Topic
Rate of convergence
About: Rate of convergence is a research topic. Over the lifetime, 31257 publications have been published within this topic receiving 795334 citations. The topic is also known as: convergence rate.
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TL;DR: In this article, the convergence rates for these optimized quadrature formulas are very fast: roughly O(3-N676 ), where N is the number of sample points or function evaluations, and convergence at a rate apparently about twice as fast, O(9.28903-N672 ), can be achieved by using a different approach: best supremum-norm rational approximants to e ≥ 0.
Abstract: Many computational problems can be solved with the aid of contour integrals containing e
z
in the integrand: examples include inverse Laplace transforms, special functions, functions of matrices and operators, parabolic PDEs, and reaction-diffusion equations. One approach to the numerical quadrature of such integrals is to apply the trapezoid rule on a Hankel contour defined by a suitable change of variables. Optimal parameters for three classes of such contours have recently been derived: (a) parabolas, (b) hyperbolas, and (c) cotangent contours, following Talbot in 1979. The convergence rates for these optimized quadrature formulas are very fast: roughly O(3-N
), where N is the number of sample points or function evaluations. On the other hand, convergence at a rate apparently about twice as fast, O(9.28903-N
), can be achieved by using a different approach: best supremum-norm rational approximants to e
z
for z∈(–∞,0], following Cody, Meinardus and Varga in 1969. (All these rates are doubled in the case of self-adjoint operators or real integrands.) It is shown that the quadrature formulas can be interpreted as rational approximations and the rational approximations as quadrature formulas, and the strengths and weaknesses of the different approaches are discussed in the light of these connections. A MATLAB function is provided for computing Cody–Meinardus–Varga approximants by the method of Caratheodory–Fejer approximation.
208 citations
TL;DR: This paper proposes a globally and quadratically convergent algorithm, which is based on smoothing and regularization methods, for solving monotone SOCCP, and studies strong semismoothness and Jacobian consistency, which play an important role in establishing quadratic convergence of the algorithm.
Abstract: The second-order cone complementarity problem (SOCCP) is a wide class of problems containing the nonlinear complementarity problem (NCP) and the second-order cone programming problem (SOCP). Recently, Fukushima, Luo, and Tseng [SIAM J. Optim., 12 (2001), pp. 436--460] extended some merit functions and their smoothing functions for NCP to SOCCP. Moreover, they derived computable formulas for the Jacobians of the smoothing functions and gave conditions for the Jacobians to be invertible. In this paper, we propose a globally and quadratically convergent algorithm, which is based on smoothing and regularization methods, for solving monotone SOCCP. In particular, we study strong semismoothness and Jacobian consistency, which play an important role in establishing quadratic convergence of the algorithm. Furthermore, we examine the effectiveness of the algorithm by means of numerical experiments.
208 citations
TL;DR: A new multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces is introduced.
Abstract: We introduce a new multiscale finite element method which is
able to accurately capture solutions of elliptic interface problems with high
contrast coefficients by using only coarse quasiuniform meshes, and without
resolving the interfaces. A typical application would be the modelling of flow
in a porous medium containing a number of inclusions of low (or high) permeability
embedded in a matrix of high (respectively low) permeability. Our
method is H^1- conforming, with degrees of freedom at the nodes of a triangular
mesh and requiring the solution of subgrid problems for the basis functions on
elements which straddle the coefficient interface but which use standard linear
approximation otherwise. A key point is the introduction of novel coefficientdependent
boundary conditions for the subgrid problems. Under moderate
assumptions, we prove that our methods have (optimal) convergence rate of
O(h) in the energy norm and O(h^2) in the L_2 norm where h is the (coarse)
mesh diameter and the hidden constants in these estimates are independent
of the “contrast” (i.e. ratio of largest to smallest value) of the PDE coefficient.
For standard elements the best estimate in the energy norm would be
O(h^(1/2−e)) with a hidden constant which in general depends on the contrast.
The new interior boundary conditions depend not only on the contrast of the
coefficients, but also on the angles of intersection of the interface with the
element edges.
207 citations
TL;DR: A variant of the nonsymmetric Lanczos method, rational Lanczos, is shown to yield a rational interpolant (multi-point Padé approximant) for the large-scale system.
Abstract: This paper presents a model reduction method for large-scale linear systems that is based on a Lanczos-type approach. A variant of the nonsymmetric Lanczos method, rational Lanczos, is shown to yield a rational interpolant (multi-point Padi approximant) for the large-scale system. An exact expression for the error in the interpolant is derived. Examples are utilized to demonstrate that the rational Lanczos method provides opportunities for significant improvements in the rate of convergence over single-point Lanczos approaches.
207 citations
TL;DR: A method is given that does a better job of computing eigenvalues from the interior of the spectrum of a large matrix and a priori bounds can be given for the accuracy of interior eigenvalue and eigenvector approximations.
207 citations