共查询到20条相似文献,搜索用时 31 毫秒
1.
We consider the construction of a special family of Runge–Kutta(RK) collocation methods based on intra-step nodal points ofChebyshev–Gauss–Lobatto type, with A-stability andstiffly accurate characteristics. This feature with its inherentimplicitness makes them suitable for solving stiff initial-valueproblems. In fact, the two simplest cases consist in the well-knowntrapezoidal rule and the fourth-order Runge–Kutta–LobattoIIIA method. We will present here the coefficients up to eighthorder, but we provide the formulas to obtain methods of higherorder. When the number of stages is odd, we have considereda new strategy for changing the step size based on the use ofa pair of methods: the given RK method and a linear multistepone. Some numerical experiments are considered in order to checkthe behaviour of the methods when applied to a variety of initial-valueproblems. 相似文献
2.
In this paper we discuss a class of numerical algorithms termed one-leg methods. This concept was introduced by Dahlquist in 1975 with the purpose of studying nonlinear stability properties of multistep methods for ordinary differential equations. Later, it was found out that these methods are themselves suitable for numerical integration because of good stability. Here, we investigate one-leg formulas on nonuniform grids. We prove that there exist zero-stable one-leg variable-coefficient methods at least up to order 11 and give examples of two-step methods of orders 2 and 3. In this paper we also develop local and global error estimation techniques for one-leg methods and implement them with the local–global step size selection suggested by Kulikov and Shindin in 1999. The goal of this error control is to obtain automatically numerical solutions for any reasonable accuracy set by the user. We show that the error control is more complicated in one-leg methods, especially when applied to stiff problems. Thus, we adapt our local–global step size selection strategy to one-leg methods. 相似文献
3.
LiPingZHANG JiYeHAN ZhengHaiHUANG 《数学学报(英文版)》2005,21(1):117-128
We propose a one-step smoothing Newton method for solving the non-linear complementarity problem with P0-function (P0-NCP) based on the smoothing symmetric perturbed Fisher function(for short, denoted as the SSPF-function). The proposed algorithm has to solve only one linear system of equations and performs only one line search per iteration. Without requiring any strict complementarity assumption at the P0-NCP solution, we show that the proposed algorithm converges globally and superlinearly under mild conditions. Furthermore, the algorithm has local quadratic convergence under suitable conditions. The main feature of our global convergence results is that we do not assume a priori the existence of an accumulation point. Compared to the previous literatures, our algorithm has stronger convergence results under weaker conditions. 相似文献
4.
In this paper we investigate POD discretizations of abstract linear–quadratic optimal control problems with control constraints.
We apply the discrete technique developed by Hinze (Comput. Optim. Appl. 30:45–61, 2005) and prove error estimates for the corresponding discrete controls, where we combine error estimates for the state and the
adjoint system from Kunisch and Volkwein (Numer. Math. 90:117–148, 2001; SIAM J. Numer. Anal. 40:492–515, 2002). Finally, we present numerical examples that illustrate the theoretical results. 相似文献
5.
This paper is concerned with the numerical solution of a symmetric indefinite system which is a generalization of the Karush–Kuhn–Tucker system. Following the recent approach of Luk?an and Vl?ek, we propose to solve this system by a preconditioned conjugate gradient (PCG) algorithm and we devise two indefinite preconditioners with good theoretical properties. In particular, for one of these preconditioners, the finite termination property of the PCG method is stated. The PCG method combined with a parallel version of these preconditioners is used as inner solver within an inexact Interior‐Point (IP) method for the solution of large and sparse quadratic programs. The numerical results obtained by a parallel code implementing the IP method on distributed memory multiprocessor systems enable us to confirm the effectiveness of the proposed approach for problems with special structure in the constraint matrix and in the objective function. Copyright © 2002 John Wiley & Sons, Ltd. 相似文献
6.
A. G. Bratsos 《Numerical Algorithms》2006,43(4):295-308
A three-time level finite-difference scheme based on a fourth order in time and second order in space approximation has been
proposed for the numerical solution of the nonlinear two-dimensional sine-Gordon equation. The method, which is analysed for
local truncation error and stability, leads to the solution of a nonlinear system. To avoid solving it, a predictor–corrector
scheme using as predictor a second-order explicit scheme is proposed. The procedure of the corrector has been modified by
considering as known the already evaluated corrected values instead of the predictor ones. This modified scheme has been tested
on the line and circular ring soliton and the numerical experiments have proved that there is an improvement in the accuracy
over the standard predictor–corrector implementation.
This research was co-funded by E.U. (75%) and by the Greek Government (25%). 相似文献
7.
In this paper we propose and analyse numerical methods for the approximation of the solution of Helmholtz transmission problems
in the half plane. The problems we deal with arise from the study of some models in photothermal science. The solutions to
the problem are represented as single layer potentials and an equivalent system of boundary integral equations is derived.
We then give abstract necessary and sufficient conditions for convergence of Petrov–Galerkin discretizations of the boundary
integral system and show for three different cases that these conditions are satisfied. We extend the results to other situations
not related to thermal science and to non-smooth interfaces. Finally, we propose a simple full discretization of a Petrov–Galerkin
scheme with periodic spline spaces and show some numerical experiments. 相似文献
8.
