Dual extremum principles and the hypercircle for biharmonic problems

The canonical Euler-Hamilton theory is used to establish the connection between extremum principles and the hypercircle for a class of biharmonic problems. An illustration of the results is provided by calculations for a clamped plate.     

Variational principles for linear mixed boundary value problems

This paper presents variational and bivariational bounds associated with the linear equation Aφ = f, with general mixed boundary conditions. The variational results bound the action f, φ> + boundary terms, while the bivariational results bound g, φ>, where g is an arbitrary function.     

On variational principles for linear initial value problems

Euler-Lagrange and Euler-Hamilton variational principles are presented for a class of linear initial value problems.     

Dual interior point methods for linear semidefinite programming problems

Dual interior point methods for solving linear semidefinite programming problems are proposed. These methods are an extension of dual barrier-projection methods for linear programs. It is shown that the proposed methods converge locally at a linear rate provided that the solutions to the primal and dual problems are nondegenerate.     

Dual coordinate step methods for linear network flow problems

We review a class of recently-proposed linear-cost network flow methods which are amenable to distributed implementation. All the methods in the class use the notion of-complementary slackness, and most do not explicitly manipulate any global objects such as paths, trees, or cuts. Interestingly, these methods have stimulated a large number of newserial computational complexity results. We develop the basic theory of these methods and present two specific methods, the-relaxation algorithm for the minimum-cost flow problem, and theauction algorithm for the assignment problem. We show how to implement these methods with serial complexities of O(N ³ logNC) and O(NA logNC), respectively. We also discuss practical implementation issues and computational experience to date. Finally, we show how to implement-relaxation in a completely asynchronous, chaotic environment in which some processors compute faster than others, some processors communicate faster than others, and there can be arbitrarily large communication delays.Supported by Grant NSF-ECS-8217668 and by the Army Research Office under grant DAAL03-86-K-0171. Thanks are due to David Castañon, Paul Tseng, and Jim Orlin for their helpful comments.     

Dual extremum principles and error bounds for a class of boundary value problems

Error bounds for a wide class of linear and nonlinear boundary value problems are derived from the theory of dual extremum principles. The results are illustrated by two examples arising in the theory of heat transfer, which involve mixed boundary conditions.     

Dual variational problems for boundary functionals of the linear theory of elasticity

Dual variational problem for use with the problem of minimization of the boundary functionals of three-dimensional theory of elasticity, is formulated using the method of orthogonal expansions at the boundary of the region constructed in /1/. Solutions of the initial and the dual problem obtained yield the estimates for the error of the approximate solutions of the boundary value problems of the theory of elasticity.     

Dual semigroups and second order linear elliptic boundary value problems

It is shown that general second order elliptic boundary value problems on bounded domains generate analytic semigroups onL ₁. The proof is based on Phillips’ theory of dual semigroups. Several sharp estimates for the corresponding semigroups inL _p, 1≦p<∞, are given.     

Complementary variational principles for linear equations

This paper presents a useful alternative to the classical complementary variational principles associated with the equation Aφ = f, where A is a bounded linear operator.     

Dual existence theorems for a special class of linear programming problems in reflexive Banach lattices

In this work the existence of dual optimal solutions for a special class of linear programming problems in a reflexive Banach space is investigated. Then these statements are applied to linear optimization problems with Noethebian operator-constraints. Finally, a maximal condition for an optimal control problem with Noehebiari operator: constraints its Derived in L _∞[0,T].

     

Minimax principles for convex eigenvalue problems

Positive solutions of the nonlinear eigenvalue problem Au = σu for a forced, convex, isotone, compact operator on a partially ordered locally convex topological vector space E are considered. Denote by $\text{σ}{}^1$ the infimum of the set of σ for which the equation has a solution u in the positive cone K of E. $\text{σ}{}^1$ is characterized as the saddle value of a functional J_A determined by A and defined on the Cartesian product of K and its dual $\text{K}{}^1$.     

Scale-independent complementary bivariational principles in a complex Hilbert space,and their application to a scattering problem

Scale-independent complementary bivariational principles in a complex Hilbert space are derived from the stationary principle. These principles consist of two scale-independent functionals which yield upper bounds and lower bounds, respectively, to both the real and the imaginary part of a particular quantity associated with an inhomogeneous linear equation. They have the advantage that one need only guess the form of solutions of the equation and its auxiliary equation, not their size. Moreover, for a given pair of trial functions, they yield better bounds than the scale-dependent complementary bivariational principles obtained by Barnsley and Baker. Their application to a scattering problem yields scale-independent complementary bivariational principles for the scattering amplitude as well as those for the total scattering cross section.     

Dual simplex method for GUB problems

The dual simplex method for generalized upper bound (GUB) problems is presented. One of the major operations in the dual simplex method is to update the elements of therth row, wherer is the index for the leaving basic variable. Those updated elements are used for the ratio test to determine the entering basic variabble. A very simple formula for therth row update for the dual simplex method for a GUB problem is derived, which is similar to the formula for the standard linear program. This derivation is based on the change key operation, which is to exchange the key column and its counterpart in the nonkey section. The change key operation is possible because of a theorem that guarantees the existence of such a counterpart.     

Positivity inheritance for linear problems

We discuss the numerical solution of positive Differential-Algebraic-Equations (DAEs). For Ordinary Differential Equations (ODEs) where the system matrix is a -M-Matrix, Runge-Kutta- or Multistep-Method are positive if the stepsize is chosen within the absolutely monotonicity radius of the considered method. We extend this concept to matrix pairs and present conditions for positivity preserving discretizations of linear, time-invariant DAEs. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some basic principles for linear coupled dynamic thermopiezoelectricity

According to the basic idea of classical yin-yang complementarity and modern dualcomplementarity, in a simple and unified way some basic principles for linear coupled dynamic thermopiezoelectricity can be established systematically. An important integral relation in terms of convolutions is given, which can be considered as the generalized principle of virtual work in mechanics. Based on this relation, it is possible not only to obtain the principle of virtual work and the reciprocal theorem in linear coupled dynamic thermopiezoelectricity, but also to derive systematically the complementary functionals for eleven-field, nine-field, six-field and three-field simplified Gurtin-type variational principles. Furthermore, with this approach, the intrinsic relationship among various principles can be explained clearly. Project supported by the National Natural Science Foundation of China (Grant No. 19672074) and Research Grand Council of Hong Kong, No.RGC97/98, HKUST 6055/97E.     

Variational principles and variational—iterative principles for a class of problems

Complementary variational principles are developed for linear equations in a function space and a variational-iterative scheme, based on these principles, is introduced for non-linear equations.     

Are linear algorithms always good for linear problems?

We exhibit linear problems for which every linear algorithm has infinite error, and show a (mildly) nonlinear algorithm with finite error. The error of this nonlinear algorithm can be arbitrarily small if appropriate information is used. We illustrate these examples by the inversion of a finite Laplace transform, a problem arising in remote sensing.     

A sign-based linear method for horizontal linear complementarity problems

Numerical Algorithms - For the horizontal linear complementarity problem, we establish a linear method based on the sign patterns of the solution of the equivalent modulus equation under the...