Nonsmooth data error estimates for damped single step methods for parabolic equations in Banach space

We consider a time discretization method for a parabolic initial boundary value problem obtained from a combination of an A-stable single step method of order p and a lower order method with good smoothing properties. Such methods, including the Crank–Nicolson method combined with the backward Euler method, were analyzed in Hilbert space by Luskin and Rannacher, and nonsmooth data error estimates of order p were obtained. We extend this result to Banach space, and also consider approximations of the time derivative. Further, we apply the results to obtain error estimates in the supremum norm for fully discrete methods obtained by discretizing the space variable by a finite element method. Received: February 1998/ Accepted: November 1998     

One step integration methods with maximum stability regions

One step integration methods, using K function evaluations, for the solution of a system of ordinary differential equations dv/dt=A?v are evaluated. A general expression for a class of methods is found for all positive integers K. This class of methods is proven to maximize the interval on the imaginary axis which is contained in the stability region such that the stability constraint is optimized. It is proven that every method with this optimal stability property has a polynomial M defined by y_1+,Δt=M?v_i for which M(iS_l)=exp(iS_lπ/2) where S_l=K?l.     

Time step integration algorithms with predetermined coefficients for second order equations

In this paper, the effect of using the predetermined coefficients in constructing time step integration algorithms suitable for linear second order differential equations based on the weighted residual method is investigated. The second order equations are manipulated directly. The displacement approximation is assumed to be in a form of polynomial in the time domain and some of the coefficients can be predetermined from the known initial conditions. The algorithms are constructed so that the approximate solutions are equivalent to the solutions given by the transformed first order equations. If there are m predetermined coefficients (in addition to the two initial conditions) and r unknown coefficients in the displacement approximation, it is shown that the formulation is consistent with a minimum order of accuracy m+r. The maximum order of accuracy achievable is m+2r. This can be related to the Padé approximations for the second order equations. Unconditionally stable algorithms equivalent to the generalized Padé approximations for the second order equations are presented. The order of accuracy is 2r−1 or 2r and it is required that m+1⩽r. The corresponding weighting parameters, weighting functions and additional weighting parameters for the Padé and generalized Padé approximations are given explicitly.     

On second order efficient robust inference

General strategies for constructing second order efficient robust distances from suitable properties of the residual adjustment functions (RAF) are discussed. Based on those properties families of estimators are constructed using the truncated polynomial, negative exponential and sigmoidal functions as RAFs and their efficiency and robustness properties are investigated. The estimators have full asymptotic efficiency, and are automatically second order efficient. Many of the proposed estimators are competitive or better than the minimum Hellinger distance estimator (MHDE) and minimum negative exponential disparity estimator (MNEDE) under the combined goals of asymptotic efficiency with strong robustness properties. Hence the proposed families give the user the flexibility to choose from a large class of robust second order efficient estimators based upon specific needs.     

Nonlinear contractivity of a class of semi-implicit multistep methods

The stability and contractivity of generalized linear multistep methods are studied for a large class of nonlinear stiff initial value problems. These methods are characterized by the fact that the coefficients of the integration formulas are matrices depending on the Jacobian or on an approximation to the Jacobian. Conditions for the parameters of such a multistep method are given which ensure that the method gives contractive numerical solutions over a large class of nonlinear dissipative systems for sufficiently small stepsizesh, where the restriction onh is not due to the stiffness of the problem. Stability and contractive properties of special methods of this class are reported.     

On first and second order box schemes

The box method for discretizing elliptic boundary value problems are discussed. Error estimates of first and second order between the Galerkin solution and the box method solution are proved. A proposal for a new second order box-like scheme is made.     

On the second order topological asymptotic expansion

The topological derivative provides the sensitivity of a given shape functional with respect to an infinitesimal (non smooth) domain perturbation at an arbitrary point of the domain. Classically, this derivative comes from the second term of the topological asymptotic expansion, dealing only with infinitesimal perturbations. However, for practical applications, we need to insert perturbations of finite size. Therefore, we consider one more term in the expansion which is defined as the second order topological derivative. In order to present these ideas, in this work we calculate first as well as second order topological derivatives for the total potential energy associated to the Laplace’s equation, when the domain is perturbed with a hole. Furthermore, we also study the effects of different boundary conditions on the hole: Neumann and Dirichlet (both homogeneous). In the Neumann’s case, the second order topological derivative depends explicitly on higher-order gradients of the state solution and also implicitly on the point where the hole is nucleated through the solution of an auxiliary problem. On the other hand, in the Dirichlet’s case, the first order topological derivative depends explicitly on the state solution as well as implicitly through the solution of an auxiliary problem, and the second order topological derivative depends only explicitly on the solution associated to the original problem. Finally, we present two simple examples showing the influence of both terms in the second order topological asymptotic expansion for each case of boundary condition on the hole.     

一类三次Bent函数的二阶非线性度*

研究了形如yTr(x5)+Tr(x3)的三次Bent函数,通过研究其导数的非线性度的下界,得到了该函数的二阶非线性度的下界,将所得结果与Carlet的结果进行了比较,结果表明,该函数的二阶非线性度大于Carlet给出的下界。     

Optimal maximum norm error estimates for some finite element methods for treating the Dirichlet problem

The goal of this paper is to establish optimalL ^∞ error estimates for a few different finite element type methods for the Dirichlet problem in a bounded domain. The methods are selected so as to avoid the necessity for imposing boundary conditions on the trial functions, usually difficult in practice. Three specific methods are treated. These are the method of interpolated boundary condition and two methods of Nitsche.     

