Adaptive algebraic smoothers

This paper will present a new method of adaptively constructing block iterative methods based on Local Sensitivity Analysis (LSA). The method can be used in the context of geometric and algebraic multigrid methods for constructing smoothers, and in the context of Krylov methods for constructing block preconditioners. It is suitable for both constant and variable coefficient problems. Furthermore, the method can be applied to systems arising from both scalar and coupled system partial differential equations (PDEs), as well as linear systems that do not arise from PDEs. The simplicity of the method will allow it to be easily incorporated into existing multigrid and Krylov solvers while providing a powerful tool for adaptively constructing methods tuned to a problem.     

The reliability/cost trade-off for a class of ODE solvers

In the numerical solution of ODEs, it is now possible to develop efficient techniques that will deliver approximate solutions that are piecewise polynomials. The resulting methods can be designed so that the piecewise polynomial will satisfy a perturbed ODE with an associated defect (or residual) that is directly controlled in a consistent fashion. We will investigate the reliability/cost trade off that one faces when implementing and using such methods, when the methods are based on an underlying discrete Runge-Kutta formula. In particular we will identify a new class of continuous Runge-Kutta methods with a very reliable defect estimator and a validity check that reflects the credibility of the estimate. We will introduce different measures of the “reliability” of an approximate solution that are based on the accuracy of the approximate solution; the maximum magnitude of the defect of the approximate solution; and how well the method is able to estimate the maximum magnitude of the defect of the approximate solution. We will also consider how methods can be implemented to detect and cope with special difficulties such as the effect of round-off error (on a single step) or the ability of a method to estimate the magnitude of the defect when the stepsize is large (as might happen when using a high-order method at relaxed accuracy requests). Numerical results on a wide selection of problems will be summarized for methods of orders five, six and eight. It will be shown that a modest increase in the cost per step can lead to a significant improvement in the quality of the approximate solutions and the reliability of the method. For example, the numerical results demonstrate that, if one is willing to increase the cost per step by 50%, then a method can deliver approximate solutions where the reported estimated maximum defect is within 1% of its true value on 95% of the steps.     

Discontinuous Galerkin derivative operators with applications to second‐order elliptic problems and stability

A discontinuous Galerkin (DG) finite‐element interior calculus is used as a common framework to describe various DG approximation methods for second‐order elliptic problems. Using the framework, symmetric interior‐penalty methods, local discontinuous Galerkin methods, and dual‐wind discontinuous Galerkin methods will be compared by expressing all of the methods in primal form. The penalty‐free nature of the dual‐wind discontinuous Galerkin method will be both motivated and used to better understand the analytic properties of the various DG methods. Consideration will be given to Neumann boundary conditions with numerical experiments that support the theoretical results. Many norm equivalencies will be derived laying the foundation for applying dual‐winding techniques to other problems. Copyright © 2015 John Wiley & Sons, Ltd.     

Numerical solution of nonlinear systems by a general class of iterative methods with application to nonlinear PDEs

A general class of multi-step iterative methods for finding approximate real or complex solutions of nonlinear systems is presented. The well-known technique of undetermined coefficients is used to construct the first method of the class while the higher order schemes will be attained by a frozen Jacobian. The point of attraction theory will be taken into account to prove the convergence behavior of the main proposed iterative method. Then, it will be observed that an m-step method converges with 2m-order. A discussion of the computational efficiency index alongside numerical comparisons with the existing methods will be given. Finally, we illustrate the application of the new schemes in solving nonlinear partial differential equations.     

Suitable iterative methods for solving the linear system arising in the three fields domain decomposition method

There are two approaches for applying substructuring preconditioner for the linear system corresponding to the discrete Steklov–Poincaré operator arising in the three fields domain decomposition method for elliptic problems. One of them is to apply the preconditioner in a common way, i.e. using an iterative method such as preconditioned conjugate gradient method [S. Bertoluzza, Substructuring preconditioners for the three fields domain decomposition method, I.A.N.-C.N.R, 2000] and the other one is to apply iterative methods like for instance bi-conjugate gradient method, conjugate gradient square and etc. which are efficient for nonsymmetric systems (the preconditioned system will be nonsymmetric). In this paper, second approach will be followed and extensive numerical tests will be presented which imply that the considered iterative methods are efficient.     

