有限元Aubin-Nitsche技巧新认识及其应用

首先回顾有限元理论中经典的Aubin-Nitsche技巧,并给出一个基于多尺度空间意义下的认识.然后基于这个新认识,重新得到一些依赖Aubin-Nitsche技巧的算法,使得Aubin-Nitsche技巧可以多次使用.作为例子,我们具体给出了在非对称椭圆方程和椭圆特征值问题方面的应用.     

On some questions arising in numerical realization of amplitude-phase methods

For computing rapidly oscillating solutions of certain second order differential equations a new version of amplitude-phase methods has recently been proposed in [11]. Error estimates were given to approximate solutions for large arguments in [15]. One of the most important points in these methods is the introduction of Prüfer transformation modified by auxiliary functions. Their appropriate choice makes the methods applicable and efficient. When implementing and applying the methods to practical problems, we face some further questions. In this paper we describe and try to answer them. Efficiency of the methods is confirmed by numerical experiments on concrete problems. This revised version was published online in June 2006 with corrections to the Cover Date.     

One-pass heuristics for large-scale unconstrained binary quadratic problems

Many significant advances have been made in recent years for solving unconstrained binary quadratic programs (UQP). As a result, the size of problem instances that can be efficiently solved has grown from a hundred or so variables a few years ago to 2000 or 3000 variables today. These advances have motivated new applications of the model which, in turn, have created the need to solve even larger problems. In response to this need, we introduce several new “one-pass” heuristics for solving very large versions of this problem. Our computational experience on problems of up to 9000 variables indicates that these methods are both efficient and effective for very large problems. The significance of problems of this size is that they not only open the door to solving a much wider array of real world problems, but also that the standard linear mixed integer formulations of the nonlinear models involve over 40,000,000 variables and three times that many constraints. Our approaches can be used as stand-alone solution methods, or they can serve as procedures for quickly generating high quality starting points for other, more sophisticated methods.     

求解非线性规划问题的两个微分方程系统

本文给出Evtushenko与Zhadan（1974）提出的求解数学规划问题微分方程系统的两个校正形式,它们可用于求解具有等式和不等式约束的非线性规化问题。第一个校正系统拓宽了Evtushenko与Zhadan微分方程方法;第二个校正系统通过引入新的方程系统导出乘子函数得到,它无需使用Evtushenko与Zhadan所用的那样强的约束规范。我们建立了这两个微分方程方法及其离散迭代方法的收敛性定理,给出了基于第二个微分方程离散格式的数值算法及其某些数值结果。     

Parallel ODE solvers based on block BVMs

In this paper we deal with Boundary Value Methods (BVMs), which are methods recently introduced for the numerical approximation of initial value problems for ODEs. Such methods, based on linear multistep formulae (LMF), overcome the stability limitations due to the well-known Dahlquist barriers, and have been the subject of much research in the last years. This has led to the definition of a new stability framework, which generalizes the one stated by Dahlquist for LMF. Moreover, several aspects have been investigated, including the efficient stepsize control [17,25,26] and the application of the methods for approximating different kinds of problems such as BVPs, PDEs and DAEs [7,23,41]. Furthermore, a block version of such methods, recently proposed for approximating Hamiltonian problems [24], is able to provide an efficient parallel solver for ODE systems [3].     

Second order Chebyshev methods based on orthogonal polynomials

Summary. Stabilized methods (also called Chebyshev methods) are explicit Runge-Kutta methods with extended stability domains along the negative real axis. These methods are intended for large mildly stiff problems, originating mainly from parabolic PDEs. The aim of this paper is to show that with the use of orthogonal polynomials, we can construct nearly optimal stability polynomials of second order with a three-term recurrence relation. These polynomials can be used to construct a new numerical method, which is implemented in a code called ROCK2. This new numerical method can be seen as a combination of van der Houwen-Sommeijer-type methods and Lebedev-type methods. Received January 14, 2000 / Revised version received November 3, 2000 / Published online May 4, 2001     

Monotone multigrid methods for elliptic variational inequalities II

Summary. We derive globally convergent multigrid methods for discrete elliptic variational inequalities of the second kind as obtained from the approximation of related continuous problems by piecewise linear finite elements. The coarse grid corrections are computed from certain obstacle problems. The actual constraints are fixed by the preceding nonlinear fine grid smoothing. This new approach allows the implementation as a classical V-cycle and preserves the usual multigrid efficiency. We give estimates for the asymptotic convergence rates. The numerical results indicate a significant improvement as compared with previous multigrid approaches. Received March 26, 1994 / Revised version received September 22, 1994     

