ω¯. The sensitivity of the first to fourth ω¯ to the random fields is analyzed, and the corresponding stochastic bands are plotted. The results indicate that the presented approach can accurately solve the generalized eigenvalue problem in stochastic free vibration; ω¯ is sensitive to random fields E11CNT and Em, and insensitive to random field E22CNT; the CNT distribution mode also affects the sensitivity.

A robust numerical strategy for finding surface waves in flows of non-Newtonian liquids

Gravity-driven flows of liquid films are frequent in nature and industry, such as in landslides, lava flow, cooling of nuclear reactors, and coating processes. In many of these cases, the liquid is non-Newtonian and has particular characteristics. In this paper, we analyze numerically the temporal stability of films of non-Newtonian liquids falling by gravity, on the onset of instability. The liquid flows over an incline, where surface waves appear under certain conditions, and we do not fix a priori its rheological behavior. For that, we made used of the Carreau–Yasuda model without assigning specific values to its constants, and we compute general stability solutions. The numerical strategy is based on expansions of Chebyshev polynomials for discretizing the Orr–Sommerfeld equation and boundary conditions, and a Galerkin method for solving the generalized eigenvalue problem. In addition, an Inverse Iteration method was implemented to increase accuracy and improve computational time. The result is a robust and light numerical tool capable of finding the critical conditions for different types of fluids, which we use to analyze some key fluids. We show that the outputs of the general code match previous solutions obtained for specific computations. Besides increasing our knowledge on surface-wave instabilities in non-Newtonian liquids, our findings provide a new tool for obtaining comprehensive solutions on the onset of instability.

Variance characteristic preserving common spatial pattern for motor imagery BCI.

The common spatial patterns (CSP) algorithm is the most popular technique for extracting electroencephalogram (EEG) features in motor imagery based brain-computer interface (BCI) systems. CSP algorithm embeds the dimensionality of multichannel EEG data to extract features of motor imagery tasks. Most previous studies focused on the optimization of the time domain and the spectrum domain of EEG signal to improve the effectiveness of CSP, whereas ignoring the constraint on the projected feature space. This study proposed a variance characteristic preserving CSP (VPCSP) that is modified by a regularization item based on graph theory. Specifically, we calculated the loss of abnormalities of the projected data while preserving the variance characteristic locally. Then the loss could be rewritten as a matrix with the introduction of the Laplace matrix, which turned it into a generalized eigenvalue problem equivalent to CSP. This study evaluated the proposed method on two public EEG datasets from the BCI competition. The modified method could extract robust and distinguishable features that provided higher classification performance. Experimental results showed that the proposed regularization improved the effectiveness of CSP significantly and achieved superior performance compared with reported modified CSP algorithms significantly. The classification accuracy of the proposed method achieved 87.88 %, 90.07 %, and 76.06 % on public dataset IV part I, III part IVa and the self-collected dataset, respectively. Comparative experiments are conducted on two public datasets and one self-collected dataset. Results showed that the proposed method outperformed the reported algorithm. The proposed method can extract robust features to increase the performance of BCI systems. And the proposal still has expandability. These results show that our proposal is a promising candidate for the performance improvement of MI-BCI.

On the conditioning of some structured generalized eigenvalue problems

This work continues the analysis of the conditioning of a Hankel structured generalized eigenvalue problem (GEP) started in [1]. The considered generalized eigenvalue problem appears in exponential analysis and sparse interpolation.
 We generalize the proof in [1] and add expressions for the relative condition numbers of two reformulations of the GEP, a reformulation as a Loewner GEP valid for general complex data, and a compression to a Hankel+Toeplitz GEP in the case of real data. Both reformulations are compared to the original Hankel GEP. The analysis is concluded with ample numerical illustrations.

Linear disturbance growth induced by viscous dissipation in Darcy–Bénard convection with throughflow

Modal and non-modal linear stability analyses are employed to investigate the effect of internal and external heating on disturbance temporal growth for the Darcy–Bénard convection with throughflow. A matrix-forming approach is employed for both purposes, where the generalised eigenvalue problem is built using the generalised integral transform technique. Although the disturbance equations are not self-adjoint, the non-modal analysis indicates that there is no transient growth. Hence, any disturbance growth in time must be induced by modal mechanisms. An absolute instability analysis reveals that viscous dissipation has a destabilising effect and introduces new modes that are eventually destabilised by increasing the Péclet number. Beyond critical values of the Péclet number, where codimension-two absolutely unstable points exist, these modes become more unstable than the classical mode found in the absence of viscous dissipation, which is stabilised by an increasing Péclet number. This internal heating mechanism generated by viscous dissipation is so strong at high enough Péclet numbers that instability becomes possible through heating from above.

