Error bounds and estimates for eigenvalues of integral equations

Summary Approximate solutions of the linear integral equation eigenvalue problem can be obtained by the replacement of the integral by a numerical quadrature formula and then collocation to obtain a linear algebraic eigenvalue problem. This method is often called the Nyström method and its convergence was discussed in [7]. In this paper computable error bounds and dominant error terms are derived for the approximation of simple eigenvalues of nonsymmetric kernels.     

On the multistep time discretization of linear\newline initial-boundary value problems and their boundary integral equations

Summary. Convergence estimates in terms of the data are shown for multistep methods applied to non-homogeneous linear initial-boundary value problems. Similar error bounds are derived for a new class of time-discrete and fully discrete approximation schemes for boundary integral equations of such problems, e.g., for the single-layer potential equation of the wave equation. In both cases, the results are obtained from convergence and stability estimates for operational quadrature approximations of convolutions. These estimates, which are also proved here, depend on bounds of the Laplace transform of the (distributional) convolution kernel outside the stability region scaled by the time stepsize, and on the smoothness of the data. Received January 18, 1993 / Revised version received September 15, 1993     

Quantifying the error of convex order bounds for truncated first moments

The concepts of convex order and comonotonicity have become quite popular in risk theory, essentially since Kaas et al. [Kaas, R., Dhaene, J., Goovaerts, M.J., 2000. Upper and lower bounds for sums of random variables. Insurance: Math. Econ. 27, 151-168] constructed bounds in the convex order sense for a sum S of random variables without imposing any dependence structure upon it. Those bounds are especially helpful, if the distribution of S cannot be calculated explicitly or is too cumbersome to work with. This will be the case for sums of lognormally distributed random variables, which frequently appear in the context of insurance and finance.In this article we quantify the maximal error in terms of truncated first moments, when S is approximated by a lower or an upper convex order bound to it. We make use of geometrical arguments; from the unknown distribution of S only its variance is involved in the computation of the error bounds. The results are illustrated by pricing an Asian option. It is shown that under certain circumstances our error bounds outperform other known error bounds, e.g. the bound proposed by Nielsen and Sandmann [Nielsen, J.A., Sandmann, K., 2003. Pricing bounds on Asian options. J. Financ. Quant. Anal. 38, 449-473].     

Sharp estimates in terms of moduli of smoothness for compound cubature rules

Summary In 1980 Dahmen-DeVore-Scherer introduced a modulus of continuity which turns out to reflect invariance properties of compound cubature rules effectively. Accordingly, sharp error bounds are derived, the existence of relevant counterexamples being a consequence of a quantitative resonance principle, established previously.     

Error bounds for the finite element approximation of an incompressible material in an unbounded domain

Summary. In this paper we design high-order local artificial boundary conditions and present error bounds for the finite element approximation of an incompressible elastic material in an unbounded domain. The finite element approximation is formulated in a bounded computational domain using a nonlocal approximate artificial boundary condition or a local one. In fact there are a family of nonlocal approximate artificial boundary conditions with increasing accuracy (and computational cost) and a family of local ones for a given artificial boundary. Our error bounds indicate how the errors of the finite element approximations depend on the mesh size, the terms used in the approximate artificial boundary condition and the location of the artificial boundary. Numerical examples of an incompressible elastic material outside a circle in the plane is presented. Numerical results demonstrate the performance of our error bounds. Received August 31, 1998 / Revised version received November 6, 2001 / Published online March 8, 2002     

A posteriori error analysis for numerical approximations of Friedrichs systems

The global error of numerical approximations for symmetric positive systems in the sense of Friedrichs is decomposed into a locally created part and a propagating component. Residual-based two-sided local a posteriori error bounds are derived for the locally created part of the global error. These suggest taking the -norm as well as weaker, dual norms of the computable residual as local error indicators. The dual graph norm of the residual is further bounded from above and below in terms of the norm of where h is the local mesh size. The theoretical results are illustrated by a series of numerical experiments. Received January 10, 1997 / Revised version received March 5, 1998     

伪凸集值映射的误差界

考虑了伪凸集值映射的误差界.证明了对于伪凸集值映射,局部误差界成立意味着整体误差界成立.通过相依导数,给出了伪凸集值映射存在误差界的一些等价叙述.     

