Nonlinear Eigenvalue Problem Research Articles

Coupled cluster degree of the Grassmannian

We determine the number of complex solutions to a nonlinear eigenvalue problem on the Grassmannian in its Plücker embedding. This is motivated by quantum chemistry, where it represents the truncation to single electrons in coupled cluster theory. We prove the formula for the Grassmannian of lines which was conjectured in earlier work with Fabian Faulstich. This rests on the geometry of the graph of a birational parametrization of the Grassmannian. We present a squarefree Gröbner basis for this graph, and we develop connections to toric degenerations from representation theory.

Accurate computation of scattering poles of acoustic obstacles with impedance boundary conditions

We propose a computation method for scattering poles of impedance obstacles. Boundary integral equations are used to formulate the problem. It is shown that the scattering poles are the eigenvalues of some integral operator. Then we employ the Nyström method to discretize the integral operator and obtain a nonlinear matrix eigenvalue problem. The eigenvalues are computed using a multistep parallel spectral indicator method. Numerical examples demonstrate the high accuracy of the proposed method and can serve as the benchmarks. Our study provides a practical approach and can be extended to other scattering problems. This paper continues our previous study on the computation method for scattering poles of sound-soft obstacles.

A contour integral-based method for nonlinear eigenvalue problems for semi-infinite photonic crystals

In this study, we introduce an efficient method for determining isolated singular points of two-dimensional semi-infinite and bi-infinite photonic crystals, equipped with perfect electric conductor and quasi-periodic mixed boundary conditions. This specific problem can be modeled by a Helmholtz equation and is recast as a generalized eigenvalue problem involving an infinite-dimensional block quasi-Toeplitz matrix. Through an intelligent implementation of cyclic structure-preserving matrix transformations, the contour integral method is elegantly employed to calculate the isolated eigenvalue and to extract a component of the associated eigenvector. Moreover, a propagation formula for electromagnetic fields is derived. This formulation enables rapid computation of field distributions across the expansive semi-infinite and bi-infinite domains, thus highlighting the attributes of edge states. The preliminary MATLAB implementation is available at https://github.com/FAME-GPU/2D_Semi-infinite_PhC.

Match-based solution of general parametric eigenvalue problems

We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters; these are then combined to obtain global approximations of the parametric eigenvalues. To gather the non-parametric data, we use non-intrusive contour-integration-based methods, which, however, cannot track eigenvalues that migrate into/out of the contour as the parameter changes. Special strategies are described for performing the combination-over-parameter step despite having only partial information on such migrating eigenvalues. Moreover, we dedicate a special focus to the approximation of eigenvalues that undergo bifurcations. Finally, we propose an adaptive strategy that allows one to effectively apply our method even without any a priori information on the behavior of the sought-after eigenvalues. Numerical tests are performed, showing that our algorithm can achieve remarkably high approximation accuracy.

On the Optimum Distribution of Multiple Helmholtz Resonators for Annular Combustors

Abstract The combustors of many modern land-based gas turbines and aero-engines are annular. This kind of combustors often suffer from thermoacoustic oscillations, with the occurrence of mostly the circumferential first-order, second-order and third-order oscillation modes in the annular combustion chamber. To solve this problem, passive control methods are widely used adding passive acoustic dampers such as Helmholtz resonators (HRs). Depending on the type and number of HRs, and the possible positions over the circumference to install them, there could easily have millions, or even billions, of possible arrangment patterns. Finding a good design is the key to solve this problem. In this paper, we perform a theoretical and numerical study of an annular combustor installed with multiple types of HRs over the circumference. Firstly, a simple annular duct with arbitrary distributions of these HRs is studied analytically by solving the nonlinear eigenvalue problem of a 1D network model. Based on the results of the analytical method, the impact of the HRs on the acoustic modes of the combustion chamber is studied and the optimum arrangement of multiple resonators is obtained. This arrangement usually gives a null (or small) mode splitting strength and a good damping effect. Finally, we apply the optimum arrangement to damp the thermoacoustic modes that have been captured by numerical simulation for a real annular combustor. We use numerical simulations based on solving the 3D Helmholtz equation in COMSOL to verify the feasibility of the optimum arrangement.

