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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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Journal ArticleDOI
TL;DR: In this article, a simple and robust approach for retrieving arbitrary complex-valued fields from three or more diffraction intensity recordings was proposed and experimentally demonstrated, and the convergence rate is rapid.
Abstract: We propose and experimentally demonstrate a simple and robust approach for retrieving arbitrary complex-valued fields from three or more diffraction intensity recordings. We need no a priori knowledge about the object field. The convergence rate is rapid. We obtained good results using experimental data with only 80 iterations (160 fast Fourier transforms). The method does not suffer any stagnation or ambiguity problem, and it also exhibits a high immunity to noise. The technique exhibits great potential in lensless phase-contrast imaging, wave-front sensing, and metrology for a wide spectral range.

196 citations

Journal ArticleDOI
TL;DR: In this paper, localized approximations of homogenized coefficients of second order divergence form elliptic operators with random statistically homogeneous coefficients, by means of "periodization" and other cut-off procedures were studied.
Abstract: This note deals with localized approximations of homogenized coefficients of second order divergence form elliptic operators with random statistically homogeneous coefficients, by means of “periodization” and other “cut-off” procedures. For instance in the case of periodic approximation, we consider a cubic sample [0 ,ρ ] d of the random medium, extend it periodically in R d and use the effective coefficients of the obtained periodic operators as an approximation of the effective coefficients of the original random operator. It is shown that this approximation converges a.s., as ρ →∞ , and gives back the effective coefficients of the original random operator. Moreover, under additional mixing conditions on the coefficients, the rate of convergence can be estimated by some negative power of ρ which only depends on the dimension, the ellipticity constant and the rate of decay of the mixing coefficients. Similar results are established for approximations in terms of appropriate Dirichlet and Neumann problems localized in a cubic sample [0 ,ρ ] d .

195 citations

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TL;DR: In this article, the authors provide a new proof of the linear convergence of the alternating direction method of multipliers (ADMM) when one of the objective terms is strongly convex, based on a framework for analyzing optimization algorithms introduced in Lessard et al.
Abstract: We provide a new proof of the linear convergence of the alternating direction method of multipliers (ADMM) when one of the objective terms is strongly convex. Our proof is based on a framework for analyzing optimization algorithms introduced in Lessard et al. (2014), reducing algorithm convergence to verifying the stability of a dynamical system. This approach generalizes a number of existing results and obviates any assumptions about specific choices of algorithm parameters. On a numerical example, we demonstrate that minimizing the derived bound on the convergence rate provides a practical approach to selecting algorithm parameters for particular ADMM instances. We complement our upper bound by constructing a nearly-matching lower bound on the worst-case rate of convergence.

195 citations

Journal ArticleDOI
TL;DR: The discontinuous Galerkin finite element method (DGFEM) for the time discretization of parabolic problems is analyzed in the context of the hp-version of theGalerkin method and it is shown that the hp's spectral convergence gives spectral convergence in problems with smooth time dependence.
Abstract: The discontinuous Galerkin finite element method (DGFEM) for the time discretization of parabolic problems is analyzed in the context of the hp-version of the Galerkin method. Error bounds which are explicit in the time steps as well as in the approximation orders are derived and it is shown that the hp-DGFEM gives spectral convergence in problems with smooth time dependence. In conjunction with geometric time partitions it is proved that the hp-DGFEM results in exponential rates of convergence for piecewise analytic solutions exhibiting singularities induced by incompatible initial data or piecewise analytic forcing terms. For the h-version DGFEM algebraically graded time partitions are determined that give the optimal algebraic convergence rates. A fully discrete hp scheme is discussed exemplarily for the heat equation. The use of certain mesh-design principles for the spatial discretizations yields exponential rates of convergence in time and space. Numerical examples confirm the theoretical results.

195 citations

Journal ArticleDOI
TL;DR: A Jacobi-like algorithm for simultaneous diagonalization of commuting pairs of complex normal matrices by unitary similarity transformations is presented, which preserves the special structure of real matrices, quaternion matrices and real symmetric matrices.
Abstract: A Jacobi-like algorithm for simultaneous diagonalization of commuting pairs of complex normal matrices by unitary similarity transformations is presented. The algorithm uses a sequence of similarity transformations by elementary complex rotations to drive the off-diagonal entries to zero. Its asymptotic convergence rate is shown to be quadratic and numerically stable. It preserves the special structure of real matrices, quaternion matrices, and real symmetric matrices.

195 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20241
2023693
20221,530
20212,129
20202,036
20191,995