Two-step Iteration Research Articles

Strong Convergence Theorems for Hybrid Mixed Type Nonlinear Mappings in Banach Spaces

Abstract In this paper, we introduce a new two-step iteration scheme of hybrid mixed type for two asymptotically nonexpansive self mappings and two asymptotically nonexpansive non-self mappings in the intermediate sense and establish some strong convergence theorems for mentioned scheme and mappings in Banach spaces. Our results extend and generalize the corresponding results recently announced by Wei and Guo [16] (Comm. Math. Res. 31(2015), 149-160) and many others.

Improved convergence theorems of the two-step modulus-based matrix splitting and synchronous multisplitting iteration methods for solving linear complementarity problems

In this paper, the convergence conditions of the two-step modulus-based matrix splitting and synchronous multisplitting iteration methods for solving linear complementarity problems of H-matrices are weakened. The convergence domain given by the proposed theorems is larger than the existing ones.

Wideband array self-calibration and DOA estimation under large position errors

For wideband array signals, existing self-calibration methods use a first order approximation to simplify the estimation procedure under the assumption of small sensor position errors. However, the performance will degrade severely as the position errors increase. In this paper, a novel self-calibration method based on the time-frequency distributions (TFDs) is proposed to relieve the small errors constraint. We firstly develop an approach for wideband signals to select the TFDs of each source. Then the TFDs are utilized to obtain the propagation delays, which have a simple linear relationship with the sensor positions. The direction of arrival (DOA) and positions are finally estimated by a two-step iteration without any approximation. Simulation results demonstrate that the proposed method can achieve accurate DOA and position estimation over a broad range of position errors with lower computational complexity, and outperforms existing self-calibration algorithms.

A two-step iteration mechanism for speckle reduction in optical coherence tomography

Abstract Optical coherence tomography (OCT) is an imaging tool that has been widely utilized for various disease diagnoses for its noninvasive and high-resolution properties. Due to the dual role of speckles in the imaging process, however, OCT images suffer from the unavoidable speckle noise, which is usually regarded to be multiplicative in nature and reduces image contrast and resolution. In this study, we propose to categorize OCT speckle noise into additive portion and multiplicative portion, and present a simple two-step iteration (TSI) mechanism to suppress such noises separately. With the augmented Lagrange minimization (ALM) method adopted to recover a low-rank image, the first step of TSI is to remove the additive Gaussian noise, while the second step of TSI is to suppress the multiplicative noise by employing a split Bregman method (SBM) to solve the total-variation (TV) de-noising problem. Extensive experiments with OCT images of the swine eye, human and rabbit retina are conducted to verify the effectiveness of the proposed method. Results show that the proposed TSI method outperforms the existing methods in different cases. Specifically, TSI helps improve the peak signal-to-noise ratio (PSNR) and structure similarity (SSIM) of the swine eye images from 17.19 dB to 33 dB and 0.12 to 0.92, respectively, with the important structural details well preserved. The clearer boundaries and higher image contrast obtained with the TSI method would largely facilitate image analyses and interpretations for the clinical applications of OCT systems.

Analysis and Minimization of Detent End Force in Linear Permanent Magnet Synchronous Machines

In this paper, the end forces caused by the longitude end effects in linear permanent magnet synchronous machines (LPMSMs) are analyzed and minimized. First, the left/right-end forces are calculated based on an analytical model and the Maxwell stress tensor, in which the optimal integration surfaces are investigated. Then, based on the spectrum analysis of the left/right-end forces, two different methods are adopted to minimize the fundamental and high-order harmonics, respectively. The optimal length of the primary iron is obtained from the phase difference of the fundamental and a two-step iteration instead of the trial-and-error with the finite element method. Furthermore, step-skewed auxiliary irons are added to the primary end to eliminate the high-order harmonics. Third, to reduce the secondary end effect when the primary moves to the secondary end, a compensation method of adding mirror permanent magnet is proposed and good results are obtained. Finally, an LPMSM prototype is manufactured and experiments are conducted. The experimental results verify the theoretical study.

Nonconvex Policy Search Using Variational Inequalities.

Policy search is a class of reinforcement learning algorithms for finding optimal policies in control problems with limited feedback. These methods have been shown to be successful in high-dimensional problems such as robotics control. Though successful, current methods can lead to unsafe policy parameters that potentially could damage hardware units. Motivated by such constraints, we propose projection-based methods for safe policies. These methods, however, can handle only convex policy constraints. In this letter, we propose the first safe policy search reinforcement learner capable of operating under nonconvex policy constraints. This is achieved by observing, for the first time, a connection between nonconvex variational inequalities and policy search problems. We provide two algorithms, Mann and two-step iteration, to solve the above problems and prove convergence in the nonconvex stochastic setting. Finally, we demonstrate the performance of the algorithms on six benchmark dynamical systems and show that our new method is capable of outperforming previous methods under a variety of settings.