A class of methods for diffusion-convection equations

A class of method with a free parameter(s) is considered for the solution of the diffusion–convection equation. This class is obtained using interpolation function approach and so, some classic methods appears when s take specific values. In order to obtain a non–oscillatory numerical solution some restrictions will be placed on s. Choosing two suitable methods of that class, an algorithm will be established and computational requirements will be considered     

An Initial Value Problem for Parabolic Monge-AmpOere Equation from Investment Theory

In this paper the upwind discontinuous Galerkin methods with triangle meshes for two dimensional neutron transport equations will be studied. The stability for both of the semi-discrete and full-discrete method will be proved.     

Upwind Discontinuous Galerkin Methods for Two Dimensional Neutron Transport Equations

In this paper the upwind discontinuous Galerkin methods with triangle meshes for two dimensional neutron transport equations will be studied. The stability for both of the semi-discrete and full-discrete method will be proved.     

Supermemory descent methods for unconstrained minimization

The supermemory gradient method of Cragg and Levy (Ref. 1) and the quasi-Newton methods with memory considered by Wolfe (Ref. 4) are shown to be special cases of a more general class of methods for unconstrained minimization which will be called supermemory descent methods. A subclass of the supermemory descent methods is the class of supermemory quasi-Newton methods. To illustrate the numerical effectiveness of supermemory quasi-Newton methods, some numerical experience with one such method is reported.The authors are indebted to Dr. H. Y. Huang for his helpful criticism of this paper.     

Application of the Adomian decomposition method for the Fokker–Planck equation

In this work we will discuss the solution of an initial value problem of parabolic type. The main objective is to propose an alternative method of solution, one not based on finite difference or finite element or spectral methods. The aim of the present paper is to investigate the application of the Adomian decomposition method for solving the Fokker–Planck equation and some similar equations. This method can successfully be applied to a large class of problems. The Adomian decomposition method needs less work in comparison with the traditional methods. This method decreases considerable volume of calculations. The decomposition procedure of Adomian will be obtained easily without linearizing the problem by implementing the decomposition method rather than the standard methods for the exact solutions. In this approach the solution is found in the form of a convergent series with easily computed components. In this work we are concerned with the application of the decomposition method for the linear and nonlinear Fokker–Planck equation. To give overview of methodology, we have presented several examples in one and two dimensional cases.     

Boundary element methods—An overview

Variational methods for boundary integral equations deal with the weak formulations of boundary integral equations. Their numerical discretizations are known as the boundary element methods. This paper gives an overview of the method from both theoretical and numerical point of view. It summarizes the main results obtained by the author and his collaborators over the last 30 years. Fundamental theory and various applications will be illustrated through simple examples. Some numerical experiments in elasticity as well as in fluid mechanics will be included to demonstrate the efficiency of the methods.     

A parallel algorithm for solving Toeplitz linear systems

Numerical methods of solution are considered for systems which are Toeplitz and symmetric. In our case, the coefficient matrix is essentially tridiagonal and sparse. There are two distinct approaches to be considered each of which is efficient in its own way. Here we will combine the two approaches which will allow application of the cyclic reduction method to coefficient matrices of more general forms. The convergence of the approximations to the exact solution will also be examined. Solving linear systems by the adapted cyclic reduction method can be parallel processed.     

Cesaro Summability and Marchaud Inequality

For Fourier expansions it will be shown that sharp Marchaud-type inequalities follow from the positivity of the Cesaro summability of some order. Some results of sharp Marchaud-type will be derived using this method when other known methods cannot be used.     

