Eigenvalue Problem Research Articles

Breaking Internal Waves and Ocean Diapycnal Diffusivity in a High‐Resolution Regional Ocean Model: Evidence of a Wave‐Turbulence Cascade

AbstractIt is generally understood that the origin of ocean diapycnal diffusivity is primarily associated with the stratified turbulence produced by breaking internal (gravity) waves (IW). However, it requires significant effort to verify diffusivity values in ocean general circulation models in any particular geographical region of the ocean due to the scarcity of microstructure measurements. Recent analyses of downscaled IW fields from an internal‐wave‐admitting global ocean simulation into higher‐resolution regional configurations northwest of Hawaii have demonstrated a much‐improved fit of the simulated IW spectra to the in‐situ profiler measurements such as the Garrett‐Munk (GM) spectrum. Here, we employ this dynamically downscaled ocean simulation to directly analyze the nature of the IW‐breaking and the wave‐turbulence cascade in this region. We employ a modified version of the Kappa Profile Parameterization (KPP) to infer what the horizontally averaged vertical profile of diapycnal diffusivity should be, and compare this to the background profile that would be employed in the ocean component of a low‐resolution coupled climate model such as the Community Earth System Model (CESM) of the US National Center for Atmospheric Research (NCAR). In pursuing this goal, we also demonstrate that the wavefield in the high‐resolution regional domain is dominated by a well‐resolved spectrum of low‐mode IWs that are predictable by solving an appropriate eigenvalue problem for stratified flow. We finally suggest a new tentative approach to improve the KPP parameterization.

The Eigenvalue Properties of a Kind of Singular Differential Equations in 3-Dimensional Space

In this paper, we consider the eigenvalue problem of the singular differential equation [see formula in PDF] in a bounded open ball with Dirichlet boundary condition in 3-dimensional space, where, [see formula in PDF][see formula in PDF], [see formula in PDF] is a given constant[see formula in PDF]. And we have made a detailed characterization of the weak solution space. Furthermore, the existence of the minimum eigenvalue and the fundamental gap are provided.

Effects of variable gravity on stability in couple-stress fluids across various conducting boundaries

This work aims to inspect the effect of variable gravity field on the convective stability of couple stress fluid for different combinations of bounding surfaces. Both linear and nonlinear analyses are conducted to obtain eigenvalue problems. For this analysis, three different combinations of bounding surfaces have been considered. Normal mode analysis is used for linear analysis, while the energy method is used for nonlinear analysis. The numerical values of critical Rayleigh numbers are calculated using the single-term Galerkin technique. It is found that the Rayleigh numbers for the two analyses coincide, suggesting global stability. It is found that the increase in the value of couple stresses stabilize the system. The variable gravity can enhance or reduce the stability of the system depending on the direction of gravity variation which can be easily controlled by its parameter values. It is also observed that fluid confined in rigid-rigid bounding surface is more thermally stable, which is suitable for convection in couple stress fluid.

Linear and non-linear stability analysis of double-diffusive Hadley-Prats flow through horizontal porous layer with non-uniform thermal and solutal gradients

A scientific model is devised to understand the onset of double-diffusion Hadley-Prats flow through an infinite parallel permeable channel with non-uniform thermal, solutal gradients, and concentration based internal heat generation under the influence of convection conditions is considered. Darcy’s model is implemented in the context of a porous medium, which is characterized as isotropic and homogeneous. A linear and nonlinear stability analysis is conducted with the help of energy functional method and longitudinal roll perturbations are examined. The emerged dimensionless eigenvalue problem in the linear and nonlinear stability analysis is solved numerically by utilizing fourth order Runge-Kutta method (RK-4) method and shooting scheme. The critical values of wave and vertical thermal Rayleigh numbers are examined. A comprehensive analysis of solutions concerning the onset of convection mechanism is conducted. The produced results are discussed for the various values of emerged physical parameters. It is evident that, the presence of non-uniform thermal and solutal gradients and the internal heat generation mechanism causes the significant variations in the critical values of vertical thermal Rayleigh number in the channel regime.

