Asymptotic methods for spherically symmetric MHD α2-dynamos

We consider two models of spherically-symmetric MHD α²–dynamos; one with idealized boundary conditions (BCs); and one with physically realistic BCs. As it has been shown in our previous work, the eigenvalues λ of a model with idealized BCs and constant α–profile α₀ are linear functions of α₀ and form a mesh in the (α₀, λ)–plane. The nodes of the spectral mesh correspond to double-degenerate eigenvalues of algebraic and geometric multiplicity 2 (diabolical points). It was found that perturbations of the constant α –profile lead to a resonant unfolding of the diabolical points with selection rules of the resonant unfolding defined by the Fourier coefficients of the perturbations. In the present contribution we present new exact results on the spectrum of the model with physically realistic BCs and constant α. For non-degenerate (simple) eigenvalues perturbation gradients are found at any particular α₀. We briefly discuss the spectral behavior of the α²–dynamo operator over a family of homotopic deformations of the BCs between idealized ones and physically realistic ones. Furthermore, we demonstrate that although the spectral singularities are lifted, a memory about their locations remains deeply imprinted in the homotopic family of spectral deformations due to a hidden underlying invariance. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dependence of eigenvalues of Dirac system on the parameters

The dependence of eigenvalues of Dirac system with general boundary conditions is studied. It is shown that the eigenvalues of Dirac operators depend not only continuously but also smoothly on the coefficients, the boundary conditions, and the endpoints of the problem. Furthermore, the differential expressions of the eigenvalues as regards these parameters are given. The results obtained in this paper would provide theoretical support for the numerical calculations of eigenvalues of the corresponding problems.     

具有混合型边界条件的左定Sturm-Liouville问题特征值的下标计算(英文)

本文研究具有混合型边界条件的左定Sturm-Liouvile问题特征值的下标计算问题.首先给出具有分离型边界条件和混合型边界条件的左定Sturm-Liouville问题的特征值之间的不等式;然后利用这个结果给出一种计算混合型边界条件下左定Sturm-Liouville问题特征值下标的方法.     

正则化HSS预处理鞍点矩阵的特征值估计

最近,Bai和Benzi针对鞍点问题提出了一类正则化HSS（Regularized Hermitian and skew-Hermitian splitting,RHSS）预处理子（BIT Numer.Math.,57（2017）287-311）.为了进一步分析RHSS预处理子的效果,本文重点研究了RHSS预处理鞍点矩阵特征值的估计,分析了复特征值实部和模的上下界、实特征值的上下界,还给出了特征值均为实数的充分条件.当正则化矩阵取为零矩阵时,RHSS预处理子退化为HSS预处理子,分析表明本文给出的复特征值实部的界比已有的结果更精确.数值算例验证了本文给出的理论结果.     

Inertia‐based spectrum slicing for symmetric quadratic eigenvalue problems

In the quadratic eigenvalue problem (QEP) with all coefficient matrices symmetric, there can be complex eigenvalues. However, some applications need to compute real eigenvalues only. We propose a Lanczos‐based method for computing all real eigenvalues contained in a given interval of large‐scale symmetric QEPs. The method uses matrix inertias of the quadratic polynomial evaluated at different shift values. In this way, for hyperbolic problems, it is possible to make sure that all eigenvalues in the interval have been computed. We also discuss the general nonhyperbolic case. Our implementation is memory‐efficient by representing the computed pseudo‐Lanczos basis in a compact tensor product representation. We show results of computational experiments with a parallel implementation in the SLEPc library.     

Asymptotic Correction of More Sturm–Liouville Eigenvalue Estimates

The asymptotic correction technique of Paine, de Hoog and Anderssen can dramatically improve the accuracy of finite difference or finite element eigenvalues at negligible extra cost if closed form expressions are available for the errors in a simpler related problem. This paper gives closed form expressions for the errors in the eigenvalues of certain Sturm–Liouville problems obtained by various methods, thereby increasing the range of problems for which asymptotic correction can achieve maximum efficiency. It also investigates implementation of the method for more general problems.     

