A Jacobi–Davidson method for two‐real‐parameter nonlinear eigenvalue problems arising from delay‐differential equations

The critical delays of a delay‐differential equation can be computed by solving a nonlinear two‐parameter eigenvalue problem. The solution of this two‐parameter problem can be translated to solving a quadratic eigenvalue problem of squared dimension. We present a structure preserving QR‐type method for solving such quadratic eigenvalue problem that only computes real‐valued critical delays; that is, complex critical delays, which have no physical meaning, are discarded. For large‐scale problems, we propose new correction equations for a Newton‐type or Jacobi–Davidson style method, which also forces real‐valued critical delays. We present three different equations: one real‐valued equation using a direct linear system solver, one complex valued equation using a direct linear system solver, and one Jacobi–Davidson style correction equation that is suitable for an iterative linear system solver. We show numerical examples for large‐scale problems arising from PDEs. Copyright © 2012 John Wiley & Sons, Ltd.     

A Jacobi-Davidson method for two real parameter nonlinear eigenvalue problems arising from delay differential equations

The critical delays of a delay-differential equation can be computed by solving a nonlinear two-parameter eigenvalue problem. For large scale problems, we propose new correction equations for a Jacobi-Davidson type method, that also forces real valued critical delays. We present two different equations: one complex valued equation using a direct linear system solver, and one Jacobi-Davidson style correction equation which is suitable for an iterative linear system solver. A numerical example of a large scale problem arising from PDEs shows the effectiveness of the method. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Long-term optimal operation of series parallel reservoirs for critical period with specified monthly generation and average monthly storage

We present in this paper an efficient approach for solving the problem of planning the long-term (multiyear) operation of a multireservoir hydroelectric power system for the critical period with a monthly variable load. This load is equal to a certain percentage of the total generation at the end of the year, subject to satisfying a number of constraints on the hydrosystem, using the minimum norm formulation.The proposed method is efficient in computing time and in calculating the total expected benefits from the system during the critical period. Numerical results are reported for a real system in operation consisting of two rivers. Each river has two series reservoirs.This work was supported by the Natural Science and Engineering Research Council of Canada, Grant No. A4146.     

Modelling and optimization of parallel reservoirs having nonlinear storage curves under critical water conditions for long-term regulation

This paper deals with the problem of parallel reservoirs having nonlinear storage-elevation curves (quadratic functions) for long-term regulation under critical water conditions using the minimum norm formulation. To overcome these nonlinearities, we introduce a set of pseudo-state variables. A set of optimizing equations is obtained. The proposed method is efficient in computing time and in calculating the expected benefits of generation from the system during the critical period. Numerical results are reported for a real system in operation consisting of two rivers; each river has two reservoirs in series.This work was supported by the National Research Council of Canada, Grant No. A4146. The authors would like to acknowledge data obtained from B. C. Hydro.     

High-energy positive solutions for a critical growth Dirichlet problem in noncontractible domains

In this paper, we consider the Dirichlet problem for a critical growth elliptic system of two equations in a noncontractible domain. By means of variational method, we establish the existence of positive solutions with high energy.     

Determination of the equilibrium boundaries of a two-layer system convection stability : PMM vol. 43, no. 6, 1979, pp. 1008–1013

The method which permits the determination of a large number of critical Rayleigh numbers and respective critical motions is applied to the determination of convection stability boundaries of a two-layer system. The method consists essentially of reducing the problem to the algebraic problem of eigenvalues by discretizing the equations by the method of finite elements or finite differences. Only the scheme of the method based on discretization by finite elements is then presented.     

A differential equation approach to nonlinear programming

A new method is presented for finding a local optimum of the equality constrained nonlinear programming problem. A nonlinear autonomous system is introduced as the base of the theory instead of usual approaches. The relation between critical points and local optima of the original optimization problem is proved. Asymptotic stability of the critical points is also proved. A numerical algorithm which is capable of finding local optima systematically at the quadratic rate of convergence is developed from a detailed analysis of the nature of trajectories and critical points. Some numerical results are given to show the efficiency of the method.     

