Adaptive finite element methods for computing band gaps in photonic crystals

In this paper we propose and analyse adaptive finite element methods for computing the band structure of 2D periodic photonic crystals. The problem can be reduced to the computation of the discrete spectra of each member of a family of periodic Hermitian eigenvalue problems on a unit cell, parametrised by a two-dimensional parameter - the quasimomentum. These eigenvalue problems involve non-coercive elliptic operators with generally discontinuous coefficients and are solved by adaptive finite elements. We propose an error estimator of residual type and show it is reliable and efficient for each eigenvalue problem in the family. In particular we prove that if the error estimator converges to zero then the distance of the computed eigenfunction from the true eigenspace also converges to zero and the computed eigenvalue converges to a true eigenvalue with double the rate. We also prove that if the distance of a computed sequence of approximate eigenfunctions from the true eigenspace approaches zero, then so must the error estimator. The results hold for eigenvalues of any multiplicity. We illustrate the benefits of the resulting adaptive method in practice, both for fully periodic structures and also for the computation of eigenvalues in the band gap of structures with defect, using the supercell method.     

Reduced energy functionals for a three-dimensional fast rotating Bose Einstein condensates

We prove that in the fast rotating regime, the three-dimensional Gross–Pitaevskii energy describing the state of a Bose Einstein condensate can be reduced to a two-dimensional problem and that the vortex lines are almost straight. Additionally, we prove that the minimum of this two-dimensional problem can be sought in a reduced space corresponding to the first eigenspace of an elliptic operator. This space is called the Lowest Landau level and is of infinite dimension     

Interaction of faults under slip‐dependent friction. Non‐linear eigenvalue analysis

We analyse the evolution of a system of finite faults by considering the non‐linear eigenvalue problems associated to static and dynamic solutions on unbounded domains. We restrict our investigation to the first eigenvalue (Rayleigh quotient). We point out its physical significance through a stability analysis and we give an efficient numerical algorithm able to compute it together with the corresponding eigenfunction. We consider the anti‐plane shearing on a system of finite faults under a slip‐dependent friction in a linear elastic domain, not necessarily bounded. The static problem is formulated in terms of local minima of the energy functional. We introduce the non‐linear (static) eigenvalue problem and we prove the existence of a first eigenvalue/eigenfunction characterizing the isolated local minima. For the dynamic problem, we discuss the existence of solutions with an exponential growth, to deduce a (dynamic) non‐linear eigenvalue problem. We prove the existence of a first dynamic eigenvalue and we analyse its behaviour with respect to the friction parameter. We deduce a mixed finite element discretization of the non‐linear spectral problem and we give a numerical algorithm to approach the first eigenvalue/eigenfunction. Finally we give some numerical results which include convergence tests, on a single fault and a two‐faults system, and a comparison between the non‐linear spectral results and the time evolution results. Copyright © 2004 John Wiley & Sons, Ltd.     

A full multigrid method for nonlinear eigenvalue problems

We introduce a type of full multigrid method for the nonlinear eigenvalue problem. The main idea is to transform the solution of the nonlinear eigenvalue problem into a series of solutions of the corresponding linear boundary value problems on the sequence of finite element spaces and nonlinear eigenvalue problems on the coarsest finite element space. The linearized boundary value problems are solved by some multigrid iterations. Besides the multigrid iteration, all other efficient iteration methods for solving boundary value problems can serve as the linear problem solver. We prove that the computational work of this new scheme is truly optimal, the same as solving the linear corresponding boundary value problem. In this case, this type of iteration scheme certainly improves the overfull efficiency of solving nonlinear eigenvalue problems. Some numerical experiments are presented to validate the efficiency of the new method.     

Linear system solution by null‐space approximation and projection (SNAP)

Solutions of large sparse linear systems of equations are usually obtained iteratively by constructing a smaller dimensional subspace such as a Krylov subspace. The convergence of these methods is sometimes hampered by the presence of small eigenvalues, in which case, some form of deflation can help improve convergence. The method presented in this paper enables the solution to be approximated by focusing the attention directly on the ‘small’ eigenspace (‘singular vector’ space). It is based on embedding the solution of the linear system within the eigenvalue problem (singular value problem) in order to facilitate the direct use of methods such as implicitly restarted Arnoldi or Jacobi–Davidson for the linear system solution. The proposed method, called ‘solution by null‐space approximation and projection’ (SNAP), differs from other similar approaches in that it converts the non‐homogeneous system into a homogeneous one by constructing an annihilator of the right‐hand side. The solution then lies in the null space of the resulting matrix. We examine the construction of a sequence of approximate null spaces using a Jacobi–Davidson style singular value decomposition method, called restarted SNAP‐JD, from which an approximate solution can be obtained. Relevant theory is discussed and the method is illustrated by numerical examples where SNAP is compared with both GMRES and GMRES‐IR. Copyright © 2006 John Wiley & Sons, Ltd.     