Adaptive differential correspondence imaging based on sorting technique

We develop an adaptive differential correspondence imaging (CI) method using a sorting technique. Different from the conventional CI schemes, the bucket detector signals (BDS) are first processed by a differential technique, and then sorted in a descending (or ascending) order. Subsequently, according to the front and last several frames of the sorted BDS, the positive and negative subsets (PNS) are created by selecting the relative frames from the reference detector signals. Finally, the object image is recovered from the PNS. Besides, an adaptive method based on two-step iteration is designed to select the optimum number of frames. To verify the proposed method, a single-detector computational ghost imaging (GI) setup is constructed. We experimentally and numerically compare the performance of the proposed method with different GI algorithms. The results show that our method can improve the reconstruction quality and reduce the computation cost by using fewer measurement data.

On different results for new three step iteration process in Banach spaces.

In this paper we propose a new iteration process, called AK iteration process, for approximation of fixed points for contraction mappings. We show that our iteration process is faster than the leading Vatan Two-step iteration process for contraction mappings. Numerical examples are given to support the analytic proofs. Stability of AK iteration process and data dependence result for contraction mappings by employing AK iteration process are also discussed.

Convergence Rate of Some Two-Step Iterative Schemes in Banach Spaces

This article proves some theorems to approximate fixed point of Zamfirescu operators on normed spaces for some two-step iterative schemes, namely, Picard-Mann iteration, Ishikawa iteration, S-iteration, and Thianwan iteration, with their errors. We compare the aforementioned iterations using numerical approach; the results show that S-iteration converges faster than other iterations followed by Picard-Mann iteration, while Ishikawa iteration is the least in terms of convergence rate. These results also suggest the best among two-step iterative fixed point schemes in the literature.

Indestructibility properties of remarkable cardinals

Remarkable cardinals were introduced by Schindler, who showed that the existence of a remarkable cardinal is equiconsistent with the assertion that the theory of $${L(\mathbb R)}$$L(R) is absolute for proper forcing (Schindler in Bull Symbolic Logic 6(2):176---184, 2000). Here, we study the indestructibility properties of remarkable cardinals. We show that if ? is remarkable, then there is a forcing extension in which the remarkability of ? becomes indestructible by all <?-closed ≤?-distributive forcing and all two-step iterations of the form $${Add(\kappa,\theta)*\dot{\mathbb R}}$$Add(?,?)?R?, where $${\dot{\mathbb R}}$$R? is forced to be <?-closed and ≤?-distributive. In the process, we introduce the notion of a remarkable Laver function and show that every remarkable cardinal carries such a function. We also show that remarkability is preserved by the canonical forcing of the GCH.

Semiparametric Pseudo-Likelihoods in Generalized Linear Models With Nonignorable Missing Data

We consider identifiability and estimation in a generalized linear model in which the response variable and some covariates have missing values and the missing data mechanism is nonignorable and unspecified. We adopt a pseudo-likelihood approach that makes use of an instrumental variable to help identifying unknown parameters in the presence of nonignorable missing data. Explicit conditions for the identifiability of parameters are given. Some asymptotic properties of the parameter estimators based on maximizing the pseudo-likelihood are established. Explicit asymptotic covariance matrix and its estimator are also derived in some cases. For the numerical maximization of the pseudo-likelihood, we develop a two-step iteration algorithm that decomposes a nonconcave maximization problem into two problems of maximizing concave functions. Some simulation results and an application to a dataset from cotton factory workers are also presented.

Viscosity iteration method in CAT(0) spaces without the nice projection property

A complete CAT(0) space X is said to have the nice projection property (property $\mathcal{N}$ for short) if its metric projection onto a geodesic segment preserves points on each geodesic segment, that is, for any geodesic segment L in X and $x,y\in X$ , $m\in[x,y]$ implies $P_{L}(m)\in[P_{L}(x), P_{L}(y)]$ , where $P_{L}$ denotes the metric projection from X onto L. In this paper, we prove a strong convergence theorem of a two-step viscosity iteration method for nonexpansive mappings in CAT(0) spaces without the condition on the property $\mathcal{N}$ . Our result gives an affirmative answer to a problem raised by Piatek (Numer. Funct. Anal. Optim. 34:1245-1264, 2013).

Two-Step Modulus-Based Synchronous Multisplitting Iteration Methods for Linear Complementarity Problems

Two-Step Modulus-Based Synchronous Multisplitting Iteration Methods for Linear Complementarity Problems

Learning a Tracking and Estimation Integrated Graphical Model for Human Pose Tracking.

We investigate the tracking of 2-D human poses in a video stream to determine the spatial configuration of body parts in each frame, but this is not a trivial task because people may wear different kinds of clothing and may move very quickly and unpredictably. The technology of pose estimation is typically applied, but it ignores the temporal context and cannot provide smooth, reliable tracking results. Therefore, we develop a tracking and estimation integrated model (TEIM) to fully exploit temporal information by integrating pose estimation with visual tracking. However, joint parsing of multiple articulated parts over time is difficult, because a full model with edges capturing all pairwise relationships within and between frames is loopy and intractable. In previous models, approximate inference was usually resorted to, but it cannot promise good results and the computational cost is large. We overcome these problems by exploring the idea of divide and conquer, which decomposes the full model into two much simpler tractable submodels. In addition, a novel two-step iteration strategy is proposed to efficiently conquer the joint parsing problem. Algorithmically, we design TEIM very carefully so that: 1) it enables pose estimation and visual tracking to compensate for each other to achieve desirable tracking results; 2) it is able to deal with the problem of tracking loss; and 3) it only needs past information and is capable of tracking online. Experiments are conducted on two public data sets in the wild with ground truth layout annotations, and the experimental results indicate the effectiveness of the proposed TEIM framework.