An eigenvalue problem for self-similar patterns in Hele-Shaw flows

Hele-Shaw problems are prototypes to study the interface dynamics. Linear theory suggests the existence of self-similar patterns in a Hele-Shaw flow. That is, with a specific injection flux the interface shape remains unchanged while its size increases. In this paper, we explore the existence of self-similar patterns in the nonlinear regime and develop a nonlinear theory characterizing their fundamental features. Using a boundary integral formulation, we pose the question of self-similarity as a generalized nonlinear eigenvalue problem, involving two nonlinear integral operators. The nonlinear flux constant Cf is the eigenvalue and the corresponding self-similar pattern x̃ is the eigenvector. We develop a quasi-Newton method to solve the problem and show the existence of nonlinear shapes with k-fold dominated symmetries. Nonlinear results are compared with the established linear theory, demonstrating a divergence between the two due to non-linear effects absent in the linear stability analysis. Further, we investigate how sensitive the shape of the interface is to the viscosity. Additionally, we conduct numerous numerical experiments using a wide range of initial guesses and initial flux constants. Through these experiments, one is able to obtain a diagram of self-similar shapes and the corresponding flux. It could be used to verify possible self-similar shapes with a proper initial guess and initial flux constant. Our results go beyond the predictions of linear theory and establish a bridge between the linear theory and simulations.

Variational approach for the two-body problem in a multiband extended-Hubbard model

Considering a spin-up and a spin-down fermion in a generic tight-binding lattice with a multi-site basis, we investigate the two-body problem using a multiband extended-Hubbard model with finite-ranged hopping and interaction parameters. We derive a linear eigenvalue problem for the entire two-body spectrum, alongside a nonlinear eigenvalue problem for the bound states in the form of a self-consistency equation. Our results, based on an exact variational approach, suggest potential applications across various lattice geometries. As an illustration, we apply them to the linear-chain model and show that the resultant spin singlet and triplet bound states align well with the existing literature.

Adaptive choice of near-optimal expansion points for interpolation-based structure-preserving model reduction

Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the iterative rational Krylov algorithm (IRKA), which computes optimal interpolation points through iterative updates by solving linear eigenvalue problems. However, in the case of preserving internal system structures, optimal interpolation points are unknown, and heuristics based on nonlinear eigenvalue problems result in numbers of potential interpolation points that typically exceed the reasonable size of reduced-order systems. In our work, we propose a projection-based iterative interpolation method inspired by IRKA for generally structured systems to adaptively compute near-optimal interpolation points as well as an appropriate size for the reduced-order system. Additionally, the iterative updates of the interpolation points can be chosen such that the reduced-order model provides an accurate approximation in specified frequency ranges of interest. For such applications, our new approach outperforms the established methods in terms of accuracy and computational effort. We show this in numerical examples with different structures.

Error estimates for a mixed finite element method for the Maxwell’s transmission eigenvalue problem

In this paper, we analyze a numerical method combining the Ciarlet-Raviart mixed finite element formulation and an iterative algorithm for the Maxwell’s transmission eigenvalue problem. The eigenvalue problem is first written as a nonlinear quad-curl eigenvalue problem. Then the real transmission eigenvalues are proved to be the roots of a non-linear function. They are the generalized eigenvalues of a related linear self-adjoint quad-curl eigenvalue problem. These generalized eigenvalues are computed by a mixed finite element method. We derive the error estimates using the spectral approximation of compact operators, the theory of mixed finite element method for quad-curl problems, and the derivatives of eigenvalues.

Nonlinear Eigenvalue Problems for (p, 2)-Equations and Laplace Equations with Perturbations

We consider two types of nonlinear eigenvalue problems involving Laplace and p-Laplace operators (p>2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(p>2)$$\\end{document}. The main result establishes the existence of at least two nontrivial weak solutions in the case of the perturbed equation and the existence of a continuous spectrum in the case of the (p, 2)-equation. In both cases, variational methods play a central role in our arguments.

