Projected Tikhonov Regularization of Large-Scale Discrete Ill-Posed Problems

The solution of linear discrete ill-posed problems is very sensitive to perturbations in the data. Confidence intervals for solution coordinates provide insight into the sensitivity. This paper presents an efficient method for computing confidence intervals for large-scale linear discrete ill-posed problems. The method is based on approximating the matrix in these problems by a partial singular value decomposition of low rank. We investigate how to choose the rank. Our analysis also yields novel approaches to the solution of linear discrete ill-posed problems with solution norm or residual norm constraints.     

Shape optimization of continua using NURBS as basis functions

The present paper introduces a numerical solution to shape optimization problems of domains in which boundary value problems of partial differential equations are defined. In the present paper, the finite element method using NURBS as basis functions in the Galerkin method is applied to solve the boundary value problems and to solve a reshaping problem generated by the H1 gradient method for shape optimization, which has been developed as a general solution to shape optimization problems. Numerical examples of linear elastic continua illustrate that this solution works as well as using the conventional finite element method.     

Error estimation and control for ODEs

This article is about the numerical solution of initial value problems for systems of ordinary differential equations. At first these problems were solved with a fixed method and constant step size, but nowadays the general-purpose codes vary the step size, and possibly the method, as the integration proceeds. Estimating and controlling some measure of error by variation of step size/method inspires some confidence in the numerical solution and makes possible the solution of hard problems. Common ways of doing this are explained briefly in the article.     

A novel finite element method for two-point boundary value problems

A finite element method is presented for solving boundary value problems for ordinary differential equations in which the general solution of the differential equation is computed first, followed by a selection procedure for the particular solution of the boundary value problem from the general solution. In this method, the discrete representation of the differential equation is a singular matrix equation, which is solved by using generalized matrix inversion. The technique is applied to both linear and nonlinear boundary value problems and to boundary value problems requiring eigenvalue evaluation. The solution of several examples involving different types of two-point boundary value problems is presented.     

Error Estimation and Control for ODEs

This article is about the numerical solution of initial value problems for systems of ordinary differential equations. At first these problems were solved with a fixed method and constant step size, but nowadays the general-purpose codes vary the step size, and possibly the method, as the integration proceeds. Estimating and controlling some measure of error by variation of step size/method inspires some confidence in the numerical solution and makes possible the solution of hard problems. Common ways of doing this are explained briefly in the article     

Solution of mixed boundary value problems with local error bound by the finite element method

A method of analysis using finite element techniques is presented for second order, mixed boundary value problems in the plane. The technique focuses computational effort on specific points in the domain and provides absolute solution error bounds at those points by applying the hypercircle method. Solution error of less than 0.0003% and solution error bounds of ± 0.012% are obtained in sample problems. The solution accuracy is notably superior to what is obtained in the traditional finite element method with equivalent discretization. Two problems are presented to illustrate both the strengths and weaknesses of the method.     

Comment on “Stability of linear stationary systems with time delay”

Quadratic programming (QP) has previously been applied to the computation of optimal controls for linear systems with quadratic cost criteria. This paper extends the application of QP to non-linear problems through quasi-linearization and the solution of a sequence of linear-quadratic sub-problems whose solutions converge to the solution of the original non-linear problem. The method is called quasi-linearization-quadratic programming or Q-QP.

The principal advantage of the Q-QP method lies in the ease with which optimal controls can be computed when saturation constraints are imposed on the control signals and terminal constraints are imposed on the state vector. Use of a bounded-variable QP algorithm permits solution of constrained problems with a negligible increase in computing time over the corresponding unconstrained problems. Numerical examples show how the method can be applied to certain problems with non-analytic objective functions and illustrate the facility of the method on problems with constraints. The Q-QP method is shown to be competitive with other methods in computation time for unconstrained problems and to be essentially unaffected in speed for problems having saturation and terminal constraints     

A new algorithm for numerical solution of dynamic elastic-plastic hardening and softening problems

The objective of this paper is to develop a new algorithm for numerical solution of dynamic elastic-plastic strain hardening/softening problems, particularly for the implementation of the gradient dependent model used in solving strain softening problems. The new algorithm for the solution of dynamic elastic-plastic problems is derived based on the parametric variational principle. The gradient dependent model is employed in the numerical model to overcome the mesh-sensitivity difficulty in dynamic strain softening or strain localization analysis. The precise integration method, which has been used for the solution of linear problems, is adopted and improved for the solution of dynamic non-linear equations. The new algorithm is proposed by taking the advantages of the parametric quadratic programming method and the precise integration method. Results of numerical examples demonstrate the validity and the advantages of the proposed algorithm.     