Analysis of Manifold and its Application

Manifold learning is a field of study in machine learning and statistics that is closely associated with dimensionality reduction algorithmic techniques is gaining popularity these days. There are two types of manifold learning approaches: linear and nonlinear. Principal component analysis (PCA) and multidimensional scaling (MDS) are two examples of linear techniques that have long been staples in the statistician's arsenal for evaluating multivariate data. Nonlinear manifold learning, which encompasses diffusion maps, Laplacian Eigenmaps, Hessian Eigenmaps, Isomap, and local linear embedding, has seen a surge in research effort recently. A few of these methods are nonlinear extensions of linear approaches. A nearest search, the definition of distances or affinities between points (a crucial component of these methods' effectiveness), and an Eigen problem for embedding high-dimensional points into a lower dimensional space make up the algorithmic process of the majority of these techniques. The strengths and weaknesses of the new method are briefly reviewed in this article. In the field of computer graphics, we utilize a particular manifold learning method was first presented in statistics and machine learning to create a global, Spectral-based shape descriptor.

Numerical solution of the multi-order fractional differential equation using Legendre wavelets and eigenfunction approach

An eigenfunction approach is implemented in this article to solve the multi-order fractional differential equations (FDEs) with boundary conditions. The approximate unknown solution is expressed as a linear combination of eigenfunctions in the present paper. The proposed approach solved the problem via complex and real eigenfunction. By using the eigenvalue problem Dαϕ(x)=λϕ(x) subjected to u(0)=u(1)=0, first we calculate the eigenvalues and their corresponding eigenfunction and then use the output of this problem in the present scheme to simplify. Exact fractional integration has been utilized to solve the eigenvalue problem. Collocation points have also been used to reduce the problem to a system of algebraic equations. The obtained solutions are compared with the exact solution. In addition, the numerical convergence has also been studied of the proposed technique through three examples. These examples also demonstrate the applicatory and simplicity of the proposed scheme.

Fast hardware-aware matrix-free algorithms for higher-order finite-element discretized matrix multivector products on distributed systems

Recent hardware-aware matrix-free algorithms for higher-order finite-element (FE) discretized matrix-vector multiplications reduce floating point operations and data access costs compared to traditional sparse matrix approaches. In this work, we address a critical gap in existing matrix-free implementations which are not well suited for the action of FE discretized matrices on very large number of vectors. In particular, we propose efficient matrix-free algorithms for evaluating FE discretized matrix-multivector products on both multi-node CPU and GPU architectures. To this end, we employ batched evaluation strategies, with the batchsize tailored to underlying hardware architectures, leading to better data locality and enabling further parallelization. On CPUs, we utilize even-odd decomposition, SIMD vectorization, and overlapping computation and communication strategies. On GPUs, we develop strategies to overlap compute with data movement for achieving efficient pipelining and reduced data accesses through the use of GPU-shared memory, constant memory and kernel fusion. Our implementation outperforms the baselines for Helmholtz operator action on 1024 vectors, achieving up to 1.4x improvement on one CPU node and up to 2.8x on one GPU node, while reaching up to 4.4x and 1.5x improvement on multiple nodes for CPUs (3072 cores) and GPUs (24 GPUs), respectively. We further benchmark the performance of the proposed implementation for solving a model eigenvalue problem for 1024 smallest eigenvalue-eigenvector pairs by employing the Chebyshev Filtered Subspace Iteration method, achieving up to 1.5x improvement on one CPU node and up to 2.2x on one GPU node while reaching up to 3.0x and 1.4x improvement on multi-node CPUs (3072 cores) and GPUs (24 GPUs), respectively.

An efficient optimization method for transcendental eigenvalue problems based on mode count constraints and heuristic algorithm

An efficient optimization method for transcendental eigenvalue problems based on mode count constraints and heuristic algorithm

Numerical solution of eigenvalue problems for a compact integral operator with Green’s kernels

Numerical solution of eigenvalue problems for a compact integral operator with Green’s kernels

R I biorthogonal polynomials of Hahn type

ABSTRACT A finite family of R I polynomials is introduced and studied. It consists in a set of polynomials of 3 F 2 form whose biorthogonality to an ensemble of rational functions is spelled out. These polynomials are shown to satisfy two generalized eigenvalue problems: in addition to their recurrence relation of R I type, they are also found to obey a difference equation. Underscoring this bispectrality is a triplet of operators with tridiagonal actions. The algebra associated to these operators is provided.