Nonlinear elliptic eigenvalue problems in cylindrical domains becoming unbounded in one direction

The aim of this work is to characterize the asymptotic behaviour of the first eigenfunction of the generalised p-Laplace operator with mixed (Dirichlet and Neumann) boundary conditions in cylindrical domains when the length of the cylindrical domains tends to infinity. This generalises an earlier work of Chipot et al. (Asymptot. Anal. 85(3–4) (2013) 199–227) where the linear case p = 2 is studied. Asymptotic behavior of all the higher eigenvalues of the linear case and the second eigenvalues of general case (using topological degree) for such problems is also studied.

Nonlinearity-induced topological phase transition characterized by the nonlinear Chern number

As first demonstrated by the characterization of the quantum Hall effect by the Chern number, topology provides a guiding principle to realize the robust properties of condensed-matter systems immune to the existence of disorder. The bulk–boundary correspondence guarantees the emergence of gapless boundary modes in a topological system whose bulk exhibits non-zero topological invariants. Although some recent studies have suggested a possible extension of the notion of topology to nonlinear systems, the nonlinear counterpart of a topological invariant has not yet been understood. Here we propose a nonlinear extension of the Chern number based on the nonlinear eigenvalue problems in two-dimensional systems and show the existence of bulk–boundary correspondence beyond the weakly nonlinear regime. Specifically, we find nonlinearity-induced topological phase transitions, in which the existence of topological edge modes depends on the amplitude of oscillatory modes. We propose and analyse a minimal model of a nonlinear Chern insulator whose exact bulk solutions are analytically obtained. The model exhibits the amplitude dependence of the nonlinear Chern number, for which we confirm the nonlinear extension of the bulk–boundary correspondence. Thus, our result reveals the existence of genuinely nonlinear topological phases that are adiabatically disconnected from the linear regime.

Nonlinear postbuckling and snap-through instability of movable simply supported BDFG porous plates rested on elastic foundations

This study presents, for the first time, a novel mathematical model that can address the nonlinear buckling, postbuckling, and snap-through of bidirectional functionally graded porous simply supported plates rested on elastic foundation and subjected to uniaxial/biaxial compressive loads. The parabolic shear deformation plate theory and the von Kármán strain nonlinearity are employed to derive governing equilibrium equations relative to neutral surface plane. The governing equations that consist of four nonlinear-coupled variable-coefficients partial differential equations are discretized by using the differential/integral quadrature method. The solution methodology depends on whether the discretized nonlinear algebraic system is homogeneous (with zero force vector) or nonhomogeneous. A homogeneous system can be formulated and solved as a nonlinear eigenvalue problem. Pseudo-arc-length continuation is implemented with a proposed iterative method to predict the load-deflection paths whether the system is homogeneous or not. Theoretical analysis and numerical results indicate that the type of porosity and position of in-plane loading have significant effects on the pitchfork-bifurcation or snap-through instability response of the bidirectional functionally graded porous plate. Parametric studies are presented to illustrate the impact of gradation indices, geometrical properties, porosity, and foundation constants on the postbuckling responses of bidirectional functionally graded porous plates. The proposed model may be used in designing nuclear, marine, aerospace, and civil structures with bi-directional functionally graded material constituents.

Band structure calculations of three-dimensional solid-fluid coupling phononic crystals using dual reciprocity boundary element method and wavelet compression method

The algorithm for calculating the band structures of two-dimensional phononic crystals (PCs) using the boundary element method (BEM) has been proposed for many years. However, it has not yet been extended to three-dimensional (3D) PCs because the fundamental solutions of 3D dynamics are complex and are related to angular frequency. In this study, the BEM is applied to calculate the band structures of 3D solid-fluid coupling PCs by combining the dual reciprocity method. The use of the dual reciprocity method can help avoid nonlinear eigenvalue problems. To address the limitation of fully populated matrices in BEM, the wavelet compression method based on B-spline wavelet on the interval is adopted. Some small matrix entries, which are generated by the vanishing moment and local support characteristics, are set to zero using the provided truncation technique. This process results in the production of sparse matrices. The constructed generalized linear eigenvalue equations are modified to tackle the ill-conditioned matrix issues arising from the distinct fundamental solutions of solids and fluids. The results show that this method is superior to the finite element method in terms of calculation efficiency. Compared to conventional BEM, the current wavelet BEM can not only generate sparse matrices but also reduce the integration calculation time when handling large-scale problems.

