A Posteriori Error Bounds for Gaussian Elimination

Explicit bounds are constructed for the error in the solutionof a system of linear algebraic equations obtained by Gaussianelimination using floating-point arithmetic. The bounds takeaccount of inherent errors in the data and all abbreviations(choppings or roundings) introduced during the process of solution.The bounds are strict and agree with the estimate for the maximumerror obtained by linearized perturbation theory. The formulationof the bounds avoids the need for specially directed roundingprocedures in the hardware or software; in consequence the boundscan be evaluated on most existing computers. The cost of computingthe bounds is comparable with the cost of computing the originalsolution.     

A Priori and a Posteriori Error Analysis of a Wavelet-Based Stabilization for the Mixed Finite Element Method

We use Galerkin least-squares terms and biorthogonal wavelet bases to develop a new stabilized dual-mixed finite element method for second-order elliptic equations in divergence form with Neumann boundary conditions. The approach introduces the trace of the solution on the boundary as a new unknown that acts also as a Lagrange multiplier. We show that the resulting stabilized dual-mixed variational formulation and the associated discrete scheme defined with Raviart–Thomas spaces are well-posed and derive the usual a priori error estimates and the corresponding rate of convergence. Furthermore, a reliable and efficient residual-based a posteriori error estimator and a reliable and quasi-efficient one are provided.     

A Posteriori Error Bounds for Linear Elliptic Equations of Potential Type

Second-order Linear elliptic partial differential equations of potential type with Dirichlet (type 1) or Neumann (type II)boundary conditions on a simply-connected two-dimensional domainare considered. Conjugate problems, that is, a pair of one type1 and one type II problem, are introduced along with an auxiliaryeppiptic system of two equations in such a way that the energiesof the given problem, its conjugate problem, and the auxiliarysystem add to a known constant. There result two-sided boundsfor the energy of the given problem and, as a consequence, aposteriori error bounds for the norm of the difference of anapproximate solution and the exact solution of the problem.A method by which the amount of computation required to obtainthe a posteriori error bounds can be almost halved in many casesof practical interest is given. A posteriori error bounds forapproximate solutions of the auxiliary system are also given.     

Guaranteed Error Bounds for Eigenvalues of Singular Sturm‐Liouville Problems

We develop a simple oscillation theory for singular Sturm‐Liouville problems and combine it with recent asymptotic results, and with the AWA interval‐arithmetic code for integration of initial value problems with guaranteed error bounds, to obtain eigenvalue approximations with guaranteed error bounds for a class of singular Sturm‐Liouville problems. We believe that this is the first time that this has been achieved for singular eigenvalue problems.     

A Priori and A Posteriori: A Bootstrapping Relationship

The distinction between a priori and a posteriori knowledge has been the subject of an enormous amount of discussion, but the literature is biased against recognizing the intimate relationship between these forms of knowledge. For instance, it seems to be almost impossible to find a sample of pure a priori or a posteriori knowledge. In this paper, it will be suggested that distinguishing between a priori and a posteriori is more problematic than is often suggested, and that a priori and a posteriori resources are in fact used in parallel. We will define this relationship between a priori and a posteriori knowledge as the bootstrapping relationship. As we will see, this relationship gives us reasons to seek for an altogether novel definition of a priori and a posteriori knowledge. Specifically, we will have to analyse the relationship between a priori knowledge and a priori reasoning, and it will be suggested that the latter serves as a more promising starting point for the analysis of aprioricity. We will also analyse a number of examples from the natural sciences and consider the role of a priori reasoning in these examples. The focus of this paper is the analysis of the concepts of a priori and a posteriori knowledge rather than the epistemic domain of a posteriori and a priori justification.     

Iterative Corrections and a Posteriori Error Estimate for Integral Equations

Starting from a well known operator identity we obtain a recurrence formula, i.e., an iterative correction scheme, for the integral equations with computable kernel. From this we can increase the order of convergence step by step, say, from 4th to 8th to 12th. What is more interesting in this scheme, besides its fast acceleration, is its weak requirement on the integral kernel: the regularity of the kernel will not be strengthened during the correction procedure.     