Optimal long-time $L_p}(0,T)$ stability and semidiscrete error estimates for the Volterra formulation of the linear quasistatic viscoelasticity problem

The purpose of this article is to show how the solution of the linear quasistatic (compressible) viscoelasticity problem, written in Volterra form with fading memory, may be sharply bounded in terms of the data if certain physically reasonable assumptions are satisfied. The bounds are derived by making precise assumptions on the memory term which then make it possible to avoid the Gronwall inequality, and use instead a comparison theorem which is more sensitive to the physics of the problem. Once the data-stability estimates are established we apply the technique also to deriving a priori error bounds for semidiscrete finite element approximations. Our bounds are derived for viscoelastic solids and fluids under the small strain assumption in terms of the eigenvalues of a certain matrix derived from the stress relaxation tensor. For isotropic materials we can be explicit about the form of these bounds, while for the general case we give a formula for their computation.     

A fractional version of the peano-sard theorem

Sard's classical generalization of the Peano kernel theorem provides an extremely useful method for expressing and calculating sharp bounds for approximation errors. The error is expressed in terms of a derivative of the underlying function. However, we can apply the theorem only if the approximation is exact on a certain set of polynomials.

In this paper, we extend the Peano-Sard theorem to the case that the approximation is exact for a class of generalized polynomials (with non-integer exponents). As a result, we obtain an expression for the remainder in terms of a fractional derivative of the function under consideration. This expression permits us to give sharp error bounds as in the classical situation. An application of our results to the classical functional (vanishing on polynomials) gives error bounds of a new type involving weighted Sobolev-type spaces. In this way, we may state estimates for functions with weaker smoothness properties than usual.

The standard version of the Peano-Sard theory is contained in our results as a special case.     

Lower Bounds for Fixed Spectrum Frequency Assignment

Determining lower bounds for the sum of weighted constraint violations in fixed spectrum frequency assignment problems is important in order to evaluate the performance of heuristic algorithms. It is well known that, when adopting a binary constraints model, clique and near-clique subproblems have a dominant role in the theory of lower bounds for minimum span problems. In this paper we highlight their importance for fixed spectrum problems. We present a method based on the linear relaxation of an integer programming formulation of the problem, reinforced with constraints derived from clique-like subproblems. The results obtained are encouraging both in terms of quality and in terms of computation time.     

A posteriori error estimators for convection-diffusion equations

Summary. We derive a posteriori error estimators for convection-diffusion equations with dominant convection. The estimators yield global upper and local lower bounds on the error measured in the energy norm such that the ratio of the upper and lower bounds only depends on the local mesh-Peclet number. The estimators are either based on the evaluation of local residuals or on the solution of discrete local Dirichlet or Neumann problems. Received February 10, 1997 / Revised version received November 4, 1997     

定常Navier-Stokes方程流函数形式两重网格算法的残量型后验误差估计

运用七种两重网格协调元方法得出了不可压Navier-Stokes方程流函数形式的残量型后验误差估计．对比标准有限元方法的后验误差估计,两重网格算法的后验误差估计多了一些额外项(三线性项)．说明了这些额外项在误差估计中对研究离散解渐近性的重要性,推出了对于最优网格尺寸,这些额外项的收敛阶不高于标准离散解的收敛阶．     