Consensus-based distributed algorithm for GEP

The generalized eigenvalue problem (GEP) plays a significant role in signal processing and machine learning. This paper proposes a consensus-based distributed algorithm for GEP in multi-agent systems, where the data are distributively stored across agents. The distributed GEP is reformulated as a consensus optimization, but the presence of its quadratic inseparable constraint makes the considered problem more challenging. To deal with it, a sequential method combined with the alternating direction method of multipliers is proposed, which requires communication between multiple pairs of nodes. Theoretical analysis shows the proposed algorithm will converge to the set of stationary solutions. And the numerical experiments on synthetic and real-world datasets validate that the approximated solution is competitive to the centered results.

On flexible block Chebyshev-Davidson method for solving symmetric generalized eigenvalue problems

On flexible block Chebyshev-Davidson method for solving symmetric generalized eigenvalue problems

Validation of the Townsend criterion for ignition of volume gas discharges

An eigenvalue problem governing ignition of volume discharges is formulated in the drift approximation with account of photoionization and the presence of multiple ion species with various reactions. The Townsend ignition (self-sustainment) criterion is deduced by partial integration of this eigenvalue problem neglecting the photoionization, and, in the case of electronegative gases, also the detachment. The Townsend criterion is applied to three examples of discharge ignition in high-pressure air, two of them referring to negative coronas in concentric-cylinder and point-to-plane configurations and one to a configuration with weakly nonuniform electric field. The comparison of the results obtained from different forms of the Townsend criterion with those given by the accurate numerical solution of the full eigenvalue problem shows that the neglect of the diffusion of the charged particles produces an error in the ignition voltage of the order of or less. The effect of photoionization for these configurations is also weak as expected. The lower and upper estimates of the effect of negative ions over the ignition voltage are obtained from different forms of the Townsend criterion. A form of the Townsend criterion, which employs an effective attachment coefficient in air and takes into account, in an approximate way, also the detachment, gives a virtually exact value of the inception voltage in all the cases considered. Practical recommendations are given on when one can use the Townsend self-sustainment criterion, and when one should resort to a numerical solution of the general eigenvalue problem governing discharge inception.

Calculation of linear stability of fluid flow in a flat channel with walls wavy across the flow

Using the full Navier-Stokes equations, the linear stability of a plane Poiseuille flow in a channel with a corrugated bottom wall is considered. The wall is corrugated across the flow, and the main flow has one velocity component. The perturbations of the velocity and pressure fields are three-dimensional with two wave numbers. The generalized eigenvalue problem is solved numerically. It is found that the critical Reynolds number, above which time-increasing disturbances appear, depends in a complex way on the dimensionless amplitude and the corrugation period. The magnitude of the ratio of the amplitude and the corrugation period divides the area of the dimensionless amplitude of the corrugation into two, where the dependences of the critical Reynolds number on the parameters of the corrugation are qualitatively different.

Max-Min Fairness Precoding for Physical Layer Security With Partial Channel Knowledge

We propose secure max-min fairness (MMF) precoding for a downlink multi-antenna system with multiple users and eavesdroppers. Since the problem is non-convex and non-smooth and partial channel knowledge is assumed, it is highly challenging to solve. To resolve the difficulties, we first reformulate the problem with conditional average rates, allowing to exploit the partial channel knowledge. Then, we derive the original problem as an approximate smoothing problem using a LogSumExp approach. We identify the first-order optimality condition and cast it to a generalized eigenvalue problem. Solving the problem via a power iteration offers the best local optimal solution. Simulations validate the proposed secure-MMF precoding method.

Variational quantum algorithm for generalized eigenvalue problems and its application to the finite-element method

Variational quantum algorithm for generalized eigenvalue problems and its application to the finite-element method

Efficient approximation of molecular kinetics using random Fourier features.

