Some Convergence Properties of a Random Closed Set Sequence

The so-called "adaptive control problem" has received attention by theoreticians and practitioners alike for the past twenty-five years. Elegant and useful theoretical advances have been made in the last decade, and especially in the past three years, that have unified diverse approaches. However, despite the fact that the central question of global asymptotic stability of the overall feedback adaptive loop has been resolved for a substantial class of adaptive controllers--at least in the deterministic case--the mechanism of adaptation and the convergence properties in general of already existing algorithms are still far from being well understood. As a consequence, there is a significant gap between the available (theoretical) methodologies and the potential applications of such adaptive schemes. This paper contains the first results of a research effort whose long range objective is in developing a methodology of design for adaptive control systems by attempting to unify promising concepts based primarily upon hyperstability theory and stochastic optimal control, respectively, with some common sense engineering techniques. In this context, special emphasis is given on the convergence properties--in terms of rates and transient behavior--of a large class of adaptive control algorithms.

Impact of HVDC line on the convergence property of AC/DC power flow calculation

This study focuses on the solution of implicit diff erence equations, which are very difficult to compute in parallel for the diffusion equation. For improving the convergence rates and the properties of gradual-approach convergence of Segment-Classic-Implicit-Iterative (SCII) and Segment-Crank-Nicolson-Iterative (SCNI) algorithms realizing efficient iterative computation in parallel by segmenting grid domains, SCII and SCNI algorithms with accelerated convergence are studied and improved through inserting classic impl icit schemes and Crank-Nicolson schemes into them respectively. The SCII and SCNI algorithms with accelerated convergence, which can be decomposed into smaller strictly tri-diagonally dom inant subsystems, are solved by using a double - sweep algorithm. In the present study, general stru ctures of SCII and SCNI algorithms with accelerated convergence are constructed with matrix forms. The convergent rates are estimated and properties of gradual-approach convergence about diffusion equation are described by a splitting coefficient matrix in detail. These algorithms improve the convergence rates in iteration while making the properties of gradual-approach convergence reach two ranks. The efficiency of computation is greatly enhanced. In addition, the algorithms are extended to the case o f two-dimensional problem by studying Peaceman- Rachford scheme into which classic implicit schemes are inserted alternately. Numerical computations employing SCII and SCNI algorithms with accelerated convergence are made to SGL/Challenge L with 8 CPUs as examples. Theoretical analyses and numerical exemplifications show that the parallel iterative algorithms with accelerated convergence f or solving one-dimensional diffusion equations are more efficient in computation and have much better convergent rates and properties of gradual- approach convergence.

The minimal polynomial and reduced rank extrapolation algorithms are two acceleration of convergence methods far sequences of vectors. In a recent survey these methods were tested and compared with the scalar, vector, and topological epsilon algorithms, and were observed to be more efficient than the latter. It was also observed that the two methods have similar convergence properties. The purpose of the present work is to analyze the convergence and stability properties of these methods, and to show that they are bona fide acceleration methods when applied to a class of vector sequences that includes those sequences obtained from systems of linear equations by using matrix iterative methods.

In this study, a learning control (LC) with state feedback is proposed for a class of non-linear systems with non-parametric uncertainties to track periodic trajectory. The proposed method offers a better convergence property than other LC methods, that is, the tracking error is bounded by an envelope monotonically decreasing to zero. It also links the controller parameters to the sum of total tracking error over the entire time horizon, thus it provides an explicit way to design the controller with respect to the control performance.

This paper concerns a generalization of the design of the adaptive observer, and an investigation of its convergence properties. By application of hyperstability theory the design freedom is enlarged to permit use of time variable adaptive gains. Based on the assumption of periodic excitation, and with the aid of Floquet Theory a new convergence criterion is defined. By this means the effects of adaptive gains and input frequencies upon convergence time are evaluated. Structural properties of the adaptive observer are compared with other adaptive schemes, in particular with regard to bias due to noise.

The convergence behavior of type-0 through type-3 stack filters is investigated. It is shown that stack filters of type-0 through type-2 all possess the convergence property; that is, they filter any input signal to a root after consecutive passes of the filter under any appending strategy. A counterexample is given to show that not all type-3 stack filters have this convergence property. The rate of convergence for convergent stack filters is also shown. It is shown that stack filters of type-0 will take at most a single pass to filter any input signal to a root. The rate of convergence of type-1 and type-2 stack filters is shown to be linear in the length of the input signal. >

Recent years have witnessed a plethora of learning-based solutions for congestion control (CC) that demonstrate better performance over traditional TCP schemes. However, they fail to provide consistently good convergence properties, including <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fairness, fast convergence</i> and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">stability</i>, due to the mismatch between their objective functions and these properties. Despite being intuitive, integrating these properties into existing learning-based CC is challenging, because: 1) their training environments are designed for the performance optimization of single flow but incapable of cooperative multi-flow optimization, and 2) there is no directly measurable metric to represent these properties into the training objective function. We present Astraea, a new learning-based congestion control that ensures fast convergence to fairness with stability. At the heart of Astraea is a multi-agent deep reinforcement learning framework that explicitly optimizes these convergence properties during the training process by enabling the learning of interactive policy between multiple competing flows, while maintaining high performance. We further build a faithful multi-flow environment that emulates the competing behaviors of concurrent flows, explicitly expressing convergence properties to enable their optimization during training. We have fully implemented Astraea and our comprehensive experiments show that Astraea can quickly converge to fairness point and exhibit better stability than its counterparts. For example, Astraea achieves near-optimal bandwidth sharing (i.e., fairness) when multiple flows compete for the same bottleneck, delivers up to 8.4× faster convergence speed and 2.8× smaller throughput deviation, while achieving comparable or even better performance over prior solutions.

