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.

Effective and practical h–p‐version adaptive analysis procedures for the finite element method

Abstract: Two practical and effective, h–p-type, finite element adaptive procedures are presented. The procedures allow not only the final global energy norm error to be well estimated using hierarchic p-refinement, but in addition give a nearly optimal mesh. The design of this is guided by the local information computed on the previous mesh. The desired accuracy can always be obtained within one or at most two h–p-refinements. The rate of convergence of the adaptive h–p-version analysis procedures has been tested for some examples and found to be very strong. The presented procedures can easily be incorporated into existing p- or h-type code structures.

Tensor SVD: Statistical and Computational Limits

Abstract: In this paper, we propose a general framework for tensor singular value decomposition (tensor singular value decomposition (SVD)), which focuses on the methodology and theory for extracting the hidden low-rank structure from high-dimensional tensor data. Comprehensive results are developed on both the statistical and computational limits for tensor SVD. This problem exhibits three different phases according to the signal-to-noise ratio (SNR). In particular, with strong SNR, we show that the classical higher-order orthogonal iteration achieves the minimax optimal rate of convergence in estimation; with weak SNR, the information-theoretical lower bound implies that it is impossible to have consistent estimation in general; with moderate SNR, we show that the non-convex maximum likelihood estimation provides optimal solution, but with NP-hard computational cost; moreover, under the hardness hypothesis of hypergraphic planted clique detection, there are no polynomial-time algorithms performing consistently in general.

Continuous Mesh Framework Part II: Validations and Applications

Abstract: This paper gives a numerical validation of the continuous mesh framework introduced in Part I [A. Loseille and F. Alauzet, SIAM J. Numer. Anal., 49 (2011), pp. 38-60]. We numerically show that the interpolation error can be evaluated analytically once analytical expressions of a mesh and a function are given. In particular, the strong duality between discrete and continuous views for the interpolation error is emphasized on two-dimensional and three-dimensional examples. In addition, we show the ability of this framework to predict the order of convergence, given a specific adaptive strategy defined by a sequence of continuous meshes. The continuous mesh concept is then used to devise an adaptive strategy to control the $\mathbf{L}^p$ norm of the continuous interpolation error. Given the $\mathbf{L}^p$ norm of the continuous interpolation error, we derive the optimal continuous mesh minimizing this error. This exemplifies the potential of this framework, as we use a calculus of variations that is not defined on the space of discrete meshes. Anisotropic adaptations on analytical functions correlate the optimal predicted theoretical order of convergence. The extension to a solution of nonlinear PDEs is also given. Comparisons with experiments show the efficiency and the accuracy of this approach.

Finito: A faster, permutable incremental gradient method for big data problems

Abstract: Recent advances in optimization theory have shown that smooth strongly convex finite sums can be minimized faster than by treating them as a black box "batch" problem. In this work we introduce a new method in this class with a theoretical convergence rate four times faster than existing methods, for sums with sufficiently many terms. This method is also amendable to a sampling without replacement scheme that in practice gives further speed-ups. We give empirical results showing state of the art performance.