Newton additive and multiplicative Schwarz iterative methods

Daniel B. Szyld^¶ Department of Mathematics, Temple University, Philadelphia, PA 19122, USA ^{Convergence properties are presented for Newton additive andmultiplicative Schwarz (AS and MS) iterative methods for thesolution of nonlinear systems in several variables. These methodsconsist of approximate solutions of the linear Newton step usingeither AS or MS iterations, where overlap between subdomainscan be used. Restricted versions of these methods are also considered.These Schwarz methods can also be used to precondition a Krylovsubspace method for the solution of the linear Newton steps.Numerical experiments on parallel computers are presented, indicatingthe effectiveness of these methods.}     

Predicting overflow in an emergency department

G. B. Byrnes Centre for Molecular, Environmental, Genetic and Analytic Epidemiology, Department of Public Health, The University of Melbourne, Victoria, Australia C. A. Bain Directorate Office, Western and Central Melbourne Integrated Cancer Service, Victoria, Australia M. Fackrell Department of Mathematics and Statistics, The University of Melbourne, Victoria, Australia C. Brand Clinical Epidemiology and Health Service Evaluation Unit, Melbourne Health, Victoria, Australia D. A. Campbell Department of Medicine, Southern Clinical School, Monash University, Victoria, Australia P. G. Taylor Department of Mathematics and Statistics, The University of Melbourne, Victoria, Australia Email: l.au{at}ms.unimelb.edu.au Received on 9 October 2007. Accepted on 4 February 2008. Ambulance bypass occurs when the emergency department (ED) ofa hospital becomes so busy that ambulances are requested totake their patients elsewhere, except in life-threatening cases.It is a major concern for hospitals in Victoria, Australia,and throughout most of the western world, not only from thepoint of view of patient safety but also financially—hospitalslose substantial performance bonuses if they go on ambulancebypass too often in a given period. We show that the main causeof ambulance bypass is the inability to move patients from theED to a ward. In order to predict the onset of ambulance bypass,the ED is modelled as a queue for treatment followed by a queuefor a ward bed. The queues are assumed to behave as inhomogeneousPoisson arrival processes. We calculate the probability of reachingsome designated capacity C within time t, given the currenttime and number of patients waiting.     

Asymptotic behaviour of a finite-volume scheme for the transient drift-diffusion model

Francis Filbet Université de Lyon, Université Lyon 1, CNRS, UMR 5208, Institut Camille Jordan, 43 Boulevard du 11 Novembre 1918, 69622 Villeurbanne cedex, France Received on 28 June 2006. Revised on 6 December 2006. In this paper, we propose a finite-volume discretization formultidimensional nonlinear drift-diffusion system. Such a systemarises in semiconductors modelling and is composed of two parabolicequations and an elliptic one. We prove that the numerical solutionconverges to a steady state when time goes to infinity. Severalnumerical tests show the efficiency of the method.     

Randomized Smolyak algorithms based on digital sequences for multivariate integration

Gunther Leobacher ^{In this paper, we consider Smolyak algorithms based on quasi-MonteCarlo rules for high-dimensional numerical integration. Thequasi-Monte Carlo rules employed here use digital (t, , ß,, d)-sequences as quadrature points. We consider the worst-caseerror for multivariate integration in certain Sobolev spacesand show that our quadrature rules achieve the optimal rateof convergence. By randomizing the underlying digital sequences,we can also obtain a randomized Smolyak algorithm. The boundon the worst-case error holds also for the randomized algorithmin a statistical sense. Further, we also show that the randomizedalgorithm is unbiased and that the integration error can beapproximated as well.}     

An efficient algorithm for the Schrödinger–Poisson eigenvalue problem

We present a new implementation of the two-grid method for computing extremum eigenpairs of self-adjoint partial differential operators with periodic boundary conditions. A novel two-grid centered difference method is proposed for the numerical solutions of the nonlinear Schrödinger–Poisson (SP) eigenvalue problem.We solve the Poisson equation to obtain the nonlinear potential for the nonlinear Schrödinger eigenvalue problem, and use the block Lanczos method to compute the first k   eigenpairs of the Schrödinger eigenvalue problem until they converge on the coarse grid. Then we perform a few conjugate gradient iterations to solve each symmetric positive definite linear system for the approximate eigenvector on the fine grid. The Rayleigh quotient iteration is exploited to improve the accuracy of the eigenpairs on the fine grid. Our numerical results show how the first few eigenpairs of the Schrödinger eigenvalue problem are affected by the dopant in the Schrödinger–Poisson (SP) system. Moreover, the convergence rate of eigenvalue computations on the fine grid is O(h³)$\mathrm{O}(h^3)$.     