Upper Bounds for the Laplacian Graph Eigenvalues

We first apply non-negative matrix theory to the matrix K = D A, where D and A are the degree-diagonal and adjacency matrices of a graph G, respectively, to establish a relation on the largest Laplacian eigenvalue λ1 (G) of G and the spectral radius p(K) of K. And then by using this relation we present two upper bounds for λ1(G) and determine the extremal graphs which achieve the upper bounds.     

Some A Priori Bounds for Flows through Heterogeneous Media

Under various conditions, analytical bounds on important displacementcharacteristics are derived for incompressible flows throughheterogeneous rectangular regions. They rely on bounds on theintermediate-scale Darcy permeability and also on its rate ofvariation with position, and can be calculated prior to fieldor laboratory displacement tests. It is suggested that thesebounds can be used in a probabilistic manner. Their limitations,applications, and possible extensions are discussed.     

Improved Error Bounds for a Chebysheff-Halley-Type Method

Using the majorant method we find sufficient conditions for the convergence of a Chebysheff-Halley-type method in a Banach space. Our results improve all our previous results as well as those of others.     

双调和算子本征值的上界

设Ω是 Rn中的有界区域 ,其边界足够光滑 ,λk为双调和算子在自由边界条件下的第 k个本征值 ,利用变分原理及 Fourier变换 ,给出了本征值部分和 ∑kj=1λj的一个上界 ,该上界仅依赖于区域的体积 .     

Hermite矩阵特征值的新扰动界

本文研究了Hermite矩阵特征值的任意扰动,给出了新的绝对和相对扰动界．所给出的界改进了Hoffman-Wielandt和Kahan早期的结果．     

Bounds for the Eigenvalues of Non-selfadjoint Differential Operators

A generalization of the Gerschgorin circle theorem is appliedto an infinite matrix representation of a differential operatorto obtain bounds for its eigenvalues.     

Extrinsic Bounds for Eigenvalues of the Dirac Operator

We derive upper eigenvalue bounds for the Dirac operator of a closed hypersurface in a manifold with Killing spinors such as Euclidean space, spheres or hyperbolic space. The bounds involve the Willmore functional. Relations with the Willmore inequality are briefly discussed. In higher codimension we obtain bounds on the eigenvalues of the Dirac operator of the submanifold twisted with the spinor bundle of the normal bundle.     

A Posteriori Error Estimates for Axisymmetric and Nonlinear Problems

We propose and examine the primal and dual finite element method for solving an axially symmetric elliptic problem with mixed boundary conditions. We derive an a posteriori error estimate and generalize the method used for a nonlinear elliptic problem. Finally, an a posteriori error estimate for a nonlinear parabolic problem based on the concept of hierarchical finite element basis functions is introduced.     

A Priori and A Posteriori Error Estimates of Streamline Diffusion Finite Element Method for Optimal Control Problem Governed by Convection Dominated Diffusion Equation

In this paper,we investigate a streamline diffusion finite element approxi- mation scheme for the constrained optimal control problem governed by linear con- vection dominated diffusion equations.We prove the existence and uniqueness of the discretized scheme.Then a priori and a posteriori error estimates are derived for the state,the co-state and the control.Three numerical examples are presented to illustrate our theoretical results.     

达尔文模型的有限元后验误差估计

在三维有界单连通区域里,当没有高频现象或电流变化不快时,达尔文模型是麦克斯韦方程组的-个很好的逼近模型.本文考虑达尔文模型的自适应算法,这种方法以有限元后验误差分析为理论基础.本文提供了基于后验误差估计子的上界估计.     

高阶椭圆算子Dirichlet本征值的下界

设Ω是R^m(m≥2）中的一个有界区域,其边界足够光滑,考察2p(p≥1）阶椭圆算子（－1）^p ∑│α│＝│β│＝pa^α(Aαβa^β)在Dirichlet边界条件下的本征值问题,给出了其本征值的一个下界,该下界除与维数m有关外仅依赖于区域Ω的体积。     

矩阵Fan积特征值的界

设Z_n为非对角元素都为非正实数的n阶方阵的集合,令A_k∈Z_n,k∈{1,…,m},给出矩阵Fan积最小特征值的一个新下界,其中p_k>0且and (sum from k=1 to m)1/p_k≥1,这个下界改进了文献中的相关结果.