Asymptotic expansions of the error of spline Galerkin boundary element methods

Summary. In this paper we analyse the existence of asymptotic expansions of the error of Galerkin methods with splines of arbitrary degree for the approximate solution of integral equations with logarithmic kernels. These expansions are obtained in terms of an interpolation operator and are useful for the application of Richardson extrapolation and for obtaining sharper error bounds. We also present and analyse a family of fully discrete spline Galerkin methods for the solution of the same equations. Following the analysis of Galerkin methods, we show the existence of asymptotic expansions of the error. Received May 18, 1995 / Revised version received April 11, 1996     

Upper norm bounds for the inverse of locally doubly strictly diagonally dominant matrices with its applications in linear complementarity problems

Numerical Algorithms - In this paper, we present two error bounds for the linear complementarity problems (LCPs) of locally doubly strictly diagonally dominant (LDSDD) matrices. The error bounds...     

A-priori error estimates of Galerkin backward differentiation methods in time-inhomogeneous parabolic problems

Summary Backward differentiation methods up to orderk=5 are applied to solve linear ordinary and partial (parabolic) differential equations where in the second case the space variables are discretized by Galerkin procedures. Using a mean square norm over all considered time levels a-priori error estimates are derived. The emphasis of the results lies on the fact that the obtained error bounds do not depend on a Lipschitz constant and the dimension of the basic system of ordinary differential equations even though this system is allowed to have time-varying coefficients. It is therefore possible to use the bounds to estimate the error of systems with arbitrary varying dimension as they arise in the finite element regression of parabolic problems.     

Backward error and perturbation bounds for high order Sylvester tensor equation

In this article, we investigate the backward error and perturbation bounds for the high order Sylvester tensor equation (STE). The bounds of the backward error and three types of upper bounds for the perturbed STE with or without dropping the second order terms are presented. The classic perturbation results for the Sylvester equation are extended to the high order case.     

Matrix numerical integration algorithms for rapidly oscillating functions

Efficient general quadrature formulas with nodes of arbitrary multiplicity are constructed for numerical integration of rapidly oscillating functions. The quadrature weights of these formulas are derived in explicit form in terms of easily evaluated integrals of products of a rapidly oscillating function and elementary basis functions and also in terms of elements of the inverse of the Vandermonde matrix. Error bounds are established for the quadrature formulas, which depend both on the integration increment and on the oscillation parameters. Necessary conditions are proved when the relative error is bounded and tends to zero with the increase of the oscillation parameters. Two-sided interpolation polynomials are applied to obtain easily computed posterior error bounds for quadrature formulas.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 75, pp. 12–20, 1991.     

Pivoting strategies leading to diagonal dominance by rows

Pivoting strategies for Gaussian elimination leading to upper triangular matrices which are diagonally dominant by rows are studied. Forward error analysis of triangular systems whose coefficient matrices are diagonally dominant by rows is performed. We also obtain small bounds of the backward errors for the pivoting strategies mentioned above. Our examples of matrices include H-matrices and some generalizations of diagonally dominant matrices, and scaled partial pivoting for the 1-norm is an example of these pivoting strategies. In the case of an M-matrix, a pivoting strategy of computational complexity is proposed, which satisfies all the results of the paper. Received June 6, 1997 / Revised version received October 27, 1997     

Coderivative Conditions for Error Bounds of γ-paraconvex Multifunctions

In this paper, error bounds for ??-paraconvex multifunctions are considered. Characterizations of a ??-paraconvex multifunction are given. In terms of normal cone and coderivative, some results on the existence of error bounds are presented.     

Residual-based a posteriori error estimates of nonconforming finite element method for elliptic problems with Dirac delta source terms

Two residual-based a posteriori error estimators of the nonconforming Crouzeix-Raviart element are derived for elliptic problems with Dirac delta source terms.One estimator is shown to be reliable and efficient,which yields global upper and lower bounds for the error in piecewise W1,p seminorm.The other one is proved to give a global upper bound of the error in Lp-norm.By taking the two estimators as refinement indicators,adaptive algorithms are suggested,which are experimentally shown to attain optimal convergence orders.