Spectral projective newton-methods for quasilinear elliptic boundary value problems

We consider Newton-like methods for the solution of quasilinear elliptic boundary value problems. The quasilinear problems are linearized by a Newton-method and the linear problems are approximately solved by a spectral projection method (e.g., the Ritz-Galerkin or the collocation method). convergence results are derived that show the spectral accuracy of this method. The results are of a local type which means that we assume the starting approximation to be sufficiently near to the exact solution.     

混沌优化算法及其在组合优化问题中的应用

混沌优化方法(COA)是针对数值优化问题提出的,在解决数值优化问题上具有一定的普遍性,能够很快地搜索到全局最优解,而利用COA解决组合优化问题存在一定的难度,该文提出了混沌优化算法解决组合优化问题的方法,该方法先产生组合优化问题的初始解,再利用混沌变量产生新解或对原解进行混沌扰动,产生新解,然后在解空间中进行最优搜索。将该方法应用到2个典型的组合优化问题(TSP问题,0/1背包问题)的求解中,仿真实验表明了该方法的有效性。     

Exact fuzzy optimal solution of fully fuzzy linear programming problems with unrestricted fuzzy variables

Kumar et al. (Appl. Math. Model. 35:817?C823, 2011) pointed out that there is no method in literature to find the exact fuzzy optimal solution of fully fuzzy linear programming (FFLP) problems and proposed a new method to find the fuzzy optimal solution of FFLP problems with equality constraints having non-negative fuzzy variables and unrestricted fuzzy coefficients. There may exist several FFLP problems with equality constraints in which no restriction can be applied on all or some of the fuzzy variables but due to the limitation of the existing method these types of problems can not be solved by using the existing method. In this paper a new method is proposed to find the exact fuzzy optimal solution of FFLP problems with equality constraints having non-negative fuzzy coefficients and unrestricted fuzzy variables. The proposed method can also be used to solve the FFLP problems with equality constraints having non-negative fuzzy variables and unrestricted fuzzy coefficients. To show the advantage of the proposed method over existing method the results of some FFLP problems with equality constraints, obtained by using the existing and proposed method, are compared. Also, to show the application of proposed method a real life problem is solved by using the proposed method.     

Quasi-trefftz spectral method for separable linear elliptic equations

A new numerical method QTSM which was earlier suggested for solving boundary value problems with elliptic operators with constant coefficients is applied to separable, second-order elliptic equations with varying coefficients. The problems in irregular domains are considered. The method combines the properties of the boundary methods with the spectral representation of the solution in the form of expansion over the eigenfunctions of some Sturm-Liouville problem. The method is tested on several one- and two-dimensional problems with exact analytic solution. The possibilities of further developments of the method are discussed.     

Spline solutions for system of second-order boundary-value problems

We use quadratic spline functions to develop a numerical method for computing approximate solution of a system of second order boundary value problems associated with obstacle, unilateral and contact problems. The present method outperforms other available techniques and it has extra added advantage of approximating the first derivative of the solution. Numerical example is presented to illustrate the applicability of the new method.     

A new bounded real lemma representation for the continuous-timecase

A differential linear matrix inequality (DLMI) approach is introduced for the solution of various linear continuous-time control problems. The proposed method permits the application of linear matrix inequalities (LMIs) to the solution of control design problems under uncertainty. These problems are solved for finite horizon linear systems while considerably reducing the overdesign inherent in previous methods. The new approach also allows for the solution of the output-feedback control problem for systems belonging to a finite set of uncertain plants with hardly any overdesign. Four examples are given to demonstrate the applicability of the new method     

An efficient dynamical systems method for solving singularly perturbed integral equations with noise

In this paper we apply the dynamical systems method (DSM) proposed by A. G. Ramm, and the variational regularization method (VRM), to obtain numerical solution to some singularly perturbed ill-posed problems contaminated by noise. The results obtained by these methods are compared to the exact solution for the model problems. It is found that the dynamical systems method is preferable because it is easier to apply, highly stable, robust, and it always converges to the solution even for large size models.     