Eigenvalues and switching algorithms for Quasi-Newton updates

Two switching algorithms QNSWl and QNSW2 are proposed in this paper. These algorithms are developed based on the eigenvalues of matrices which are inertial to the symmetric rank-one (SR1) updates and the BFGS updates. First, theoretical results on the eigenvalues and condition numbers of these matrices are presented. Second, switch-ing mechanisms are then developed based on theoretical results obtained so that each proposed algorithm has the capability of applying appropriate updating formulae at each iterative point during the whole minimization process. Third, the performance of

each of the proposed algorithms is evaluated over a wide range of test problems with variable dimensions. These results are then compared to the results obtained by some well-known minimization packages. Comparative results show that among the tested methods, the QNSW2 algorithm has the best overall performance for the problems examined. In some cases, the number of iterations and the number function/gradient calls required by certain existing methods are more than a four-fold increase over that required by the proposed switching algorithms     

Treatment of Dirichlet-type boundary conditions in the spline-based wavelet Galerkin method employing multiple point constraints

The wavelet methods have been extensively adopted and integrated in various numerical methods to solve partial differential equations. The wavelet functions, however, do not satisfy the Kronecker delta function properties, special treatment methods for imposing the Dirichlet-type boundary conditions are thus required. It motivates us to present in this paper a novel treatment technique for the essential boundary conditions (BCs) in the spline-based wavelet Galerkin method (WGM), taking the advantages of the multiple point constraints (MPCs) and adaptivity. The linear B-spline scaling function and multilevel wavelet functions are employed as basis functions. The effectiveness of the present method is addressed, and in particular the applicability of the MPCs is also investigated. In the proposed technique, MPC equations based on the tying relations of the wavelet basis functions along the essential BCs are developed. The stiffness matrix is degenerated based on the MPC equations to impose the BCs. The numerical implementation is simple, and no additional degrees of freedom are needed in the system of linear equations. The accuracy of the present formulation in treating the BCs in the WGM is high, which is illustrated through a number of representative numerical examples including an adaptive analysis.     

Solving Hamiltonian systems arising from ODE eigenproblems

This paper presents an algorithm for solving a linear Hamiltonian system arising in the study of certain ODE eigenproblems. The method follows the phase angles of an associated unitary matrix, which are essential for correct indexing of the eigenvalues of the ODE. Compared to the netlib code SL11F [11] the new method has the property that on many important problems – in particular, on matrix–vector Schrödinger equations – the cost of the integration is bounded independently of the eigenparameter λ. This allows large eigenvalues to be found much more efficiently. Numerical results show that our implementation of the new algorithm is substantially faster than the netlib code SL11F.     

The extrapolation of numerical eigenvalues by finite elements for differential operators

This paper discusses the extrapolation of numerical eigenvalues by finite elements for differential operators and obtains the following new results: (a) By extending a theorem of eigenvalue error estimate, which was established by Osborn, a new expansion of eigenvalue error is obtained. Many achievements, which are about the asymptotic expansions of finite element methods of differential operator eigenvalue problems, are brought into the framework of functional analysis. (b) The Richardson extrapolation of nonconforming finite elements for multiple eigenvalues and splitting extrapolation of finite elements based on domain decomposition of non-selfadjoint differential operators for multiple eigenvalues are achieved. In addition, numerical examples are provided to support the theoretical analysis.     

Relations among eigenvalues of Sturm-Liouville problems with different types of leading coefficient functions

For any Sturm-Liouville problem with a separable boundary condition and whose leading coefficient function changes sign (exactly once), we first give a geometric characterization of its eigenvalues λn using the eigenvalues of some corresponding problems with a definite leading coefficient function. Consequences of this characterization include simple proofs of the existence of the λn's, their Prüfer angle characterization, and a way for determining their indices from the zeros of their eigenfunctions. Then, interlacing relations among the λn's and the eigenvalues of the corresponding problems are obtained. Using these relations, a simple proof of asymptotic formulas for the λn's is given.     

Convergence Analysis for the Iterated Defect Correction Scheme of Finite Element Methods on Rectangle Grids

This paper develops a new method to analyze convergence of the iterated defect correction scheme of finite element methods on rectangular grids in both two and three dimensions. The main idea is to formulate energy inner products and energy (semi)norms into matrix forms. Then, two constants of two key inequalities involved are min and max eigenvalues of two associated generalized eigenvalue problems, respectively. Local versions on the element level of these two generalized eigenvalue problems are exactly solved to obtain sharp (lower) upper bounds of these two constants. This and some essential observations for iterated solutions establish convergence in 2D and the monotone decreasing property in 3D. For two dimensions the results herein improve those in literature; for three dimensions the results herein are new. Numerical results are presented to examine theoretical results.     