: The paper began as a study of the convergence properties of an algorithm of H. S. Witsenhausen. His algorithm deals with a linear differential system driven by a bounded perturbation and a bounded control, with a cost that is a convex function of the state reached at a given final time. The controller receives exact samples of the current state of the system at a finite number of sampling times and seeks to minimize the supremum (over-all possible perturbations) of the cost. Witsenhausen proposes a sequence of approximate algorithms, all related to the boundary X of a certain d-dimensional compact convex set K associated with the problem, and shows that the sequence has desirable convergence properties for all points of a certain subset X sub e of X. For the procedure to be fully applicable, X sub e should be all of X, and he shows that this is the case if K is polyhedral or strictly convex. Here we show that X sub e = X when d = 2, thus proving a conjecture of J. B. Kruskal, but that the situation is more complicated when d = or > 3. Specifically, X sub e must be a dense G sub delta subset of X but its (d-1)-dimensional measure may be zero. Thus Witsenhausen's algorithm has good convergence properties with respect to category but not necessarily with respect to measure. (Author)

&lt;p&gt;&lt;b&gt;Metric Learning has proved valuable in information retrieval and classification problems, with many algorithms using statistical formulations to their advantage. Optimal Transport, a powerful framework for computing distances between probability distributions, has seen significant uptake in Machine Learning and some applications in Metric Learning. The Bures-Wasserstein distance, a particular formulation of Optimal Transport, has seen limited application in Metric Learning despite being well suited to the field. In this thesis Linear Metric Learning algorithms incorporating the Bures-Wasserstein distance are developed, utilising its novel properties. The first set of algorithms we contribute use the Bures-Wasserstein distance for learning linear metrics using Cyclic Projections, Riemannian barycenters and gradient-based algorithms. The convergence properties and classification performance of these algorithms are then compared to benchmark Linear Metric Learning algorithms. Our algorithms achieve competitive classification performance on feature and image datasets and converge reliability.&lt;/b&gt;&lt;/p&gt; &lt;p&gt;The second set of contributions extend Linear Metric Learning with the Bures-Wasserstein distance to a low-rank context to address issues related to scalability. An extension of the previous Cyclic Projections algorithm and a Riemannian optimisation method are developed. Convergence and classification experiments analogous to the full-rank case are conducted to examine how the reduced rank affects both of these performance metrics. Both algorithms converge reliably under a variety of synthetic conditions. Reduction in rank generally decreased the training time for algorithms until effects caused by the size of the training data dominated. Reducing rank either left classification error unchanged or increased it. With respect to other low-rank Linear Metric Learning algorithms our algorithms again performed competitively with respect to classification error. &lt;/p&gt; &lt;p&gt;The last set of contributions consider Linear Metric Learning on Symmetric Positive Definite (SPD) matrices. Some datasets are expressed as or are suited to a SPD representation. As such, Metric Learning algorithms that respect the structure of these matrices have found success when adapted to this kind of data. Another extension of the Cyclic Projections algorithm is developed to use the Generalised Bures-Wasserstein distance on SPD matrices. Two other methods based on a ground metric formulation of the Bures-Wasserstein distance for learning linear metrics on SPD data are also introduced. One ground metric formulation uses an approximation (inspired by the Sinkhorn distance) to increase its scalability. Convergence experiments on synthetic problems adapted to SPD matrices are conducted on the ground metric algorithms to establish their convergence properties. Classification experiments are also conducted with respect to several types of SPD data. While the Cyclic Projections method is competitive, the ground metric algorithms have poor performance. Despite their poor classification performance, the approximation method does evaluate faster than the Generalised Bures-Wasserstein distance, potentially offering some scalability improvements.&lt;/p&gt;

This paper presents a method for accurately treating active power losses in linear (DC) optimal power flow models. Quadratic nodal losses are approximated by iteratively adding linear constraints. In each iteration a linear programming problem is solved, and constraints on the nodal loss functions are built as linearizations around the current system operating state. The performance of the proposed model - in terms of computational time and convergence properties - is demonstrated on the IEEE 118 bus test system.

Evolutionary Multi-objective Optimization(EMO) is becoming a hot research area and quite a few aspects of Multi-objective Evolutionary algorithms(MOEAs) have been studied and discussed. However there are still few literatures discussing the roles of search and selection operators in MOEAs. This paper studied their roles by a representative combinatorial Multi-objective Problem(MOP): Multi-objective TSP. In the new MOEA, We adopt an efficient search operator, which has the properties of both crossover and mutation, to generate the new individuals and chose two kinds of selection operators: Family Competition and Population Competition with probabilities to realize selection. The simulation experiments showed that this new MOEA could get good uniform solutions representing the Pareto Front and outperformed SPEA in almost every simulation run on this problem. Furthermore, we analyzed its convergence property using Finite Markov Chain and proved that it could converge to Pareto Front with probability 1.We also find that the convergence property of MOEAs has much relationship with search and selection operators.

Accurate numerical, integral methods for computing drift-kinetic Trubnikov-Rosenbluth potentials

Local stress in periodic composites via the Riesz summability method

We consider the streamline-diffusion finite element method with finite elements of lowest order for solving convection-diffusion problems. Our investigations cover both conforming and nonconforming finite element approximations on triangular and quadrilateral meshes. Although the considered finite elements are of the same interpolation order their stability and approximation properties are quite different. We give a detailed overview on the stability and the convergence properties in the L 2- and in the streamline–diffusion norm. Numerical experiments show that often the theoretical predictions on the convergence properties are sharp.