Slow kinetic processes in molecular systems can be analyzed by computing the dominant eigenpairs of the Koopman operator or its generator. In this context, the Variational Approach to Markov Processes (VAMP) provides a rigorous way of discerning the quality of different approximate models. Kernel methods have been shown to provide accurate and robust estimates for slow kinetic processes, but they are sensitive to hyper-parameter selection and require the solution of large-scale generalized eigenvalue problems, which can easily become computationally demanding for large data sizes. In this contribution, we employ a stochastic approximation of the kernel based on random Fourier features (RFFs) to derive a small-scale dual eigenvalue problem that can be easily solved. We provide an interpretation of this procedure in terms of a finite, randomly generated basis set. By combining the RFF approach and model selection by means of the VAMP score, we show that kernel parameters can be efficiently tuned and accurate estimates of slow molecular kinetics can be obtained for several benchmarking systems, such as deca alanine and the NTL9 protein.

Numerical Computation of Spectral Solutions for Sturm-Liouville Eigenvalue Problems

This paper focuses on the study of Sturm-Liouville eigenvalue problems. In the classical Chebyshev collocation method, the Sturm-Liouville problem is discretized to a generalized eigenvalue problem where the functions represent interpolants in suitably rescaled Chebyshev points. We are concerned with the computation of high-order eigenvalues of Sturm-Liouville problems using an effective method of discretization based on the Chebfun software algorithms with domain truncation. We solve some numerical Sturm-Liouville eigenvalue problems and demonstrate the efficiency of computations.

Fast Complementary Dynamics via Skinning Eigenmodes

We propose a reduced-space elastodynamic solver that is well suited for augmenting rigged character animations with secondary motion. At the core of our method is a novel deformation subspace based on Linear Blend Skinning that overcomes many of the shortcomings prior subspace methods face. Our skinning subspace is parameterized entirely by a set of scalar weights, which we can obtain through a small, material-aware and rig-sensitive generalized eigenvalue problem. The resulting subspace can easily capture rotational motion and guarantees that the resulting simulation is rotation equivariant. We further propose a simple local-global solver for linear co-rotational elasticity and propose a clustering method to aggregate per-tetrahedra nonlinear energetic quantities. The result is a compact simulation that is fully decoupled from the complexity of the mesh.

Implementation of Basic Operations for Sparse Matrices when Solving a Generalized Eigenvalue Problem in the ACELAN-COMPOS Complex

Introduction. The widespread use of piezoelectric materials in various industries stimulates the study of their physical characteristics and determines the urgency of such research. In this case, modal analysis makes it possible to determine the operating frequency and the coefficient of electromechanical coupling of piezoelectric elements of various devices. These indicators are of serious theoretical and applied interest. The study was aimed at the development of numerical methods for solving the problem of determining resonance frequencies in a system of elastic bodies. To achieve this goal, we needed new approaches to the discretization of the problem based on the finite element method and the execution of the software implementation of the selected method in C# on the .net platform. Current solutions were created in the context of the ACELAN-COMPOS class library. The known methods of solving the generalized eigenvalue problem based on matrix inversion are not applicable to large-dimensional matrices. To overcome this limitation, the presented scientific work implemented the logic of constructing mass matrices and created software interfaces for exchanging data on eigenvalue problems with pre- and postprocessing modules.Materials and Methods. A platform was used to implement numerical methods .net and the C# programming language. Validation of the research results was carried out through comparing the values found with solutions obtained in well-known SAE packages (computer-aided engineering). The created routines were evaluated in terms of performance and applicability for large-scale tasks. Numerical experiments were carried out to validate new algorithms in small-dimensional problems that were solved by known methods in MATLAB. Next, the approach was tested on tasks with a large number of unknowns and taking into account the parallelization of individual operations. To avoid finding the inverse matrix, a modified Lanczos method was programmatically implemented. We examined the formats for storing matrices in RAM: triplets, CSR, СSC, Skyline. To solve a system of linear algebraic equations (SLAE), an iterative symmetric LQ method adapted to these storage formats was used.Results. New calculation modules integrated into the class library of the ACELAN-COMPOS complex were developed. Calculations were carried out to determine the applicability of various formats for storing sparse matrices in RAM and various methods for implementing operations with sparse matrices. The structure of stiffness matrices constructed for the same task, but with different renumbering of nodes of a finite element grid, was graphically visualized. In relation to the problem of the theory of electroelasticity, data on the time required to perform basic operations with stiffness matrices in various storage formats were summarized and presented in the form of a table. It has been established that the renumbering of grid nodes gives a significant increase in performance even without changing the internal structure of the matrix in memory. Taking into account the objectives of the study, the advantages and weaknesses of known matrix storage formats were named. Thus, CSR was optimal when multiplying a matrix by a vector, SKS was optimal when inverting a matrix. In problems with the number of unknowns of the order of 103, iterative methods for solving a generalized eigenvalue problem won in speed. The performance of the software implementation of the Lanczos method was evaluated. The contribution of all operations to the total solution time was measured. It has been found that the operation of solving SLAE takes up to 95% of the total time of the algorithm. When solving the SLAE by symmetric LQ method, the greatest computational costs were needed to multiply the matrix by a vector. To increase the performance of the algorithm, parallelization with shared memory was resorted to. When using eight threads, the performance gain increased by 40–50%.Discussion and Conclusion. The software modules obtained as part of the scientific work were implemented in the ACELAN-COMPOS package. Their performance for model problems with quasi-regular finite element grids was estimated. Taking into account the features of the structures of the stiffness and mass matrices obtained through solving the generalized eigenvalue problem for an electroelastic body, the preferred methods for their processing were determined.