An integral method for solving nonlinear eigenvalue problems

We propose a numerical method for computing all eigenvalues (and the corresponding eigenvectors) of a nonlinear holomorphic eigenvalue problem that lie within a given contour in the complex plane. The method uses complex integrals of the resolvent operator, applied to at least k column vectors, where k is the number of eigenvalues inside the contour. The theorem of Keldysh is employed to show that the original nonlinear eigenvalue problem reduces to a linear eigenvalue problem of dimension k. No initial approximations of eigenvalues and eigenvectors are needed. The method is particularly suitable for moderately large eigenvalue problems where k is much smaller than the matrix dimension. We also give an extension of the method to the case where k is larger than the matrix dimension. The quadrature errors caused by the trapezoid sum are discussed for the case of analytic closed contours. Using well known techniques it is shown that the error decays exponentially with an exponent given by the product of the number of quadrature points and the minimal distance of the eigenvalues to the contour.     

Equivalence of communication and projective boundedness properties for monotone and homogeneous functions

This work concerns the eigenvalue problem for a monotone and homogeneous self-mapping f of a finite-dimensional positive cone. A communication criterion is formulated such that it is equivalent to the projective boundedness of the upper eigenspaces associated with f, a property that yields the existence of a nonlinear eigenvalue. Using the idea of dual function, a similar result is obtained for lower eigenspaces.     

A posteriori error control for finite element approximations of elliptic eigenvalue problems

We develop a new approach to a posteriori error estimation for Galerkin finite element approximations of symmetric and nonsymmetric elliptic eigenvalue problems. The idea is to embed the eigenvalue approximation into the general framework of Galerkin methods for nonlinear variational equations. In this context residual-based a posteriori error representations are available with explicitly given remainder terms. The careful evaluation of these error representations for the concrete situation of an eigenvalue problem results in a posteriori error estimates for the approximations of eigenvalues as well as eigenfunctions. These suggest local error indicators that are used in the mesh refinement process.     

Numerical approximation of anisotropic geometric evolution equations in the plane

Harald Garcke Naturwissenchaftliche Fakultät I' Mathematik, Universität Regensburg, 93040 Regensburg, Germany Robert Nürnberg Department of Mathematics, Imperial College London, London SW7 2AZ, UK Received on 13 April 2006. Revised on 20 February 2007. We present a variational formulation of fully anisotropic motionby surface diffusion and mean curvature flow, as well as relatedflows. The proposed scheme covers both the closed-curve caseand the case of curves that are connected via triple junctionpoints. On introducing a parametric finite-element approximation,we prove stability bounds and report on numerical experiments,including regularized crystalline mean curvature flow and regularizedcrystalline surface diffusion. The presented scheme has verygood properties with respect to the distribution of mesh pointsand, if applicable, area conservation.     

Critical orbits of symmetric functionals

Let G be a compact Lie group and V a G-module, i.e. a finite-dimensional real vector space on which G acts orthogonally. We are interested in finding G-orbits of critical points of G-invariant C²-functionals f: SV→—, SV the unit sphere of V. Using a generalization of the Borsuk-Ulam theorem by Komiya [15] we give lower bounds for the number of critical orbits with a given orbit type. These results are applied to nonlinear eigenvalue problems which are symmetric with respect to an action of O(3) or a closed subgroup of O(3).     

Analysis of supply contracts from a supplier's perspective

Hexin Wang and Khairy A. H. Kobbacy Centre for Operational Research and Applied Statistics, University of Salford, Salford, M5 4WT, UK Email: w.wang{at}salford.ac.uk Received on 9 May 2006. Accepted on 22 December 2006. Incentive structure and demand uncertainty may cause supplychains to operate at a low efficiency. Therefore, many supplycontracts are employed in practice to improve the performanceof supply chains, i.e. to benefit all members involved in thechain. Supply chain contracts provide mechanisms to change theincentive structures of the supply chain members so that theirdecisions can improve the supply chain efficiency, while alsoprotect their own interests. It is important to understand theimpacts of supply contracts and their differences from a supplier'sperspective, since it is often the supplier who initiates asupply contract. This paper reports on a comprehensive analysisof supply contracts from a supplier's perspective. Six commonlyused supply contracts are analysed and the contract parametersare optimized to maximize the supplier's expected profit withconsideration to improve the retailer's profit. This case hasnot been thoroughly investigated in literature to date. Therisk-sharing mechanism and the division of the increased profitbetween the retailer and supplier for some of the contractsare also investigated in detail.     

