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.
Papers published on a yearly basis
Papers
More filters
TL;DR: It is shown that, although parallel computers are the main motivation, polynomial smoothers are often surprisingly competitive with Gauss-Seidel smoothers on serial machines.
202 citations
TL;DR: A full-wave solver to model large-scale and complex multiscale structures using the augmented electric field integral equation (A-EFIE), which includes both the charge and the current as unknowns to avoid the imbalance between the vector potential and the scalar potential in the conventional EFIE.
Abstract: We describe a full-wave solver to model large-scale and complex multiscale structures. It uses the augmented electric field integral equation (A-EFIE), which includes both the charge and the current as unknowns to avoid the imbalance between the vector potential and the scalar potential in the conventional EFIE. The formulation proves to be stable in the low-frequency regime with the appropriate frequency scaling and the enforcement of charge neutrality. To conquer large-scale and complex problems, we solve the equation using iterative methods, design an efficient constraint preconditioning, and employ the mixed-form fast multipole algorithm (FMA) to accelerate the matrix-vector product. Numerical tests on various examples show high accuracy and fast convergence. At last, complex interconnect and packaging problems with over one million integral equation unknowns can be solved without the help of a parallel computer.
201 citations
TL;DR: In this paper, the estimates for parabolic Bellman's equations with variable coefficients were obtained for constant and variable coefficients, respectively, and they were extended to the case of variable coefficients.
Abstract: The estimates presented here for parabolic Bellman's equations with variable coefficients extend the ones earlier obtained for constant coefficients.
200 citations
TL;DR: Numerical evidence indicates that when m > n/2 3, where n is the problem dimension, CBB is locally superlinearly convergent, and it is proved that the convergence rate is no better than linear, in general.
Abstract: In the cyclic Barzilai–Borwein (CBB) method, the same Barzilai–Borwein (BB) stepsize is reused for m consecutive iterations. It is proved that CBB is locally linearly convergent at a local minimizer with positive definite Hessian. Numerical evidence indicates that when m > n/2 3, where n is the problem dimension, CBB is locally superlinearly convergent. In the special case m = 3 and n = 2, it is proved that the convergence rate is no better than linear, in general. An implementation of the CBB method, called adaptive cyclic Barzilai–Borwein (ACBB), combines a non-monotone line search and an adaptive choice for the cycle length m. In numerical experiments using the CUTEr test problem library, ACBB performs better than the existing BB gradient algorithm, while it is competitive with the well-known PRP+ conjugate gradient algorithm.
200 citations
TL;DR: In this article, the authors presented new immersed finite element (IFE) methods for solving the popular second order elliptic interface problems on structured Cartesian meshes even if the involved interfaces have nontrivial geometries.
Abstract: This article presents new immersed finite element (IFE) methods for solving the popular second order elliptic interface problems on structured Cartesian meshes even if the involved interfaces have nontrivial geometries. These IFE methods contain extra stabilization terms introduced only at interface edges for penalizing the discontinuity in IFE functions. With the enhanced stability due to the added penalty, not only can these IFE methods be proven to have the optimal convergence rate in an energy norm provided that the exact solution has sufficient regularity, but also numerical results indicate that their convergence rates in both the $H^1$-norm and the $L^2$-norm do not deteriorate when the mesh becomes finer, which is a shortcoming of the classic IFE methods in some situations. Trace inequalities are established for both linear and bilinear IFE functions that are not only critical for the error analysis of these new IFE methods but are also of a great potential to be useful in error analysis for other...
200 citations