Upper bound and stability of scaled pseudoinverses

Summary. For given matrices and where is positive definite diagonal, a weighed pseudoinverse of is defined by and an oblique projection of is defined by . When is of full column rank, Stewart [3] and O'Leary [2] found sharp upper bound of oblique projections which is independent of , and an upper bound of weighed pseudoinverse by using the bound of . In this paper we discuss the sharp upper bound of over a set of positive diagonal matrices which does not depend on the upper bound of , and the stability of over . Received September 29, 1993 / Revised version received October 31, 1994     

Is the polynomial so perfidious?

Summary. Wilkinson, in [1], has given a comprehensive account of the numerical difficulties of working with polynomials on a floating point computer. The object of this note is to attempt to rehabilitate the polynomial to a certain extent. In particular it is shown here that polynomial deflation can be performed satisfactorily by a method akin to `backward recursion'. Error analyses and examples are given to illustrate the stability of the process. Received October 4, 1993 / Revised version received July 14, 1993     

Asymptotic estimation for the condition numbers in BEM

Summary. We apply the boundary element methods (BEM) to the interior Dirichlet problem of the two dimensional Laplace equation, and its discretization is carried out with the collocation method using piecewise linear elements. In this paper, some precise asymptotic estimations for the discretization matrix (where denotes the division number) are investigated. We show that the maximum norm of and the condition number of have the forms: and , respectively, as , where the constants and are explicitly given in the proof. Although these estimates indicate illconditionedness of this numerical computation, the -convergence of this scheme with maximum norm is proved as an application of the main results. Received January 25, 1993 / Revised version received March 13, 1995     

The convergence rate of the minimal residual method for the Stokes problem

Summary. Discretisation of the classical Stokes problem gives rise to symmetric indefinite matrices with eigenvalues which, in a precise way, are not symmetric about the origin, but which do depend on a mesh size parameter. Convergence estimates for the Conjugate Residual or Minimum Residual iterative solution of such systems are given by best minimax polynomial approximations on an inclusion set for the eigenvalues. In this paper, an analytic convergence estimate for such problems is given in terms of an asymptotically small mesh size parameter. Received November 16, 1993 / Revised version received August 2, 1994     

The index of general nonlinear DAEs

Summary. In the last few years there has been considerable research on differential algebraic equations (DAEs) where is identically singular. Much of the mathematical effort has focused on computing a solution that is assumed to exist. More recently there has been some discussion of solvability of DAEs. There has historically been some imprecision in the use of the two key concepts of solvability and index for DAEs. The index is also important in control and systems theory but with different terminology. The consideration of increasingly complex nonlinear DAEs makes a clear and correct development necessary. This paper will try to clarify several points concerning the index. After establishing some new and more precise terminology that we need, some inaccuracies in the literature will be corrected. The two types of indices most frequently used, the differentiation index and the perturbation index, are defined with respect to solutions of unperturbed problems. Examples are given to show that these indices can be very different for the same problem. We define new "maximum indices," which are the maxima of earlier indices in a neighborhood of the solution over a set of perturbations and show that these indices are simply related to each other. These indices are also related to an index defined in terms of Jacobians. Received November 15, 1993 / Revised version received December 23, 1994     

Nonlinear Galerkin methods and mixed finite elements: two-grid algorithms for the Navier-Stokes equations

Summary. A nonlinear Galerkin method using mixed finite elements is presented for the two-dimensional incompressible Navier-Stokes equations. The scheme is based on two finite element spaces and for the approximation of the velocity, defined respectively on one coarse grid with grid size and one fine grid with grid size and one finite element space for the approximation of the pressure. Nonlinearity and time dependence are both treated on the coarse space. We prove that the difference between the new nonlinear Galerkin method and the standard Galerkin solution is of the order of $H^2$, both in velocity ( and pressure norm). We also discuss a penalized version of our algorithm which enjoys similar properties. Received October 5, 1993 / Revised version received November 29, 1993     

The number of centered forms for a polynomial

A general centered form for polynomials is defined. This centered form is based on the concept of an arrangement of the variables of the polynomial. A formula for the number of arrangements of a term and of a polynomial is given. Some examples are computed showing that even polynomials with moderately many variables may have a very large number of possible centered forms.     

Forms for meromorphic functions

An infinite circular arithmetic is defined and used to define power sum and centered forms for meromorphic functions. The properties of these forms are investigated and it is shown that both the power sum and the centered forms form including chains with respect to their orders and that they possess a kind of quadratic convergence with respect to the domain circle. An example of the use of the forms is given.     

A Cholesky dual method for proximal piecewise linear programming

Summary. A quadratic programming method is given for minimizing a sum of piecewise linear functions and a proximal quadratic term, subject to simple bounds on variables. It may be used for search direction finding in nondifferentiable optimization algorithms. An efficient implementation is described that updates a Cholesky factorization of active constraints and provides good accuracy via iterative refinement. Numerical experience is reported for some large problems. Received March 29, 1993 / Revised version received December 18, 1993     

Block monotone iterative methods for numerical solutions of nonlinear elliptic equations

Summary. Two block monotone iterative schemes for a nonlinear algebraic system, which is a finite difference approximation of a nonlinear elliptic boundary-value problem, are presented and are shown to converge monotonically either from above or from below to a solution of the system. This monotone convergence result yields a computational algorithm for numerical solutions as well as an existence-comparison theorem of the system, including a sufficient condition for the uniqueness of the solution. An advantage of the block iterative schemes is that the Thomas algorithm can be used to compute numerical solutions of the sequence of iterations in the same fashion as for one-dimensional problems. The block iterative schemes are compared with the point monotone iterative schemes of Picard, Jacobi and Gauss-Seidel, and various theoretical comparison results among these monotone iterative schemes are given. These comparison results demonstrate that the sequence of iterations from the block iterative schemes converges faster than the corresponding sequence given by the point iterative schemes. Application of the iterative schemes is given to a logistic model problem in ecology and numerical ressults for a test problem with known analytical solution are given. Received August 1, 1993 / Revised version received November 7, 1994     