Duality algorithms for nonconvex variational principles

This paper describes, and analyzes, a method of successive approximations for finding critical points of a function which can be written as the difference of two convex functions. The method is based on using a non-convex duality theory. At each iteration one solves a convex, optimization problem. This alternates between the primal and the dual variables. Under very general structural conditions on the problem, we prove that the resulting sequence is a descent sequence, which converges to a critical point of the problem. To illustrate the method, it is applied to some weighted eigenvalue problems, to a problem from astrophysics, and to some semilinear elliptic equations.     

A Bilevel p-median model for the planning and protection of critical facilities

The bilevel p-median problem for the planning and protection of critical facilities involves a static Stackelberg game between a system planner (defender) and a potential attacker. The system planner determines firstly where to open p critical service facilities, and secondly which of them to protect with a limited protection budget. Following this twofold action, the attacker decides which facilities to interdict simultaneously, where the maximum number of interdictions is fixed. Partial protection or interdiction of a facility is not possible. Both the defender’s and the attacker’s actions have deterministic outcome; i.e., once protected, a facility becomes completely immune to interdiction, and an attack on an unprotected facility destroys it beyond repair. Moreover, the attacker has perfect information about the location and protection status of facilities; hence he would never attack a protected facility. We formulate a bilevel integer program (BIP) for this problem, in which the defender takes on the leader’s role and the attacker acts as the follower. We propose and compare three different methods to solve the BIP. The first method is an optimal exhaustive search algorithm with exponential time complexity. The second one is a two-phase tabu search heuristic developed to overcome the first method’s impracticality on large-sized problem instances. Finally, the third one is a sequential solution method in which the defender’s location and protection decisions are separated. The efficiency of these three methods is extensively tested on 75 randomly generated instances each with two budget levels. The results show that protection budget plays a significant role in maintaining the service accessibility of critical facilities in the worst-case interdiction scenario.     

On the bottleneck assignment problem

In this paper, a new approach for solving the bottleneck assignment problem is presented. The problem is treated as a special class of permutation problems which we call max-min permutation problems. By defining a suitable neighborhood system in the space of permutations and designating certain permutations as critical solutions, it is shown that any critical solution yields a global optimum. This theorem is then used as a basis to develop a general method to solve max-min permutation problems.This work was carried out by the junior author while holding a Purdue University Fellowship.     

The problem of the equilibrium of a plate reinforced with stiffeners

The problem of the equilibrium of a non-linear plate reinforced with stiffeners is considered. The idea of a generalized solution of the problem as a critical point of the energy functional of an elastic system is introduced and the existence of a generalized solution of the problem is proved. The convergence of Ritz' method within the framework of this problem and also of the conformal versions of the finite-element method, constructed on the basis of Ritz' method, is validated. Similar problems were discussed in [1–3].     

临界群与二阶差分方程解的多重性

研究了一类二阶非线性差分方程两点边值问题解的多重性.当该问题的非线性项在无穷远点具有特殊的渐近线性性质时,利用变分方法,结合临界群与Morse理论,同时考虑正、负能量泛函的临界点,不论该问题是否发生共振,均证明了它至少存在两个非零解.     

Laguerre collocation solutions to boundary layer type problems

In this paper a Laguerre collocation type method based on usual Laguerre functions is designed in order to solve high order nonlinear boundary value problems as well as eigenvalue problems, on semi-infinite domain. The method is first applied to Falkner–Skan boundary value problem. The solution along with its first two derivatives are computed inside the boundary layer on a fine grid which cluster towards the fixed boundary. Then the method is used to solve a generalized eigenvalue problem which arise in the study of the stability of the Ekman boundary layer. The method provides reliable numerical approximations, is robust and easy implementable. It introduces the boundary condition at infinity without any truncation of the domain. A particular attention is payed to the treatment of boundary conditions at origin. The dependence of the set of solutions to Falkner–Skan problem on the parameter embedded in the system is reproduced correctly. For Ekman eigenvalue problem, the critical Reynolds number which assure the linear stability is computed and compared with existing results. The leftmost part of the spectrum is validated using QZ as well as some Jacobi–Davidson type methods.     

