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.

Riemannian Medians and Means With Applications to Radar Signal Processing

Abstract: We develop a new geometric approach for high resolution Doppler processing based on the Riemannian geometry of Toeplitz covariance matrices and the notion of Riemannian p -means. This paper summarizes briefly our recent work in this direction. First of all, we introduce radar data and the problem of target detection. Then we show how to transform the original radar data into Toeplitz covariance matrices. After that, we give our results on the Riemannian geometry of Toeplitz covariance matrices. In order to compute p-means in practical cases, we propose deterministic and stochastic algorithms, of which the convergence results are given, as well as the rate of convergence and error estimates. Finally, we propose a new detector based on Riemannian median and show its advantage over the existing processing methods.

The Direct Discontinuous Galerkin (DDG) Methods for Diffusion Problems

Abstract: A new discontinuous Galerkin finite element method for solving diffusion problems is introduced. Unlike the traditional local discontinuous Galerkin method, the scheme called the direct discontinuous Galerkin (DDG) method is based on the direct weak formulation for solutions of parabolic equations in each computational cell and lets cells communicate via the numerical flux $\widehat{u_x}$ only. We propose a general numerical flux formula for the solution derivative, which is consistent and conservative; and we then introduce a concept of admissibility to identify a class of numerical fluxes so that the nonlinear stability for both one-dimensional (1D) and multidimensional problems are ensured. Furthermore, when applying the DDG scheme with admissible numerical flux to the 1D linear case, $k$th order accuracy in an energy norm is proven when using $k$th degree polynomials. The DDG method has the advantage of easier formulation and implementation and efficient computation of the solution. A series of numerical examples are presented to demonstrate the high order accuracy of the method. In particular, we study the numerical performance of the scheme with different admissible numerical fluxes.

Position-Transitional Particle Swarm Optimization-incorporated Latent Factor Analysis

Abstract: High-dimensional and sparse (HiDS) matrices are frequently found in various industrial applications. A latent factor analysis (LFA) model is commonly adopted to extract useful knowledge from an HiDS matrix, whose parameter training mostly relies on a stochastic gradient descent (SGD) algorithm. However, an SGD-based LFA model's learning rate is hard to tune in real applications, making it vital to implement its self-adaptation. To address this critical issue, this study firstly investigates the evolution process of a particle swarm optimization algorithm with care, and then proposes to incorporate more dynamic information into it for avoiding accuracy loss caused by premature convergence without extra computation burden, thereby innovatively achieving a novel position-transitional particle swarm optimization (P2SO) algorithm. It is subsequently adopted to implement a P2SO-based LFA (PLFA) model that builds a learning rate swarm applied to the same group of LFs. Thus, a PLFA model implements highly efficient learning rate adaptation as well as represents an HiDS matrix precisely. Experimental results on four HiDS matrices emerging from real applications demonstrate that compared with an SGD-based LFA model, a PLFA model no longer suffers from a tedious and expensive tuning process of its learning rate to achieve higher prediction accuracy for missing data.

Second Order Methods for Optimal Control of Time-Dependent Fluid Flow

Abstract: Second order methods for open loop optimal control problems governed by the two-dimensional instationary Navier--Stokes equations are investigated Optimality systems based on a Lagrangian formulation and adjoint equations are derived The Newton and quasi-Newton methods as well as various variants of SQP methods are developed for applications to optimal flow control, and their complexity in terms of system solves is discussed Local convergence and rate of convergence are proved A numerical example illustrates the feasibility of solving optimal control problems for two-dimensional instationary Navier--Stokes equations by second order numerical methods in a standard workstation environment