Spectral methods in linear minimax estimation

We consider the minimax-linear estimator in a linear regression model with circular constraints. Two necessary and sufficient conditions for the optimality of an estimator, the socalled left spectral equation and the right spectral equation (Girko spectral equation), are derived. For the special case of a simple maximal eigenvalue and a single eigenspace explicit estimation formulas are derived. These formulas also show some of the shortcomings of the minimax-linear estimator (MILE). Finally, the relation with Bayesian analysis and the Hoffmann-Läuter estimator is outlined.     

Dependence of Eigenvalues of Certain Closely Discrete Sturm-Liouville Problems

This paper is concerned with dependence of eigenvalues of certain closely discrete Sturm-Liouville problems. Topologies and geometric structures on various spaces of such problems are firstly introduced. Then, relationships between the analytic and geometric multiplicities of an eigenvalue are discussed. It is shown that all problems sufficiently close to a given problem have eigenvalues near each eigenvalue of the given problem. So, all the simple eigenvalues live in so-called continuous simple eigenvalue branches over the space of problems, and all the eigenvalues live in continuous eigenvalue branches over the space of self-adjoint problems. The analyticity, differentiability and monotonicity of continuous eigenvalue branches are further studied.     

Global bifurcation results for semilinear elliptic boundary value problems with indefinite weights and nonlinear boundary conditions

We investigate the global nature of bifurcation components of positive solutions of a general class of semilinear elliptic boundary value problems with nonlinear boundary conditions and having linear terms with sign-changing coefficients. We first show that there exists a subcontinuum, i.e., a maximal closed and connected component, emanating from the line of trivial solutions at a simple principal eigenvalue of a linearized eigenvalue problem. We next consider sufficient conditions such that the subcontinuum is unbounded in some space for a semilinear elliptic problem arising from population dynamics. Our approach to establishing the existence of the subcontinuum is based on the global bifurcation theory proposed by López-Gómez. We also discuss an a priori bound of solutions and deduce from it some results on the multiplicity of positive solutions.     

A note on Fiedler vectors interpreted as graph realizations

The second smallest eigenvalue of the Laplace matrix of a graph and its eigenvectors, also known as Fiedler vectors in spectral graph partitioning, carry significant structural information regarding the connectivity of the graph. Using semidefinite programming duality, we offer a geometric interpretation of this eigenspace as optimal solution to a graph realization problem. A corresponding interpretation is also given for the eigenspace of the maximum eigenvalue of the Laplacian.     

$p(x)$-Laplacian方程Robin边界条件下的特征值问题

本文研究了Robin边界条件下$p(x)$-Laplacian方程特征值问题. 利用变指数Sobolev空间理论, 我们用Luxemburg范数来定义Rayleigh商, 并给出该Rayleigh商的最小值点对应的Euler-Lagrange方程. 根据Ljusternik-Schnirelman原理, 我们证明了Robin边值问题存在无穷多特征值序列, 其中最小的特征值存在且是严格大于零的, 并且与最小的特征值相对应的特征函数不变号.     

Eigenvalue Gap Theorems for a Class of Nonsymmetric Elliptic Operators on Convex Domains

Adapting the method of Andrews-Clutterbuck we prove an eigenvalue gap theorem for a class of non symmetric second order linear elliptic operators on a convex domain in euclidean space. The class of operators includes the Bakry-Emery laplacian with potential and any operator with second order term the laplacian whose first order terms have coefficients with compact support in the open domain. The eigenvalue gap is bounded below by the gap of an associated Sturm-Liouville problem on a closed interval.     

On the modification of an eigenvalue problem that preserves an eigenspace

Eigenvalue problems arise in many application areas ranging from computational fluid dynamics to information retrieval. In these fields we are often interested in only a few eigenvalues and corresponding eigenvectors of a sparse matrix. In this paper, we comment on the modifications of the eigenvalue problem that can simplify the computation of those eigenpairs. These transformations allow us to avoid difficulties associated with non-Hermitian eigenvalue problems, such as the lack of reliable non-Hermitian eigenvalue solvers, by mapping them into generalized Hermitian eigenvalue problems. Also, they allow us to expose and explore parallelism. They require knowledge of a selected eigenvalue and preserve its eigenspace. The positive definiteness of the Hermitian part is inherited by the matrices in the generalized Hermitian eigenvalue problem. The position of the selected eigenspace in the ordering of the eigenvalues is also preserved under certain conditions. The effect of using approximate eigenvalues in the transformation is analyzed and numerical experiments are presented.     

Nonlinear Eigenvalue Problems in the Stability Analysis of Morphogen Gradients

This paper is concerned with several eigenvalue problems in the linear stability analysis of steady state morphogen gradients for several models of Drosophila wing imaginal discs including one not previously considered. These problems share several common difficulties including the following: (a) The steady state solution which appears in the coefficients of the relevant differential equations of the stability analysis is only known qualitatively and numerically. (b) Though the governing differential equations are linear, the eigenvalue parameter appears nonlinearly after reduction to a problem for one unknown. (c) The eigenvalues are determined not only as solutions of a homogeneous boundary value problem with homogeneous Dirichlet boundary conditions, but also by an alternative auxiliary condition to one of the Dirichlet conditions allowed by a boundary condition of the original problem. Regarding the stability of the steady state morphogen gradients, we prove that the eigenvalues must all be positive and hence the steady state morphogen gradients are asymptotically stable. The other principal finding is a novel result pertaining to the smallest (positive) eigenvalue that determines the slowest decay rate of transients and the time needed to reach steady state. Here we prove that the smallest eigenvalue does not come from the nonlinear Dirichlet eigenvalue problem but from the complementary auxiliary condition requiring only to find the smallest zero of a rational function. Keeping in mind that even the steady state solution needed for the stability analysis is only known numerically, not having to solve the nonlinear Dirichlet eigenvalue problem is both an attractive theoretical outcome and a significant computational simplification.     