On symmetric Runge-Kutta methods of high order

The usual characterization of symmetry for Runge-Kutta methods is that given by Stetter. In this paper an equivalent characterization of symmetry based on theW-transformation of Hairer and Wanner is proposed. Using this characterization it is simple to show symmetry for some well-known classes of high order Runge-Kutta methods which are based on quadrature formulae. It can also be used to construct a one-parameter family of symmetric and algebraically stable Runge-Kutta methods based on Lobatto quadrature. Methods constructed in this way and presented in this paper extend the known class of implicit Runge-Kutta methods of high order.     

One step integration methods of third-fourth order accuracy with large hyperbolic stability limits

One step integration methods of third and fourth order accuracy that use K function evaluations to solve the system of differential equations $\text{d}\text{y}\text{d}\text{t}\text{= A · y}$ are proposed. These methods are shown to have a hyperbolic stability limit of y (K ? 1)² ? 1 which approaches the theoretical maximum limit of K ? 1 at large K obtained for methods of lower order accuracy.     

On visual servoing based on efficient second order minimization

This paper deals with the Efficient Second order Minimization (ESM) and the image-based visual servoing schemes. In other words, it deals with the minimization based on the pseudo-inverse of the mean of the Jacobians or on the mean of Jacobian Pseudo-inverses. Chronologically, it has been noticed in Tahri and Chaumette (2003) [22] that ESM generally improves the system behavior when compared with the system in which only the simple Jacobian Pseudo-inverses are used. Subsequently, a mathematical explanation has been given in Malis (2004) [12]. In this paper, the proofs given by Malis are discussed and it will be shown that there is a limitation to the validity of the ESM. We will also show that the use of ESM does not necessarily ensure a better system behavior, especially in the situations where large rotational motions are involved. Further, a new appropriate formula of the ESM is proposed and validated using several kinds of features.     

On minimal order stabilization of single loop plants

It is shown by example that a single loop plant of order n, relative degree 1, with one right half plane pole and one right half plane zero may not be stabilizable by a compensator of order n−2. A further example shows that a minimum phase plant of order n, relative degree r, with one right half plane pole may not be stabilizable by a compensator of order r−2.     

拟Newton法在高阶矩阵中的应用——求解最大特征值及特征向量

将求解高阶矩阵的最大特征值及其对应的特征向量问题转化为高阶非线性方程组的求解问题。在此基础上,提出了求解矩阵最大特征值及其对应特征向量的拟Newton法,给出求解矩阵最大特征值及其单位化向量重新整理后的Broyden方法公式、BFS方法公式、DFP方法公式及其对应的Broyden算法,BFS算法,DFP算法。以层次分析法中高阶判断矩阵为例验证了该方法的可行性,说明了该方法相对收敛速度快的优势。     

On the maximum order of optimal B-convergence of Lobatto III C schemes

The maximum orderp for which ans-stage Lobatto III C method,s>-4, is optimallyB-convergent for problems with a negative one-sided Lipschitz constantm is derived; it turns out to bes?1.     

Comparison of collocation methods for the solution of second order non-linear boundary value problems

This article concerns the application of cubic spline collocation tau-method for solving non-linear second order ordinary differential equations. Three collocation methods [Taiwo, O.A., 1986, A computational method for ordinary differential equations and error estimation. MSc dissertation, University of Ilorin, Nigeria (unpublished); Taiwo, O.A., 2002, Exponential fitting for the solution of two point boundary value problem with cubic spline collocation tau-method. International Journal of Computer Mathematics, 79(3), 229–306.] are discussed and applied to some second order non-linear problems. They are standard collocation, perturbed collocation, and exponentially fitted collocation. Numerical examples are given to illustrate the accuracy, efficiency and computational cost.     

On high order strong stability preserving runge-kutta and multi step time discretizations

Strong stability preserving (SSP) high order time discretizations were developed for solution of semi-discrete method of lines approximations of hyperbolic partial differential equations. These high order time discretization methods preserve the strong stability properties-in any norm or seminorm—of the spatial discretization coupled with first order Euler time stepping. This paper describes the development of SSP methods and the recently developed theory which connects the timestep restriction on SSP methods with the theory of monotonicity and contractivity. Optimal explicit SSP Runge-Kutta methods for nonlinear problems and for linear problems as well as implicit Runge-Kutta methods and multi step methods will be collected.     

Higher order eigensensitivity analysis of damped systems with repeated eigenvalues

A simplified method for the computation of first-, second- and higher-order derivatives of eigenvalues and eigenvectors associated with repeated eigenvalues is presented. Adjacent eigenvectors and orthonormal conditions are used to compose an algebraic equation. The algebraic equation which is developed can be used to compute derivatives of eigenvalues and eigenvectors simultaneously. Since the coefficient matrix in the proposed algebraic equation is non-singular, symmetric and based on N-space, it is numerically stable and very efficient compared to previous methods. To verify the efficiency of the proposed method, the finite element model of the cantilever beam and a mechanical system in the case of a non-proportionally damped system are considered.