Modal reduction for exterior acoustic and vibroacoustic problems

Modal analysis, mode superposition and modal reduction are considered standard approaches for interior acoustic problems as well as for pure vibration problems. Things are different for exterior problems. Only a few methods are known to formulate a linearized eigenvalue problem for unbounded acoustic problems. Even fewer techniques on modal superposition and reduction are found in the literature. This talk will review such techniques which are either based on conjugate Astley-Leis infinite elements or on boundary elements for the unbounded region. While the results appear to be promising, a number of open questions need to be solved in this context. Among others, it is yet unclear which modes are actually required for a modal reduction and how to find a method to just evaluate these modes and the related eigenvalues. The situation is different for vibroacoustic radiation problems with clear resonances. In such cases, a modal reduction is able to substantially accelerate the solution process. Future work might lead to modal reduction using modes from non-linear eigenvalue problems which are known from the numerous papers on the boundary element methods published over the last decade.

Massively parallel implementation of iterative eigensolvers in large-scale plane-wave density functional theory

The Kohn-sham density functional theory (DFT) is a powerful method to describe the electronic structures of molecules and solids in condensed matter physics, computational chemistry and materials science. However, large and accurate DFT calculations within plane waves process a cubic-scaling computational complexity, which is usually limited by expensive computation and communication costs. The rapid development of high performance computing (HPC) on leadership supercomputers brings new opportunities for developing plane-wave DFT calculations for large-scale systems. Here, we implement parallel iterative eigensolvers in large-scale plane-wave DFT calculations, including Davidson, locally optimal block preconditioned conjugate gradient (LOBPCG), projected preconditioned conjugate gradient (PPCG) and the Chebyshev subspace iteration (CheFSI) algorithms, and analyze the performance of these algorithms in massively parallel plane-wave computing tasks. We adopt a two-level parallelization strategy that combines the message passing interface (MPI) with open multi-processing (OpenMP) parallel programming to handle data exchange and matrix operations in the construction and diagonalization of large-scale Hamiltonian matrix within plane waves. Numerical results illustrate that these iterative eigensolvers can scale up to 42,592 processing cores with high peak performance of 30% on leadship supercomputers to study the electronic structures of bulk silicon systems containing 10,648 atoms. Program summaryProgram Title: Plane wave density functional theory (PWDFT)CPC Library link to program files:https://doi.org/10.17632/c8v2mx5vn4.1Developer's repository link:https://bitbucket.org/berkeleylab/scalesLicensing provisions: BSD 3-clauseProgramming language: C++Nature of problem: PWDFT is used for electronic structure calculations based on Kohn-Sham density functional theory. The key challenge to address is a constrained energy minimization problem, which can also be formulated as a nonlinear eigenvalue problem. MPI/OpenMP-based approaches are employed to provide multi-core acceleration for the study of the chemical and material properties of larger-scale molecules and solids.Solution method: PWDFT implements self-consistent field (SCF) iterations and direct constrained minimization algorithms with various acceleration strategies. It is written in C++ and offers parallel acceleration based on MPI/OpenMP.