Instability of a weakly viscoelastic film flow on an oscillating inclined plane

The long- and finite-wavelength instabilities of weakly viscoelastic film on an oscillating inclined plane are investigated. By using the Chebyshev series solution with the Floquet theory, the combined effects of viscoelasticity and forcing amplitude on instability are described when the inclined plane oscillates in streamwise and wall-normal directions. For long-wavelength instability, the solution to the eigenvalue problem is obtained analytically by the asymptotic expansion method. Results show that the Floquet exponent is independent of the wall-normal oscillation amplitude. The effects of inclined angle, gravity and surface tension on the stability of viscoelastic liquid film are also discussed. For finite-wavelength instability, numerical results corresponding to the wall-normal oscillation disclose that with the increase of viscoelasticity, the unstable gravitational boundary moves to a higher wavenumber, and the critical amplitudes of subharmonic and harmonic instabilities are reduced. The neutral curve of gravity instability for streamwise oscillatory flow is divided into two parts, and a stable bandwidth is formed for a large value of the viscoelastic parameter. Besides, a new oscillatory mode is identified at small angles of inclination, which will be enhanced if the effect of viscoelasticity is incorporated.

Multi-objective shape optimization of TESLA-like cavities: Addressing stochastic Maxwell's eigenproblem constraints

In this paper, we present a robust formulation to address nonlinear equality-constrained and inequality-constrained multi-objective optimization problems for multi-cell accelerating cavities with TESLA-like geometry under consideration of uncertain parameters. We propose to explore a robust approach that employs the polynomial chaos expansion (PCE) to model the uncertainty. For the uncertainty quantification (UQ), the resulting random-dependent eigenvalue problem is solved using the stochastic collocation method (SCM) combined with PCE. As a result, the shape optimization of the studied multi-cell cavity is defined in terms of statistical moments, which serve as the cost functionals in the multi-objective (MO) steepest descent algorithm. Two versions of this method are developed and implemented. Finally, the optimization results and their implications for the operation of a multi-cell cavity are discussed.

A one-dimensional high-order dynamic model for twin-cell box girders with deformable cross-section

A one-dimensional high-order dynamic model for single-box twin-cell box girders is presented together with the pattern recognition algorithm. The model takes into account the deformable cross-section and can accurately predict its 3D dynamic behaviors. The cross-section deformation is captured by basis functions satisfying displacement continuity condition, which is essential to construct the initial model formulation based on the Hamilton principle. The axial variation patterns of generalized coordinates are decoupled by solving the eigenvalue problem. On this basis, the combinations of basis functions are obtained to bring out cross-section deformation. The cross-section deformation, hierarchically organized and physically meaningful, are used to update the basis functions in the reconstructed high-order model. Numerical analysis has verified the accuracy and applicability of the reconstructed one-dimensional high-order model.

New vector transport operators extending a Riemannian CG algorithm to generalized Stiefel manifold with low-rank applications

This paper proposes two innovative vector transport operators, leveraging the Cayley transform, for the generalized Stiefel manifold embedded with a non-standard metric. Specifically, it introduces the differentiated retraction and an approximation of the Cayley transform to the differentiated matrix exponential. These vector transports are demonstrated to satisfy the Ring–Wirth non-expansive condition under non-standard metrics, and one of them is also isometric. Building upon the novel vector transport operators, we extend the modified Polak–Ribière–Polyak (PRP) conjugate gradient method to the generalized Stiefel manifold. Under a non-monotone line search condition, we prove our algorithm globally converges to a stationary point. The efficiency of the proposed vector transport operators is empirically validated through numerical experiments involving generalized eigenvalue problems and canonical correlation analysis.

Energy stability of magnetohydrodynamic flow in channels and ducts

We study the energy stability of pressure-driven laminar magnetohydrodynamic flow in a rectangular duct with a transverse homogeneous magnetic field and electrically insulating walls. For sufficiently strong fields, the laminar velocity distribution has a uniform core and convex Hartmann and Shercliff boundary layers on the walls perpendicular and parallel to the magnetic field. The problem is discretized by a double expansion in Chebyshev polynomials in the cross-stream coordinates. The linear eigenvalue problem for the critical Reynolds number depends on the streamwise wavenumber, Hartmann number and the aspect ratio. We consider the limits of small and large aspect ratios in order to compare with stability models based on one-dimensional base flows. For large aspect ratios, we find good numerical agreement with results based on the quasi-two-dimensional approximation. The lift-up mechanism dominates in the limit of a zero streamwise wavenumber and provides a linear dependence between the critical Reynolds and Hartmann numbers in the duct. As the aspect ratio is reduced away from unity, the duct results converge to Orr's original energy stability result for spanwise uniform perturbations imposed on the plane Poiseuille base flow. We also examine different possible symmetries of eigenmodes as well as the purely hydrodynamic case in the duct geometry.