On the eigenvalues of specially low-rank perturbed matrices

We study the eigenvalues of a matrix A perturbed by a few special low-rank matrices. The perturbation is constructed from certain basis vectors of an invariant subspace of A, such as eigenvectors, Jordan vectors, or Schur vectors. We show that most of the eigenvalues of the low-rank perturbed matrix stayed unchanged from the eigenvalues of A; the perturbation can only change the eigenvalues of A that are related to the invariant subspace. Existing results mostly studied using eigenvectors with full column rank for perturbations, we generalize the results to more general settings. Applications of our results to a few interesting problems including the Google’s second eigenvalue problem are presented.     

Combined boundary integral equations for acoustic scattering‐resonance problems

In this paper, boundary integral formulations for a time‐harmonic acoustic scattering‐resonance problem are analyzed. The eigenvalues of eigenvalue problems resulting from boundary integral formulations for scattering‐resonance problems split in general into two parts. One part consists of scattering‐resonances, and the other one corresponds to eigenvalues of some Laplacian eigenvalue problem for the interior of the scatterer. The proposed combined boundary integral formulations enable a better separation of the unwanted spectrum from the scattering‐resonances, which allows in practical computations a reliable and simple identification of the scattering‐resonances in particular for non‐convex domains. The convergence of conforming Galerkin boundary element approximations for the combined boundary integral formulations of the resonance problem is shown in canonical trace spaces. Numerical experiments confirm the theoretical results. Copyright © 2016 John Wiley & Sons, Ltd.     

Symplectic effective order methods

A family of three stage symplectic Runge–Kutta methods are derived with effective order 4. The methods are constrained so that the coefficient matrix A has only real eigenvalues. This restriction enables transformations to be introduced into the implementation of the method so that, for large Hamiltonian problems, there is a significant gain in efficiency.     

Non-local boundary value problems of arbitrary order

We give a new unified method of establishing the existence ofmultiple positive solutions for a large number of non-lineardifferential equations of arbitrary order with any allowed numberof non-local boundary conditions (BCs). In particular, we areable to determine the Green's function for these problems withvery little explicit calculation, which shows that studyinga more general version of a problem with appropriate notationcan lead to a simplification in approach. We obtain existenceand non-existence results, some of which are sharp, and givenew results for both non-local and local BCs. We illustratethe theory with a detailed account of a fourth-order problemthat models an elastic beam and also determine optimal valuesof constants that appear in the theory.     

Numerical Approaches to Compute Spectra of Non-Self Adjoint Operators and Quadratic Pencils

In this article we are interested in the numerical computation of spectra of non-self adjoint quadratic operators. This leads to solve nonlinear eigenvalue problems. We begin with a review of theoretical results for the spectra of quadratic operators, especially for the Schrödinger pencils. Then we present the numerical methods developed to compute the spectra: spectral methods and finite difference discretization, in infinite or in bounded domains. The numerical results obtained are analyzed and compared with the theoretical results. The main difficulty here is that we have to compute eigenvalues of strongly non-self-adjoint operators which are very unstable.     

A Sine-Collocation Method for the Computation of the Eigenvalues of the Radial Schr?dinger Equation

A collocation scheme using sine function basis elements is developedand used to approximate the eigenvalues of the radial Schr?dingerequation. The method is shown to apply to problems with singulareigensolutions and error bounds for the approximate eigenvaluesare given. The method is applied to a few test examples to indicateboth the accuracy and the implementation of the method.     

椭圆PDE-约束优化问题的一个预条件子

针对由Galerkin有限元离散椭圆PDE-约束优化问题产生的具有特殊结构的3×3块线性鞍点系统,提出了一个预条件子并给出了预处理矩阵特征值及特征向量的具体表达形式.数值结果表明了该预条件子能够有效地加速Krylov子空间方法的收敛速率,同时也验证了理论结果.     

Extremum Problems of Laplacian Eigenvalues and Generalized Polya Conjecture(Dedicated to Professor Haim Brezis on the occasion of his 70th birthday)

In this survey on extremum problems of Laplacian-Dirichlet eigenvalues of Euclidian domains, the author briefly presents some relevant classical results and recent progress. The main goal is to describe the well-known conjecture due to Polya, its connections to Weyl's asymptotic formula for eigenvalues and shape optimizations. Many related open problems and some preliminary results are also discussed.