A homogeneous Rayleigh quotient with applications in gradient methods

Given an approximate eigenvector, its (standard) Rayleigh quotient and harmonic Rayleigh quotient are two well-known approximations of the corresponding eigenvalue. We propose a new type of Rayleigh quotient, the homogeneous Rayleigh quotient, and analyze its sensitivity with respect to perturbations in the eigenvector. Furthermore, we study the inverse of this homogeneous Rayleigh quotient as stepsize for the gradient method for unconstrained optimization. The notion and basic properties are also extended to the generalized eigenvalue problem.

Elastic buckling analysis of side-cracked thin plates by weak form quadrature elements

A weak form quadrature element formulation is established for elastic buckling analysis of thin plates with side cracks. The entire problem domain is divided into a square subdomain around the crack tip and several quadrilateral subdomains elsewhere. The stress distribution of a cracked plate is determined in pre-buckling analysis. In either pre-buckling analysis or buckling analysis of a cracked plate, the potential energy of the inner square subdomain is computed using asymptotic displacement expansion around the crack tip, while the potential energy elsewhere is computed using quadrilateral weak form quadrature elements. A generalized eigenvalue problem is dealt with to determine the critical loads and buckling modes. The first ten non-dimensional buckling loads and critical buckling modes for several typical problems are given to validate the present formulation.

Automatic performance tuning using the ATMathCoreLib tool: Two experimental studies related to dense symmetric eigensolvers

SummaryWe consider automatic performance tuning of dense symmetric eigenvalue problems using ATMathCoreLib, which is a library to assist automatic tuning. We deal with two problems, namely, automatic code selection for the symmetric generalized eigenvalue problem in distributed‐memory parallel environments and automatic parameter tuning in tridiagonalization of dense symmetric matrices on multicore processors. As for the first problem, numerical experiments show that ATMathCoreLib can choose the fastest solver for a given computing environment and problem size quickly even if the fluctuation in the execution time is as high as 40%. As for the second problem, ATMathCoreLib was able to select nearly optimal combinations of the algorithm and its parameter reliably and efficiently for various computing environments and matrix sizes. The performance of auto‐tuning was further enhanced by incorporating a user‐provided execution‐time model into ATMathCoreLib.

Real-space solution to the electronic structure problem for nearly a million electrons.

We report a Kohn-Sham density functional theory calculation of a system with more than 200 000 atoms and 800 000 electrons using a real-space high-order finite-difference method to investigate the electronic structure of large spherical silicon nanoclusters. Our system of choice was a 20nm large spherical nanocluster with 202 617 silicon atoms and 13 836 hydrogen atoms used to passivate the dangling surface bonds. To speed up the convergence of the eigenspace, we utilized Chebyshev-filtered subspace iteration, and for sparse matrix-vector multiplications, we used blockwise Hilbert space-filling curves, implemented in the PARSEC code. For this calculation, we also replaced our orthonormalization + Rayleigh-Ritz step with a generalized eigenvalue problem step. We utilized all of the 8192 nodes (458 752 processors) on the Frontera machine at the Texas Advanced Computing Center. We achieved two Chebyshev-filtered subspace iterations, yielding a good approximation of the electronic density of states. Our work pushes the limits on the capabilities of the current electronic structure solvers to nearly 106 electrons and demonstrates the potential of the real-space approach to efficiently parallelize large calculations on modern high-performance computing platforms.