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.

The convergence rate for difference approximations to general mixed initial boundary value problems

Abstract: The convergence rate is investigated for the solutions to difference schemes approximating the mixed initial boundary value problem for general systems of differential equations. It is shown that if an energy estimate holds, then the extra boundary conditions can be of one order lower accuracy without destroying the convergence rate expected from the approximation at inner points. If the maximal order of the derivatives occurring in the boundary conditions is low enough, then even lower accuracy can be permitted for the extra boundary conditions.

On the Convergence of Policy Iteration in Stationary Dynamic Programming

Abstract: The policy iteration method of dynamic programming is studied in an abstract setting. It is shown to be equivalent to the Newton-Kantorovich iteration procedure applied to the functional equation of dynamic programming. This equivalence is used to obtain the rate of convergence and error bounds for the sequence of values generated by policy iteration. These results are discussed in the context of the finite state Markovian decision problem with compact action space. An example is analyzed in detail.

Sufficient conditions for uniform convergence on layer-adapted grids

Abstract: We study convergence properties of the simple upwind difference scheme and a Galerkin finite element method on generalized Shishkin grids. We derive conditions on the mesh-characterizing function that are sufficient for the convergence of the method, uniformly with respect to the perturbation parameter. These conditions are easy to check and enable one to immediately deduce the rate of convergence. Numerical experiments support these theoretical results and indicate that the estimates are sharp. The analysis is set in one dimension, but can be easily generalized to tensor product meshes in 2D.

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.