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
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TL;DR: It is shown that the spectrum of the preconditioned system is contained in a real, positive interval and that the interval bounds can be made independent of the discretization and penalty parameters to construct bounds of the convergence rate of the GMRES method with respect to an energy norm.
Abstract: Block-triangular preconditioners for a class of saddle point problems with a penalty term are considered. An important example is the mixed formulation of the pure displacement problem in linear elasticity. It is shown that the spectrum of the preconditioned system is contained in a real, positive interval and that the interval bounds can be made independent of the discretization and penalty parameters. This fact is used to construct bounds of the convergence rate of the GMRES method with respect to an energy norm. Numerical results are given for GMRES and BI-CGSTAB.
167 citations
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TL;DR: In this article, the authors combine several novel techniques for spectrum simulation in the Eddington computer program which solves the comoving frame equation of transfer coupled with the statistical and radiative equilibrium equations.
Abstract: The study combines several novel techniques for spectrum simulation in the Eddington computer program which solves the comoving frame equation of transfer coupled with the statistical and radiative equilibrium equations. One of these is a generalization of the accelerated lambda iteration (ALI) scheme to include an approximate frequency-derivative operator. This greatly enhances the convergence rate of ALI in optically thick, high-velocity shear flows. Another is a partial linearization technique which is capable of efficiently solving a very large number of rate equations on a moderately sized computer. An expansion opacity and emissivity approximation is derived which makes it possible to determine the effect on the transfer and statistical equilibrium of a very large number of lines not explicitly represented in the frequency grid and additionally to treat line-blanketing from species not explicitly included in the rate equations. The utility of these techniques is illustrated with models of two supernovae.
167 citations
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TL;DR: In this article, a general method for constructing high-order approximation schemes for Hamilton-Jacobi-Bellman equations is given, based on a discrete version of the Dynamic Programming Principle.
Abstract: A general method for constructing high-order approximation schemes for
Hamilton-Jacobi-Bellman equations is given. The method is based on a
discrete version of the Dynamic Programming Principle. We prove a
general convergence result for this class of approximation schemes also
obtaining, under more restrictive assumptions, an estimate in
$L^\infty$
of the order of convergence and of the local truncation error. The
schemes can be applied, in particular, to the stationary linear first
order equation in
${\Bbb R}^n$
. We present several
examples of schemes
belonging to this class and with fast convergence to the solution.
166 citations
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02 May 2016TL;DR: It is demonstrated experimentally that the proposed new stochastic L-BFGS algorithm performs well on large-scale convex and non-convex optimization problems, exhibiting linear convergence and rapidly solving the optimization problems to high levels of precision.
Abstract: We propose a new stochastic L-BFGS algorithm and prove a linear convergence rate for strongly convex and smooth functions. Our algorithm draws heavily from a recent stochastic variant of L-BFGS proposed in Byrd et al. (2014) as well as a recent approach to variance reduction for stochastic gradient descent from Johnson and Zhang (2013). We demonstrate experimentally that our algorithm performs well on large-scale convex and non-convex optimization problems, exhibiting linear convergence and rapidly solving the optimization problems to high levels of precision. Furthermore, we show that our algorithm performs well for a wide-range of step sizes, often differing by several orders of magnitude.
166 citations
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TL;DR: The proposed solution is formulated to ameliorate the limited convergence properties of least-mean-square type distributed adaptive filters with colored inputs to achieve an acceptable misadjustment performance with lower computational and memory cost.
Abstract: We study the problem of distributed estimation based on the affine projection algorithm (APA), which is developed from Newton's method for minimizing a cost function. The proposed solution is formulated to ameliorate the limited convergence properties of least-mean-square (LMS) type distributed adaptive filters with colored inputs. The analysis of transient and steady-state performances at each individual node within the network is developed by using a weighted spatial-temporal energy conservation relation and confirmed by computer simulations. The simulation results also verify that the proposed algorithm provides not only a faster convergence rate but also an improved steady-state performance as compared to an LMS-based scheme. In addition, the new approach attains an acceptable misadjustment performance with lower computational and memory cost, provided the number of regressor vectors and filter length parameters are appropriately chosen, as compared to a distributed recursive-least-squares (RLS) based method.
166 citations