Earth surface effects on active faults: An eigenvalue asymptotic analysis

We study in this paper an eigenvalue problem (of Steklov type), modeling slow slip events (such as silent earthquakes, or earthquake nucleation phases) occurring on geological faults. We focus here on a half space formulation with traction free boundary condition: this simulates the earth surface where displacements take place and can be picked up by GPS measurements. We construct an appropriate functional framework attached to a formulation suitable for the half space setting. We perform an asymptotic analysis of the solution with respect to the depth of the fault. Starting from an integral representation for the displacement field, we prove that the differences between the eigenvalues and eigenfunctions attached to the half space problem and those attached to the free space problem, is of the order of d^-2, where d is a depth parameter: intuitively, this was expected as this is also the order of decay of the derivative of the Green's function for our problem. We actually prove faster decay in case of symmetric faults. For all faults, we rigorously obtain a very useful asymptotic formula for the surface displacement, whose dominant part involves a so called seismic moment. We also provide results pertaining to the analysis of the multiplicity of the first eigenvalue in the line segment fault case. Finally we explain how we derived our numerical method for solving for dislocations on faults in the half plane. It involves integral equations combining regular and Hadamard's hypersingular integration kernels.     

The First Biharmonic Steklov Eigenvalue: Positivity Preserving and Shape Optimization

We consider the Steklov problem for the linear biharmonic equation. We survey existing results for the positivity preserving property to hold. These are connected with the first Steklov eigenvalue. We address the problem of minimizing this eigenvalue among suitable classes of domains. We prove the existence of an optimal convex domain of fixed measure.     

基于辛本征空间的线性阻尼振动系统动力学分析

对于考虑阻尼项和陀螺项的一般线性动力学振动系统,建立基于辛本征空间展开求解的一般方法.基于Rayleigh商本征值的模态展开方法被广泛应用于复杂结构动力系统振动分析,但对于很多机械系统,由于其不能有效考虑陀螺效应的影响,其适用性却受到很大限制.该文首先讨论了无阻尼系统Rayleigh商本征值问题与辛本征值问题的对应关系,表明前者实际可由后者的一种退化形式给出（也即忽略陀螺效应）,而后者更具有一般性.在此基础上,进一步基于辛本征空间本征向量展开,推导了同时考虑阻尼和陀螺系统的一般线性动力学系统的有效求解方法.数值算例选取不考虑陀螺效应及考虑陀螺效应的两种线性阻尼振动系统对所提出的方法进行了验证,分析结果表明了该文所建立方法的正确性和有效性.     

Numerical analysis of generalised max-plus eigenvalue problems

This paper is concerned with the deterministic discrete-time infinite horizon optimisation problem on a compact metric space with an average cost criterion involving two functions K (the “cost”) and T (the “time”). Firstly, we collect the different characterisations of the value λ in terms of generalised max-plus eigenvalue problem and in terms of linear programming. Secondly, we prove an error bound on λ when the space is discretised.     

Condition of extremum for eigenvalues of elliptic boundary-value problems

In this paper, we consider problems of eigenvalue optimization for elliptic boundary-value problems. The coefficients of the higher derivatives are determined by the internal characteristics of the medium and play the role of control. The necessary conditions of the first and second order for problems of the first eigenvalue maximization are presented. In the case where the maximum is reached on a simple eigenvalue, the second-order condition is formulated as completeness condition for a system of functions in Banach space. If the maximum is reached on a double eigenvalue, the necessary condition is presented in the form of linear dependence for a system of functions. In both cases, the system is comprised of the eigenfunctions of the initial-boundary value problem. As an example, we consider the problem of maximization of the first eigenvalue of a buckling column that lies on an elastic foundation.     

Basis Property of the Eigenfunction System of the Gellerstedt Problem in the Elliptic Part of the Domain

The Gellerstedt eigenvalue problem with homogeneous boundary conditions on interior characteristics is studied. We prove that the eigenfunction system of this problem is a Riesz basis in the L₂ space in the elliptic domain.     

Isoperimetric bounds for the first eigenvalue of the Laplacian

In this paper, generalizing an earlier result by Payne–Rayner, we prove an isoperimetric lower bound for the first eigenvalue of the Laplacian in the fixed membrane problem on a compact minimal surface in a Euclidean space R ⁿ with weakly connected boundary. We also prove an isoperimetric upper bound for the first eigenvalue of the Laplacian of an embedded closed hypersurface in R ⁿ .