A multiphase eigenvalue problem on a stratified Lie group

AbstractWe consider a multiphase spectral problem on a stratified Lie group. We prove the existence of an eigenfunction of (2, q)-eigenvalue problem on a bounded domain. Furthermore, we also establish a Pohozaev-like identity corresponding to the problem on the Heisenberg group.

The Partition of Unity Finite Element Method for the Schrödinger Equation

Abstract A Schrödinger equation for the system’s wavefunctions in a parallelepiped unit cell subject to Bloch-periodic boundary conditions must be solved repeatedly in quantum mechanical computations to derive the materials’ properties. Recent studies have demonstrated how enriched finite element type Galerkin methods can substantially lower the number of degrees of freedom necessary to produce accurate solutions with respect to the standard plane-waves method. In particular, the flat-top partition of unity finite element method enriched with the radial eigenfunctions of the one-dimensional Schrödinger equation offers a very effective way of solving the three-dimensional Schrödinger eigenvalue problem. We investigate the theoretical properties of this approximation method, its well-posedness and stability, we prove its convergence and derive suitable bound for the ℎ- and 𝑝-refinement in the L 2 L^{2} and energy norm for both the eigenvalues and the eigenfunctions. Finally, we confirm these theoretical results by applying this method to the eigenvalue problem of the one-electron Schrödinger equation with the harmonic potential, for which the exact solution is known.

Conical-Shaped Shells of Non-Uniform Thickness Vibration Analysis Using Higher-Order Shear Deformation Theory

Conical-Shaped Shells of Non-Uniform Thickness Vibration Analysis Using Higher-Order Shear Deformation Theory

The pseudospectra of black holes in AdS

We study the stability of quasinormal modes (QNMs) in electrically charged black brane spacetimes that asymptote to AdS by means of the pseudospectrum. Methodologically, we adopt ingoing Eddington-Finkelstein coordinates to cast QNMs in terms of a generalised eigenvalue problem involving a non-selfadjoint operator; this simplifies the computation significantly in comparison with previous results in the literature. Our analysis reveals spectral instability for (neutral) scalar as well as gravitoelectric perturbations. This indicates that the equilibration process of perturbed black branes is sensitive to external perturbations. Particular attention is given on the hydrodynamic modes, which are found to be the least unstable. In contrast with computations in hyperboloidal coordinates, we find that the pseudospectral contour lines cross to the upper half plane. This indicates the existence of pseudo-resonances as well as the possibility of transient instabilities. We also investigate the asymptotic structure of pseudospectral contour levels and we find remarkable universality across all sectors, persistent in the extremal limit.

Asymptotic profiles of a nonlocal dispersal SIS epidemic model with saturated incidence

Infection mechanism plays a significant role in epidemic models. To investigate the influence of saturation effect, a nonlocal (convolution) dispersal susceptible-infected-susceptible epidemic model with saturated incidence is considered. We first study the impact of dispersal rates and total population size on the basic reproduction number. Yang, Li and Ruan (J. Differ. Equ. 267 (2019) 2011–2051) obtained the limit of basic reproduction number as the dispersal rate tends to zero or infinity under the condition that a corresponding weighted eigenvalue problem has a unique positive principal eigenvalue. We remove this additional condition by a different method, which enables us to reduce the problem on the limiting profile of the basic reproduction number into that of the spectral bound of the corresponding operator. Then we establish the existence and uniqueness of endemic steady states by a equivalent equation and finally investigate the asymptotic profiles of the endemic steady states for small and large diffusion rates to provide reference for disease prevention and control, in which the lack of regularity of the endemic steady state and Harnack inequality makes the limit function of the sequence of the endemic steady state hard to get. Finally, we find whether lowing the movements of susceptible individuals can eradicate the disease or not depends on not only the sign of the difference between the transmission rate and the recovery rate but also the total population size, which is different from that of the model with standard or bilinear incidence.