Design sensitivity analysis of modal frequencies of elastic structures submerged in an infinite fluid domain

SummaryThis paper presents a numerical approach for the design sensitivity analysis of modal frequencies of elastic structures submerged in an unbounded heavy fluid domain. Because the feedback of sound pressure onto the fluid‐loaded structures has to be taken into account in this case, a fully coupled scheme which combines the structural finite element method (FEM) and the acoustic boundary element method (BEM) is employed in the numerical simulation. The resulting nonlinear eigenvalue problem created by the coupled finite element and boundary element (FE‐BE) method is converted into a generalized eigenvalue problem (GEP) of reduced dimension by using a contour integral method. The design sensitivity formulation for non‐repeated eigenvalues is derived based on an adjoint method which uses both the left and right eigenvectors, and a small GEP is formed to calculate the gradients of repeated eigenvalues. The Burton‐Miller formulation is employed to shift the fictitious eigenfrequencies and their derivatives, and the parameter setting which is beneficial to the filtering of fictitious eigenfrequencies is investigated. Numerical examples are given to verify the accuracy and effectiveness of the developed method.

Locally unitarily invariantizable NEPv and convergence analysis of SCF

We consider a class of eigenvector-dependent nonlinear eigenvalue problems (NEPv) without the unitary invariance property. Those NEPv commonly arise as the first-order optimality conditions of a particular type of optimization problems over the Stiefel manifold, and previously, special cases have been studied in the literature. Two necessary conditions, a definiteness condition and a rank-preserving condition, on an eigenbasis matrix of the NEPv that is a global optimizer of the associated optimization problem are revealed, where the definiteness condition has been known for the special cases previously investigated. We show that, locally close to the eigenbasis matrix satisfying both necessary conditions, the NEPv can be reformulated as a unitarily invariant NEPv, the so-called aligned NEPv, through a basis alignment operation — in other words, the NEPv is locally unitarily invariantizable. Numerically, the NEPv is naturally solved by a self-consistent field (SCF)-type iteration. By exploiting the differentiability of the coefficient matrix of the aligned NEPv, we establish a closed-form local convergence rate for the SCF-type iteration and analyze its level-shifted variant. Numerical experiments confirm our theoretical results.

Existence of solutions of a nonlinear eigenvalue problem and their properties

An eigenvalue problem is considered for a nonlinear nonautonomous ordinary differential equation of the second order on a closed interval with conditions of the first type and an additional (local) condition. The nonlinearity in the equation is due to a monotonically increasing nonnegative function with powerlike growth at infinity. The existence of infinite numbers of negative and positive eigenvalues is shown. Asymptotic formulae for the eigenvalues and the maxima of eigenfunctions are found, and comparison theorems are established. Bibliography: 20 titles.

Frequency-Variant Double-Zero Single-Pole Reactive Coupling Networks for Coupled-Resonator Microwave Bandpass Filters

In this work, a family of frequency-variant reactive coupling (FVRCAFVC) networks is introduced and discussed as new building blocks for the synthesis of coupled-resonator bandpass filters with real or complex transmission zeros (TZs). The FVRCAFVC is a type of nonideal frequency-dependent inverter that has nonzero elements on the diagonal of the impedance matrix, along with a nonlinear frequency-variation profile of its transimpedance parameter. The distinctive feature of these new FVRCsAFVC is that they can be modeled with a simple bridged-T network consisting of four reactive elements. Such FVRCsAFVCs can introduce one pole and up to two TZs—so that they are referred to as double-zero single-pole (DZSP) networks. DZSP networks allow the pole to be located above, below, or in-between the TZs. Depending on the choice of the elements in the bridged-T circuit, the position of the TZs and the pole can be controlled independently. The coupling matrix for filters with DZSP networks can be found by solving an inverse-structured nonlinear eigenvalue problem (ISNEVP). Examples of possible implementations of DZSP couplingAFVC networks are provided for both lumped and quasi-lumped-element circuits, as well as for transmission-line-based and waveguide technologies. The application of DZSP coupling networks to the design of microwave bandpass filters with generalized Chebyshev-type characteristics is illustrated by means of the synthesis of three microstrips and two waveguide coupled-resonator bandpass filters. The waveguide filters are verified through electromagnetic (EM) simulations and the microstrip filters are manufactured and characterized for proof-of-concept demonstration purposes, showing a fairly close agreement between EM simulations and measurements.