Least supported bases and local linear independence

Summary. We introduce the concept of least supported basis, which is very useful for numerical purposes. We prove that this concept is equivalent to the local linear independence of the basis. For any given locally linearly independent basis we characterize all the bases of the space sharing the same property. Several examples for spline spaces are given. Received December 4, 1992 / Revised version received March 2, 1993     

Evolution-Galerkin methods and their supraconvergence

Summary. Evolution-Galerkin methods for partial differential equations of the form are characterised by (i) the use of some form of approximation to the corresponding evolution operator , and (ii) projection onto an approximation space to obtain . In this paper we concentrate on characteristic-Galerkin and Lagrange-Galerkin methods to derive basic error estimates for multidimensional convection problems. Methods covered include those using recovery techniques to improve accuracy. Many schemes exhibit a supraconvergence phenomenon and a general technique for its analysis is given, together with a number of particular examples. Received July 5, 1993 / Revised version received February 6, 1995     

On the stability of a class of convergence acceleration methods for power series

In this paper we study the problem of evaluating the sum of a power series whose terms are given numerically with a moderate accuracy. For a large class of divergent series a sum may be defined using analytic continuation. This sum may be estimated using the values of a finite number of terms. However, it is established here that the accuracy of this estimate will generally deteriorate if we use an ever-growing number of terms. A result on the stability of product quadrature is also obtained as a corollary of our main stability theorem.Dedicated to professor Germund Dahlquist, on the occasion of his 60th birthday     

Generation of higher-order polynomial basis of Nédélec H(curl) finite elements for Maxwell’s equations

The goal of this study is the automatic construction of a vectorial polynomial basis for Nédélec mixed finite elements, J.C. Nédélec (1980), in particular, the generation of finite elements without the expression of the polynomial basis functions, using symbolic calculus: the exhibition of basis functions has no practical interest.     

A second order finite difference error indicator for adaptive transonic flow computations

Summary. An adaptive finite element method for the calculation of transonic potential flows was developed. An error indicator based on first order finite differences of gradients is introduced as a local error estimator. It measures second order distributional derivatives. Estimates involving this error estimator, a residual and the error are given. The error estimator can be used as a criterion for mesh refinement. We also give some computational results. Received September 16, 1993 / Revised version received June 7, 1994     

关于局部凸空间中向量Ekeland变分原理的等价性

基于各种Ekeland变分原理的等价形式, 主要研究局部凸空间中给定有界凸子集乘以距离函数为扰动的单调半连续映射的向量Ekeand变分原理的等价性问题. 首先利用局部凸空间中的向量Ekeland变分原理证明了向量Caristi-Kirk不动点定理,向量 Takahashi非凸极小化定理和向量Oettli-Th\'{e}ra定理. 进一步研究了向量Ekeland变分原理与向量Caristi-Kirk不动点定理,向量Takahashi非凸极小化定理和向量Oettli-Th\'{e}ra定理的等价性.     

On the locking phenomenon for a class of elliptic problems

Summary. In this paper we study the numerical behaviour of elliptic problems in which a small parameter is involved and an example concerning the computation of elastic arches is analyzed using this mathematical framework. At first, the statements of the problem and its Galerkin approximations are defined and an asymptotic analysis is performed. Then we give general conditions ensuring that a numerical scheme will converge uniformly with respect to the small parameter. Finally we study an example in computation of arches working in linear elasticity conditions. We build one finite element scheme giving a locking behaviour, and another one which does not. Revised version received October 25, 1993     

Finite difference preconditioning cubic spline collocation method of elliptic equations

Summary. We discuss a finite difference preconditioner for the interpolatory cubic spline collocation method for a uniformly elliptic operator defined by in (the unit square) with homogeneous Dirichlet boundary conditions. Using the generalized field of values arguments, we discuss the eigenvalues of the preconditioned matrix where is the matrix of the collocation discretization operator corresponding to , and is the matrix of the finite difference operator corresponding to the uniformly elliptic operator given by in with homogeneous Dirichlet boundary conditions. Finally we mention a bound of -singular values of for a general elliptic operator in . Received December 11, 1995 / Revised version received June 20, 1996     

基于向量夹角余弦的加权调和平均组合预测模型的有效性

加权调和平均组合预测为一种非线性的组合预测方法.提出了基于向量夹角余弦的加权调和平均组合预测模型,针对该模型定义了优性组合预测、预测方法优超和组合预测冗余度等新的概念;探讨了非劣性组合预测和优性组合预测存在的充分条件;给出了一个冗余预测方法出现的判定定理.     

Global pointwise estimates for Green's matrix of second order elliptic systems

We establish global pointwise bounds for the Green's matrix for divergence form, second order elliptic systems in a domain under the assumption that weak solutions of the system vanishing on a portion of the boundary satisfy a certain local boundedness estimate. Moreover, we prove that such a local boundedness estimate for weak solutions of the system is equivalent to the usual global pointwise bound for the Green's matrix. In the scalar case, such an estimate is a consequence of De Giorgi-Moser-Nash theory and holds for equations with bounded measurable coefficients in arbitrary domains. In the vectorial case, one need to impose certain assumptions on the coefficients of the system as well as on domains to obtain such an estimate. We present a unified approach valid for both the scalar and vectorial cases and discuss several applications of our result.