Multi-Dimensional Quadrature of Singular and Discontinuous Functions

In many simulations of physical phenomena, discontinuous material coefficients and singular forces pose severe challenges for the numerical methods. The singularity of the problem can be reduced by using a numerical method based on a weak form of the equations. Such a method, combined with an interface tracking method to track the interfaces to which the discontinuities and singularities are confined, will require numerical quadrature with singular or discontinuous integrands. We introduce a class of numerical integration methods based on a regularization of the integrand. The methods can be of arbitrary high order of accuracy. Moment and regularity conditions control the overall accuracy.     

求总极值的区间压缩方法

对求解无约束最优化问题 ,本文给出了一个区间压缩方法 .应用此方法能使函数值按几何级数收敛于 f (x)的极小值 ,并且计算量远小于郑权等人方法的计算量     

NUMERICAL METHODS FOR MAXWELL＇S EQUATIONS IN INHOMOGENEOUS MEDIA WITH MATERIAL INTERFACES

In this paper, we will present some recent results on developing numerical methods for solving Maxwell‘s equations in inhomogeneous media with material interfaces. First,we will present a second order upwinding embedded boundary method - a Cartesian grid based finite difference method with special upwinding treatment near the material interfaces. Second, we will present a high order discontinuous spectral element with Dubinar orthogonal polynomials on triangles. Numerical results on electromagnetic scattering and photonic waveguide will be included.     

Utilizing expected improvement and generalized data envelopment analysis in multi-objective genetic algorithms

Meta-heuristic methods such as genetic algorithms (GA) and particle swarm optimization (PSO) have been extended to multi-objective optimization problems, and have been observed to be useful for finding good approximate Pareto optimal solutions. In order to improve the convergence and the diversity in the search of solutions using meta-heuristic methods, this paper suggests a new method to make offspring by utilizing the expected improvement (EI) and generalized data envelopment analysis (GDEA). In addition, the effectiveness of the proposed method will be investigated through several numerical examples in comparison with the conventional multi-objective GA and PSO methods.     

常微分方程组的整体解及两点边值问题——Ⅰ.整体解方法

众所周知,n维向量函数u(x)的一阶常微分方程组,如在某点上只给出n_1相似文献   

Relaxation strategies for nested Krylov methods

This paper studies computational aspects of Krylov methods for solving linear systems where the matrix–vector products dominate the cost of the solution process because they have to be computed via an expensive approximation procedure. In recent years, so-called relaxation strategies for tuning the precision of the matrix–vector multiplications in Krylov methods have proved to be effective for a range of problems. In this paper, we will argue that the gain obtained from such strategies is often limited. Another important strategy for reducing the work in the matrix–vector products is preconditioning the Krylov method by another iterative Krylov method. Flexible Krylov methods are Krylov methods designed for this situation. We combine these two approaches for reducing the work in the matrix–vector products. Specifically, we present strategies for choosing the precision of the matrix–vector products in several flexible Krylov methods as well as for choosing the accuracy of the variable preconditioner such that the overall method is as efficient as possible. We will illustrate this computational scheme with a Schur-complement system that arises in the modeling of global ocean circulation.     

A comparison of analytic series methods for Laplacian free boundary problems

Solutions to Laplace's equation are required for a wide range of problems. Arguably, the most difficult class of problems involves a “free” boundary, where the location of one (or more) of the boundaries is initially unknown. Analytical solutions for these problems were restricted to regular boundary geometries. However, recently the classical series method has been modified, to cater for arbitrary boundary geometries, using least squares methods. For free boundary problems, solutions can be obtained by solving a sequence of known boundary problems—at each iteration, the series coefficients can be estimated. Efficient calculation of the series coefficients becomes very important, particularly when the number of iterations is relatively high. In this paper, three methods for estimating the series coefficients will be described, in the context of a free boundary problem. The computational cost of each method will be analysed and compared, and the most appropriate method for this class of problem is indicated.