The sample solution approach for determination of the optimal shape parameter in the Multiquadric function of the Kansa method

The Kansa method with the Multiquadric-radial basis function (MQ-RBF) is inherently meshfree and can achieve an exponential convergence rate if the optimal shape parameter is available. However, it is not an easy task to obtain the optimal shape parameter for complex problems whose analytical solution is often a priori unknown. This has long been a bottleneck for the MQ-Kansa method application to practical problems. In this paper, we present a novel sample solution approach (SSA) for achieving a reasonably good shape parameter of the MQ-RBF in the Kansa method for the solution of problems whose analytical solution is unknown. The basic assumption behind the SSA is that the optimal shape parameter is considered to be largely depended on the shape of computational domain, the type of the boundary conditions, the number and distribution of nodes, and the governing equation. In the procedure of the SSA, we set up a pseudo-problem as the sample solution whose solution is known. It is not difficult to obtain the optimal parameter of the MQ-RBF in the numerical solution of the pseudo-problem. The SSA suggests that the optimal shape parameter of the pseudo-problem can also achieve an approximately optimal accuracy in the solution of the original problem. Numerical examples and comparisons are provided to verify the proposed SSA in terms of accuracy and stability in solving homogeneous problems and non-homogeneous modified Helmholtz problems in several complex domains even using chaotic distribution of collocation points.     

A modified Runge-Kutta method with phase-lag of order infinity for the numerical solution of the Schrödinger equation and related problems

A modified Runge-Kutta method with phase-lag of order infinity for the numerical solution of the Schr?dinger equation and related problems is developed in this paper. This new modified method is based on the classical Runge-Kutta method of algebraic order four. The numerical results indicate that this new method is more efficient for the numerical solution of the Schr?dinger equation and related problems than the well known classical Runge-Kutta method of algebraic order four.     

Solution of nonlinear problems in applied sciences by generalized collocation methods and mathematica

This paper deals with the developments of mathematical methods for the discretization of continuous models and the solution of nonlinear problems of interest in applied sciences. The contents refer to developments of the differential quadrature method which leads to the so-called generalized collocation methods. The method is developed and applied to the solution of initial-boundary value problems. The computational problems are technically solved with Mathematica.     

基于Radau伪谱法的非线性最优控制问题的收敛性

在过去的10年里,伪谱方法(如Legendre伪谱法、Gauss伪谱法、Radau伪谱法)逐步成为求解不同领域中非线性最优控制问题的一种高效、灵活的数值解法.本文从最优控制问题解的存在性、收敛性以及解的可行性3个方面对采用Radau伪谱法求解一般非线性最优控制问题解的收敛性进行研究.证明了原最优控制问题的离散解存在、存在收敛到原最优控制问题解上的离散解和离散形式的收敛解是原最优控制问题的最优解.在此基础上,证明了Radau伪谱法的收敛性.本文结论与现有文献相比,去掉了一些必要条件,更适合一般的非线性时不变系统.     

Genetic algorithm and large neighbourhood search to solve the cell formation problem

We first introduce a local search procedure to solve the cell formation problem where each cell includes at least one machine and one part. The procedure applies sequentially an intensification strategy to improve locally a current solution and a diversification strategy destroying more extensively a current solution to recover a new one. To search more extensively the feasible domain, a hybrid method is specified where the local search procedure is used to improve each offspring solution generated with a steady state genetic algorithm. The numerical results using 35 most widely used benchmark problems indicate that the line search procedure can reduce to 1% the average gap to the best-known solutions of the problems using an average solution time of 0.64 s. The hybrid method can reach the best-known solution for 31 of the 35 benchmark problems, and improve the best-known solution of three others, but using more computational effort.