Nonconforming finite elements and the cascadic multi-grid method

Summary. We derive sufficient conditions under which the cascadic multi-grid method applied to nonconforming finite element discretizations yields an optimal solver. Key ingredients are optimal error estimates of such discretizations, which we therefore study in detail. We derive a new, efficient modified Morley finite element method. Optimal cascadic multi-grid methods are obtained for problems of second, and using a new smoother, of fourth order as well as for the Stokes problem. Received February 12, 1998 / Revised version received January 9, 2001 / Published online September 19, 2001     

Smooth methods of multipliers for complementarity problems

This paper describes several methods for solving nonlinear complementarity problems. A general duality framework for pairs of monotone operators is developed and then applied to the monotone complementarity problem, obtaining primal, dual, and primal-dual formulations. We derive Bregman-function-based generalized proximal algorithms for each of these formulations, generating three classes of complementarity algorithms. The primal class is well-known. The dual class is new and constitutes a general collection of methods of multipliers, or augmented Lagrangian methods, for complementarity problems. In a special case, it corresponds to a class of variational inequality algorithms proposed by Gabay. By appropriate choice of Bregman function, the augmented Lagrangian subproblem in these methods can be made continuously differentiable. The primal-dual class of methods is entirely new and combines the best theoretical features of the primal and dual methods. Some preliminary computation shows that this class of algorithms is effective at solving many of the standard complementarity test problems. Received February 21, 1997 / Revised version received December 11, 1998? Published online May 12, 1999     

Determining the weights of criteria in the ELECTRE type methods with a revised Simos' procedure

In a decision aiding context, knowing the preferences of the Decision Maker (DM) and determining weights of criteria are very hard questions. Several methods can be used to give an appropriate value to the weights of criteria. J. Simos proposed a very simple procedure, using a set of cards, allowing to determine indirectly numerical values for weights. The purpose of this paper is first to explain why the above method needs to be revised, and second, the revised version we propose. This new version takes into account a new kind of information from the DM and changes certain computing rules of the former method. A software has been implemented based on the revised Simos' procedure whose main features are presented in this paper. The new method has been applied to different real-life cases (public transportation problems, water resources problems, environment problems, etc); it proved to be successful.     

Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions

Summary. Multilevel Schwarz methods are developed for a conforming finite element approximation of second order elliptic problems. We focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions. We establish a condition number estimate for the iterative operator, which is independent of the coefficients, and grows at most as the square of the number of levels. We also characterize a class of distributions of the coefficients, called quasi-monotone, for which the weighted -projection is stable and for which we can use the standard piecewise linear functions as a coarse space. In this case, we obtain optimal methods, i.e. bounds which are independent of the number of levels and subregions. We also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas. We consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods. Received April 6, 1994 / Revised version received December 7, 1994     

A new family of Runge–Kutta type methods for the numerical integration of perturbed oscillators

Summary. Our task in this paper is to present a new family of methods of the Runge–Kutta type for the numerical integration of perturbed oscillators. The key property is that those algorithms are able to integrate exactly, without truncation error, harmonic oscillators, and that, for perturbed problems the local error contains the perturbation parameter as a factor. Some numerical examples show the excellent behaviour when they compete with Runge–Kutta–Nystr?m type methods. Received June 12, 1997 / Revised version received July 9, 1998     

Synchronous approach in interactive multiobjective optimization

We introduce a new approach in the methodology development for interactive multiobjective optimization. The presentation is given in the context of the interactive NIMBUS method, where the solution process is based on the classification of objective functions. The idea is to formulate several scalarizing functions, all using the same preference information of the decision maker. Thus, opposed to fixing one scalarizing function (as is done in most methods), we utilize several scalarizing functions in a synchronous way. This means that we as method developers do not make the choice between different scalarizing functions but calculate the results of different scalarizing functions and leave the final decision to the expert, the decision maker. Simultaneously, (s)he obtains a better view of the solutions corresponding to her/his preferences expressed once during each iteration.In this paper, we describe a synchronous variant of the NIMBUS method. In addition, we introduce a new version of its implementation WWW-NIMBUS operating on the Internet. WWW-NIMBUS is a software system capable of solving even computationally demanding nonlinear problems. The new version of WWW-NIMBUS can handle versatile types of multiobjective optimization problems and includes new desirable features increasing its user-friendliness.     

Parallel Implementation of Synchronous Type Artificial Bee Colony Algorithm for Global Optimization

Evolutionary algorithms often need huge running times when solving large-scale optimization problems. One of the solutions for this issue is to introduce parallelization into the algorithm. To benefit from this approach for the artificial bee colony optimization algorithm, we present a new synchronous and parallel version of the algorithm. Performances of the proposed version and the original asynchronous algorithm are compared in terms of efficiency and speedup. Algorithms are competed to solve 20 large-scale global optimization problems. Comparative results show that the proposed parallel algorithm is still efficient as asynchronous version while it requires much less time to solve complex and large problems.     