In this paper, we first discuss how the nearly exact (NE) method proposed by Moré and Sorensen [14] for solving trust region
(TR) subproblems can be modified to solve large-scale “low-rank” TR subproblems efficiently. Our modified algorithm completely
avoids computation of Cholesky factorizations by instead relying primarily on the Sherman–Morrison–Woodbury formula for computing
inverses of “diagonal plus low-rank” type matrices. We also implement a specific version of the modified log-barrier (MLB)
algorithm proposed by Polyak [17] where the generated log-barrier subproblems are solved by a trust region method. The corresponding
direction finding TR subproblems are of the low-rank type and are then solved by our modified NE method. We finally discuss
the computational results of our implementation of the MLB method and its comparison with a version of LANCELOT [5] based
on a collection extracted from CUTEr [12] of nonlinear programming problems with simple bound constraints.
相似文献
9.
We consider three numerical methods – one based on power series, one on the Magnus series and matrix exponentials, and one
a library initial value code – for solving a linear system arising in non‐selfadjoint ODE eigenproblems. We show that in general,
none of these methods has a cost or an accuracy which is uniform in the eigenparameter, but that for certain special types
of problem, the Magnus method does yield eigenparameter‐uniform accuracy. This property of the Magnus method is explained
by a trajectory‐shadowing result which, unfortunately, does not generalize to higher order Magnus type methods such as those
in [11,12].
This revised version was published online in June 2006 with corrections to the Cover Date. 相似文献
10.
The Douglas–Peaceman–Rachford–Varga operator splitting methods (DPRV methods) are attractive methods for monotone variational
inequalities. He et al. [Numer. Math. 94, 715–737 (2003)] proposed an inexact self-adaptive operator splitting method based on DPRV. This paper relaxes the inexactness
restriction further. And numerical experiments indicate the improvement of this relaxation.
相似文献
11.
Avoiding Order Reduction of Runge–Kutta Discretizations for Linear Time-Dependent Parabolic Problems
A technique is developed in this paper to avoid order reduction when discretizing linear parabolic problems with time dependent
operator using Runge–Kutta methods in time and standard schemes in space. In an abstract framework, the boundaries of the
stages of the Runge–Kutta method which would completely avoid the order reduction are given. Then, the possible practical
implementations for the calculus of those boundaries from the given data are studied, and the full discretization is completely
analyzed. Some numerical experiments are included.
This revised version was published online in July 2006 with corrections to the Cover Date. 相似文献
12.
Silvia Bonettini Emanuele Galligani Valeria Ruggiero 《Computational Optimization and Applications》2007,37(1):1-34
This paper deals with the solution of nonlinear programming problems arising from elliptic control problems by an interior
point scheme. At each step of the scheme, we have to solve a large scale symmetric and indefinite system; inner iterative
solvers, with an adaptive stopping rule, can be used in order to avoid unnecessary inner iterations, especially when the current
outer iterate is far from the solution.
In this work, we analyse the method of multipliers and the preconditioned conjugate gradient method as inner solvers for interior
point schemes. We discuss the convergence of the whole approach, the implementation details and report the results of numerical
experimentation on a set of large scale test problems arising from the discretization of elliptic control problems. A comparison
with other interior point codes is also reported.
This research was supported by the Italian Ministry for Education, University and Research (MIUR) projects: FIRB Project:
“Parallel Nonlinear Numerical Optimization PN
2
O” (grant n. RBAU01JYPN, ) and COFIN/PRIN04 Project “Numerical Methods and Mathematical Software for Applications” (grant n. 2004012559, ). 相似文献
13.
In this paper, we propose two efficient numerical integration processes for initial value problems of ordinary differential
equations. The first algorithm is the Legendre–Gauss collocation method, which is easy to be implemented and possesses the
spectral accuracy. The second algorithm is a mixture of the collocation method coupled with domain decomposition, which can
be regarded as a specific implicit Legendre–Gauss Runge–Kutta method, with the global convergence and the spectral accuracy.
Numerical results demonstrate the spectral accuracy of these approaches and coincide well with theoretical analysis.
相似文献
14.
A Logarithmic-Quadratic Proximal Prediction-Correction Method for Structured Monotone Variational Inequalities 总被引:1,自引:0,他引:1
Inspired by the Logarithmic-Quadratic Proximal (LQP) method for variational inequalities, we present a prediction-correction
method for structured monotone variational inequalities. Each iteration of the new method consists of a prediction and a correction.
Both the predictor and the corrector are obtained easily with tiny computational load. In particular, the LQP system that
appears in the prediction is approximately solved under significantly relaxed inexactness restriction. Global convergence
of the new method is proved under mild assumptions. In addition, we present a self-adaptive version of the new method that
leads to easier implementations. Preliminary numerical experiments for traffic equilibrium problems indicate that the new
method is effectively applicable in practice.
Presented at the 6th International conference on Optimization: Techniques and Applications, Ballarat Australia, December 9–11,
2004.