非线性特征值问题的二次近似方法

本文研究求解非线性特征值问题的数值方法.基于矩阵值函数的二次近似,将非线性特征值问题转化为二次特征值问题,提出了求解非线性特征值问题的逐次二次近似方法,分析了该方法的收敛性.结合求解二次特征值问题的Arnoldi方法和Jacobi-Davidson方法,给出求解非线性特征值问题的一些二次近似方法.数值结果表明本文所给算法是有效的.     

Spectrum of the 1‐Laplacian and Cheeger's Constant on Graphs

We develop a nonlinear spectral graph theory, in which the Laplace operator is replaced by the 1 ? Laplacian Δ₁. The eigenvalue problem is to solve a nonlinear system involving a set valued function. In the study, we investigate the structure of the solutions, the minimax characterization of eigenvalues, the multiplicity theorem, etc. The eigenvalues as well as the eigenvectors are computed for several elementary graphs. The graphic feature of eigenvalues are also studied. In particular, Cheeger's constant, which has only some upper and lower bounds in linear spectral theory, equals to the first nonzero Δ₁ eigenvalue for connected graphs.     

A duality theory for some non-convex functions of matrices

Abstract We study a special class of non-convex functions which appear in nonlinear elasticity, and we prove that they have a well-defined Legendre transform. Several examples are given, and an application to a nonlinear eigenvalue problem. Keywords: Duality, Legendre transform, Nonlinear elasticity Mathematics Subject Classification (2000): 05C38, 15A15, 05A15, 15A18     

Eigenvalue problems for nonlinear third‐order m‐point p‐Laplacian dynamic equations on time scales

This work deals with the existence and uniqueness of a nontrivial solution for the third‐order p‐Laplacian m‐point eigenvalue problems on time scales. We find several sufficient conditions of the existence and uniqueness of nontrivial solution of eigenvalue problems when λ is in some interval. The proofs are based on the nonlinear alternative of Leray–Schauder. To illustrate the results, some examples are included. Copyright © 2014 John Wiley & Sons, Ltd.     

Pricing vulnerable European options with stochastic default barriers

CF Lo and KC Ku Institute of Theoretical Physics and Department of Physics, The Chinese University of Hong Kong, Shatin, Hong Kong, China Email: cho-hoi_hui{at}hkma.gov.hk Received on 31 July 2006. Accepted on 15 March 2007. This paper develops a valuation model of European options incorporatinga stochastic default barrier, which extends a constant defaultbarrier proposed in the Hull–White model. The defaultbarrier is considered as an option writer's liability. Closed-formsolutions of vulnerable European option values based on themodel are derived to study the impact of the stochastic defaultbarriers on option values. The numerical results show that negativecorrelation between the firm values and the stochastic defaultbarriers of option writers gives material reductions in optionvalues where the options are written by firms with leverageratios corresponding to BBB or BB ratings.     

Verzweigung an Eigenwerten unendlicher Vielfachheit-Bemerkungen zu einer Arbeit von H.-P.Heinz

This article is dealing with two theorems on nonlinear eigenvalue problems given by H.-P.Heinz in [2]: The theorem 3.6 on bifurcation at an eigenvalue of infinite multiplicity is shown to be empty. The theorem 3.7, concerning eigenvalue problems with a homogeneous nonlinearity, is considerably improved. This results follow easily from a classical theorem on eigenvectors of nonlinear completely continuous operators.     

Traveling Wave Solutions for Combustion in a Porous Medium

Traveling wave solutions are sought for a model of combustion in a porous medium. The problem is formulated as a nonlinear eigenvalue problem for a system of ordinary differential equations of order four, defined over an infinite interval. A shooting method is used to prove existence, and a priori bounds for the solution and parameters are obtained.     

Analyzing the convergence factor of residual inverse iteration

We will establish here a formula for the convergence factor of the method called residual inverse iteration, which is a method for nonlinear eigenvalue problems and a generalization of the well-known inverse iteration. The formula for the convergence factor is explicit and involves quantities associated with the eigenvalue to which the iteration converges, in particular the eigenvalue and eigenvector. Residual inverse iteration allows for some freedom in the choice of a vector w _k and we can use the formula for the convergence factor to analyze how it depends on the choice of w _k . We also use the formula to illustrate the convergence when the shift is close to the eigenvalue. Finally, we explain the slow convergence for double eigenvalues by showing that under generic conditions, the convergence factor is one, unless the eigenvalue is semisimple. If the eigenvalue is semisimple, it turns out that we can expect convergence similar to the simple case.