Adaptive filtering

Summary. In this paper, the adaptive filtering method is introduced and analysed. This method leads to robust algorithms for the solution of systems of linear equations which arise from the discretisation of partial differential equations with strongly varying coefficients. These iterative algorithms are based on the tangential frequency filtering decompositions (TFFD). During the iteration with a preliminary preconditioner, the adaptive test vector method calculates new test vectors for the TFFD. The adaptive test vector iterative method allows the combination of the tangential frequency decomposition and other iterative methods such as multi-grid. The connection with the TFFD improves the robustness of these iterative methods with respect to varying coefficients. Interface problems as well as problems with stochastically distributed properties are considered. Realistic numerical experiments confirm the efficiency of the presented algorithms. Received June 26, 1996 / Revised version received October 7, 1996     

ROS3P—An Accurate Third-Order Rosenbrock Solver Designed for Parabolic Problems

In this note we present a new Rosenbrock solver which is third-order accurate for nonlinear parabolic problems. Since Rosenbrock methods suffer from order reduction when they are applied to partial differential equations, additional order conditions have to be satisfied. Although these conditions have been known for a longer time, from the practical point of view only little has been done to construct new methods. Steinebach modified the well-known solver RODAS of Hairer and Wanner to preserve its classical order four for special problem classes including linear parabolic equations. His solver RODASP, however, drops down to order three for nonlinear parabolic problems. Our motivation here was to derive an efficient third-order Rosenbrock solver for the nonlinear situation. Such a method exists with three stages and two function evaluations only. A comparison with other third-order methods shows the substantial potential of our new method.This revised version was published online in October 2005 with corrections to the Cover Date.     

Valid inequalities and facets of the capacitated plant location problem

Recently, several successful applications of strong cutting plane methods to combinatorial optimization problems have renewed interest in cutting plane methods, and polyhedral characterizations, of integer programming problems. In this paper, we investigate the polyhedral structure of the capacitated plant location problem. Our purpose is to identify facets and valid inequalities for a wide range of capacitated fixed charge problems that contain this prototype problem as a substructure.The first part of the paper introduces a family of facets for a version of the capacitated plant location problem with a constant capacity for all plants. These facet inequalities depend on the capacity and thus differ fundamentally from the valid inequalities for the uncapacited version of the problem.We also introduce a second formulation for a model with indivisible customer demand and show that it is equivalent to a vertex packing problem on a derived graph. We identify facets and valid inequalities for this version of the problem by applying known results for the vertex packing polytope.This research was partially supported by Grant # ECS-8316224 from the National Science Foundation's Program in Systems Theory and Operations Research.     

Convergence estimates of nonrestarted and restarted block‐Lanczos methods

The block‐Lanczos method serves to compute a moderate number of eigenvalues and the corresponding invariant subspace of a symmetric matrix. In this paper, the convergence behavior of nonrestarted and restarted versions of the block‐Lanczos method is analyzed. For the nonrestarted version, we improve an estimate by Saad by means of a change of the auxiliary vector so that the new estimate is much more accurate in the case of clustered or multiple eigenvalues. For the restarted version, an estimate by Knyazev is generalized by extending our previous results on block steepest descent iterations and single‐vector restarted Krylov subspace iterations. The new estimates can also be reformulated and applied to invert‐block‐Lanczos methods for solving generalized matrix eigenvalue problems.     

A numerical scheme for highly oscillatory DAEs and its application to circuit simulation

A new numerical integration scheme for the simulation of differential–algebraic equations is presented. In the context of the computer-aided design of electronic circuits, the modeling of highly oscillatory circuits leads to oscillatory differential–algebraic equations mostly of index 1 or 2. Standard schemes can solve these equations neither efficiently nor reliably. The new discretiziation scheme is constructed in such a way as to overcome the problems of classical numerical methods. It uses the Principle of Coherence due to Hersch in combination with a multistep approach. A combined Maple and Fortran77 implementation of the presented integration scheme reduces the simulation time for a quartz-controlled oscillator to about 2% compared with standard methods. Therefore, it is a powerful tool for the design of highly oscillatory circuits. This revised version was published online in June 2006 with corrections to the Cover Date.     

Global optimization by controlled random search

The paper describes a new version, known as CRS2, of the author's controlled random search procedure for global optimization (CRS). The new procedure is simpler and requires less computer storage than the original version, yet it has a comparable performance. The results of comparative trials of the two procedures, using a set of standard test problems, are given. These test problems are examples of unconstrained optimization. The controlled random search procedure can also be effective in the presence of constraints. The technique of constrained optimization using CRS is illustrated by means of examples taken from the field of electrical engineering.