This author was supported by NSFC Grant 10571083, the MOEC grant 20020284027 and Jiangsu NSF grant BK2002075 相似文献
15.
Ralf Werner 《Central European Journal of Operations Research》2008,16(2):179-189
It is well known that the robust counterpart introduced by Ben-Tal and Nemirovski (Math Oper Res 23:769–805, 1998) increases
the numerical complexity of the solution compared to the original problem. Kočvara, Nemirovski and Zowe therefore introduced
in Kočvara et al. (Comput Struct 76:431–442, 2000) an approximation algorithm for the special case of robust material optimization,
called cascading. As the title already indicates, we will show that their method can be seen as an adjustment of standard exchange methods
to semi-infinite conic programming. We will see that the adjustment can be motivated by a suitable reformulation of the robust
conic problem.
相似文献
16.
S. Cafieri M. D’Apuzzo M. Marino A. Mucherino G. Toraldo 《Journal of Optimization Theory and Applications》2006,129(1):55-75
In this paper, we present an interior-point algorithm for large and sparse convex quadratic programming problems with bound
constraints. The algorithm is based on the potential reduction method and the use of iterative techniques to solve the linear
system arising at each iteration. The global convergence properties of the potential reduction method are reassessed in order
to take into account the inexact solution of the inner system. We describe the iterative solver, based on the conjugate gradient
method with a limited-memory incomplete Cholesky factorization as preconditioner. Furthermore, we discuss some adaptive strategies
for the fill-in and accuracy requirements that we use in solving the linear systems in order to avoid unnecessary inner iterations
when the iterates are far from the solution. Finally, we present the results of numerical experiments carried out to verify
the effectiveness of the proposed strategies. We consider randomly generated sparse problems without a special structure.
Also, we compare the proposed algorithm with the MOSEK solver.
Research partially supported by the MIUR FIRB Project RBNE01WBBB “Large-Scale Nonlinear Optimization.” 相似文献
17.
In this paper, we analyze the outer approximation property of the algorithm for generalized semi-infinite programming from
Stein and Still (SIAM J. Control Optim. 42:769–788, 2003). A simple bound on the regularization error is found and used to formulate a feasible numerical method for generalized semi-infinite programming with convex lower-level problems. That is, all iterates of the
numerical method are feasible points of the original optimization problem. The new method has the same computational cost
as the original algorithm from Stein and Still (SIAM J. Control Optim. 42:769–788, 2003). We also discuss the merits of this approach for the adaptive convexification algorithm, a feasible point method for standard
semi-infinite programming from Floudas and Stein (SIAM J. Optim. 18:1187–1208, 2007). 相似文献
18.
D. S. Filippychev 《Computational Mathematics and Modeling》1999,10(1):61-73
We study the problem of the behavior of a plasma bounded longitudinally by an absorbing sheath. This model contains charged
particles (electrons and ions) moving subject to a self-consistent electrostatic field. New particle pairs are generated in
the region of a distributed source. As a numerical model we used the electrostatic “particle-in-cell” method supplemented
by the Emmert model for a bulk source and the algorithm of binary Coulomb collisions using the Monte Carlo method. We give
a mathematical statement of the problem. The computations were carried out using the direct implicit method with the “explicit
limit” time step. The results of numerical simulation of this system are given. We consider the formation and evoluiton of
potential structures (multiple weak nonmonotonic double layers). Five figures. Bibliography: 35 titles.
Translated fromProblemy Matematicheskoi Fiziki, 1998, pp. 75–89. 相似文献
19.
Any real-valued nonlinear function in 0–1 variables can be rewritten as a multilinear function. We discuss classes of lower
and upper bounding linear expressions for multilinear functions in 0–1 variables. For any multilinear inequality in 0–1 variables,
we define an equivalent family of linear inequalities. This family contains the well-known system of generalized covering
inequalities, as well as other linear equivalents of the multilinear inequality that are more compact, i.e., of smaller cardinality.
In a companion paper [7]. we discuss dominance relations between various linear equivalents of a multilinear inequality, and
describe a class of algorithms for multilinear 0–1 programming based on these results.
Research supported by the National Science Foundation under grant ECS7902506 and by the Office of Naval Research under contract
N00014-75-C-0621 NR 047-048. 相似文献
20.
We derive a numerical scheme to compute invariant manifolds for time-variant discrete dynamical systems, i.e., nonautonomous
difference equations. Our universally applicable method is based on a truncated Lyapunov–Perron operator and computes invariant
manifolds using a system of nonlinear algebraic equations which can be solved both locally using (nonsmooth) inexact Newton,
and globally using continuation algorithms. Compared to other algorithms, our approach is quite flexible, since it captures
time-dependent, nonsmooth, noninvertible or implicit equations and enables us to tackle the full hierarchy of strongly stable,
stable and center-stable manifolds, as well as their unstable counterparts. Our results are illustrated using a test example
and are applied to a population dynamical model and the Hénon map. Finally, we discuss a linearly implicit Euler–Bubnov–Galerkin
discretization of a reaction diffusion equation in order to approximate its inertial manifold. 相似文献