Adaptive Critic Design Using Policy Iteration Technique for LTI Systems: A Comprehensive Performance Analysis

This paper presents adaptive critic design (ACD) using policy iteration technique for adaptive optimal control of continuous-time linear time-invariant (LTI) dynamical systems. A comprehensive performance analysis of control scheme is presented in this paper. ACD using online policy iteration technique provides an adaptive optimal control solution for an infinite horizon problem subject to the real-time dynamics of a continuous-time system. In general, ACD involves a critic and an actor, where the critic evaluates the performance of present control policy and generates a critic signal to update the control action for performance improvement, and the actor provides the control input to the system being controlled. Policy iteration technique which is based on actor–critic structure consists of two-step iteration: policy evaluation and policy improvement. In this paper, the application of control scheme is implemented for a practical example of LTI systems–load frequency control of power system. The modeling, simulation results, and analysis are presented for both of system models without and with integral control. The comparative performance investigation of adaptive critic control scheme and linear quadratic regulator is also presented.

Strong and △-convergence theorems for two asymptotically nonexpansive mappings in the intermediate sense in CAT ( 0 ) spaces

In this paper, we study strong and △-convergence for a newly defined two-step iteration process involving two asymptotically nonexpansive mappings in the intermediate sense which is wider than the class of asymptotically nonexpansive mappings in the setting of \(\operatorname{CAT}(0)\) spaces. Our results generalize, unify and extend many known results from the existing literature.

Two-step iteration scheme for nonexpansive mappings in Banach space

Two-step iteration scheme for nonexpansive mappings in Banach space

Forcing with copies of countable ordinals

Let \alpha be a countable ordinal and \P(\alpha) the collection of its subsets isomorphic to \alpha. We show that the separative quotient of the set \P (\alpha) ordered by the inclusion is isomorphic to a forcing product of iterated reduced products of Boolean algebras of the form P(\omega ^\gamma)/I(\omega ^\gamma), where \gamma is a limit ordinal or 1 and I(\omega ^\gamma) the corresponding ordinal ideal. Moreover, the poset \P(\alpha) is forcing equivalent to a two-step iteration P(\omega)/Fin * \pi, where \pi is an \omega_1-closed separative pre-order in each extension by P(\omega)/Fin and, if the distributivity number is equal to\omega_1, to P(\omega)/Fin. Also we analyze the quotients over ordinal ideals P(\omega ^\delta)/I(\omega ^\delta) and their distributivity and tower numbers.

Application of policy iteration technique based adaptive optimal control design for automatic voltage regulator of power system

This paper presents an adaptive optimal control design approach for automatic voltage regulator (AVR) system using policy iteration technique based adaptive critic scheme. The infinite horizon optimal control design using linear quadratic regulator (LQR) by solution of algebraic Riccati equation (ARE) and Hamilton–Jacobi–Bellman (HJB) equation require the complete knowledge of the system dynamics, and giving solution offline. Policy iteration technique which is based on actor–critic structure consists of two-step iteration: policy evaluation and policy improvement. The online policy iteration technique based control scheme gives online the continuous-time adaptive optimal control solution without using the complete knowledge of the system’s internal dynamics. The knowledge of systems internal dynamics (i.e. matrix A) is not needed for evaluation of cost or the update of control policy; only the knowledge of input matrix B is required for updating the control policy. Thus this control scheme becomes partially model-free. The proposed control scheme has been implemented for AVR system for its both models neglecting and including sensor dynamics. The simulation results and performance analysis are presented to justify the robustness & effectiveness of control scheme. The comparative performance analysis of adaptive critic control scheme and LQR is also presented.

Adaptive Optimal Control of Nonlinear Inverted Pendulum System Using Policy Iteration Technique

This paper presents the adaptive optimal control of nonlinear inverted pendulum-cart dynamic system using policy iteration technique based adaptive critic scheme. Inverted pendulum is a highly nonlinear unstable system which is used as a benchmark for implementing the control methods. The optimal control design using Hamilton-Jacobi-Bellman (HJB) equation requires the complete knowledge of the system dynamics with an offline solution. Policy iteration technique which is based on actor-critic structure consists of two-step iteration: policy evaluation and policy improvement. The online policy iteration technique gives online continuous-time adaptive optimal control solution without using the complete knowledge of the system's internal dynamics. In this paper the state regulation control problem of cart position control and stabilization of inverted pendulum in vertically upright position is considered. The simulation results justify the effectiveness of the control scheme.