A fast and accurate algorithm for computing radial transonic flows

An efficient algorithm is described for calculating stationary one-dimensional transonic outflow solutions of the compressible Euler equations with gravity and heat source terms. The stationary equations are solved directly by exploiting their dynamical system form. Transonic expansions are the stable manifolds of saddle-point-type critical points, and can be obtained efficiently and accurately by adaptive integration outward from the critical points. The particular transonic solution and critical point that match the inflow boundary conditions are obtained by a two-by-two Newton iteration which allows the critical point to vary within the manifold of possible critical points. The proposed Newton Critical Point (NCP) method typically converges in a small number of Newton steps, and the adaptively calculated solution trajectories are highly accurate. A sample application area for this method is the calculation of transonic hydrodynamic escape flows from extrasolar planets and the early Earth. The method is also illustrated for an example flow problem that models accretion onto a black hole with a shock.     

High energy positive solutions of Neumann problem for an elliptic system of equations with critical nonlinearities

In this paper, we consider the Neumann problem for an elliptic system of two equations involving the critical Sobolev exponents. By means of variational method, we establish the existence of positive solutions with high energy. Supported by Special Funds For Major States Basic Research Projects of China(G1999075107) and Knowledge Innovation Funds of CAS in China.     

Elliptic problems with discontinuities

In this paper we prove two existence theorems for elliptic problems with discontinuities. The first one is a noncoercive Dirichlet problem and the second one is a Neumann problem. We do not use the method of upper and lower solutions. For Neumann problems we assume that f is nondecreasing. We use the critical point theory for locally Lipschitz functionals.     

Least-squares method in the theory of ill-posed linear boundary-value problems with pulse action

We use the scheme of the classic least-squares method for the construction of an approximate pseudosolution of a linear ill-posed boundary-value problem with pulse action for a system of ordinary differential equations in the critical case. The pseudosolution obtained is represented in the form of partial sums of a generalized Fourier series.     

A subspace shift technique for nonsymmetric algebraic Riccati equations associated with an M‐matrix

The worst situation in computing the minimal nonnegative solution of a nonsymmetric algebraic Riccati equation associated with an M‐matrix occurs when the corresponding linearizing matrix has two very small eigenvalues, one with positive and one with negative real part. When both eigenvalues are exactly zero, the problem is called critical or null recurrent. Although in this case the problem is ill‐conditioned and the convergence of the algorithms based on matrix iterations is slow, there exist some techniques to remove the singularity and transform the problem to a well‐behaved one. Ill‐conditioning and slow convergence appear also in close‐to‐critical problems, but when none of the eigenvalues is exactly zero, the techniques used for the critical case cannot be applied. In this paper, we introduce a new method to accelerate the convergence properties of the iterations also in close‐to‐critical cases, by working on the invariant subspace associated with the problematic eigenvalues as a whole. We present numerical experiments that confirm the efficiency of the new method.Copyright © 2012 John Wiley & Sons, Ltd.     

Reliability Estimation of Optimal Viscoelastic Composite Shells in Critical-Time Calculations

The problem of reliability estimation for optimal viscoelastic composite shells in critical-time calculations is considered. The weight minimization of a viscoelastic composite shell, with constraints on deflections at the critical time, leads to a system deforming unstably, whose deflections grow in time with an increasing rate. A method for estimating the reliability of such shells in calculating the critical time is discussed. This time is regarded as a random variable depending on many, roughly equivalent, factors. An analysis of the reliability is carried out for different values of coefficients of variation, occurring in practice.