Convergence Of Iterative Algorithm Research Articles

A Comprehensive Study on Advancement in Hybrid Contraction and Graphical Analysis of £-Fuzzy Fixed Points with Application

Hybrid contractions serve as a flexible and versatile framework for establishing fixed-point Theorems and analyzing the convergence of iterative algorithms. This paper demonstrates the adapted form of the admissible hybrid fuzzy Z-contraction in the perspective of £-fuzzy set-valued maps for extended ♭-metric spaces. Sufficient criteria for obtaining £-fuzzy fixed points for this contraction have been established. In addition, the hypotheses of its main result are endorsed by some nontrivial supportive examples featuring graphical illustrations. Consequently, the concept of graphical extended ♭-metric spaces is introduced and a £-fuzzy fixed point result in the context of newly defined space is derived. Illustrative examples, incorporating relevant graphs, are provided with the support of a computer simulation to validate the established results, enhancing the understanding of the underlying notions and investigations. The concepts presented here not only considerably improve, enrich, and extend a number of well-known pre-existing fixed-point results but also assemble and merge several ones in the corresponding domain.

Efficient Data Augmentation Techniques for Some Classes of State Space Models

Data augmentation improves the convergence of iterative algorithms, such as the EM algorithm and Gibbs sampler by introducing carefully designed latent variables. In this article, we first propose a data augmentation scheme for the first-order autoregression plus noise model, where optimal values of working parameters introduced for recentering and rescaling of the latent states, can be derived analytically by minimizing the fraction of missing information in the EM algorithm. The proposed data augmentation scheme is then utilized to design efficient Markov chain Monte Carlo (MCMC) algorithms for Bayesian inference of some non-Gaussian and nonlinear state space models, via a mixture of normals approximation coupled with a block-specific reparametrization strategy. Applications on simulated and benchmark real data sets indicate that the proposed MCMC sampler can yield improvements in simulation efficiency compared with centering, noncentering and even the ancillarity-sufficiency interweaving strategy.

Novel Approach to Improve Stability and Convergence of Flowfield Solution Processes: Mode Multigrid

In the field of computational fluid dynamics, stability and convergence problems are often encountered when solving the governing equation. This paper studies the effect of the mode multigrid on the stability and convergence of iterative algorithms. By further analyzing the mechanism for accelerating the convergence of mode multigrid, a new adaptive mode multigrid (AMMG) is proposed, and an adaptive selection criterion is formulated for the parameters of dynamic modal decomposition, which can accurately identify and filter out the unstable modes in the flowfield, thus efficiently obtaining the accurate steady solution. In the cases of laminar flow past circular cylinder and turbulent flow past airfoil, the AMMG significantly improves the stability of the iterative algorithm; in the case of transonic flow past airfoil, the AMMG solves the problem of shock wave shaking during the iteration, and the convergence is significantly improved. In summary, the AMMG method significantly enhances the stability and convergence of the iterative algorithm, and can be used as an effective convergence-preserving computation strategy.

Sharp global convergence guarantees for iterative nonconvex optimization with random data

We consider a general class of regression models with normally distributed covariates, and the associated nonconvex problem of fitting these models from data. We develop a general recipe for analyzing the convergence of iterative algorithms for this task from a random initialization. In particular, provided each iteration can be written as the solution to a convex optimization problem satisfying some natural conditions, we leverage Gaussian comparison theorems to derive a deterministic sequence that provides sharp upper and lower bounds on the error of the algorithm with sample splitting. Crucially, this deterministic sequence accurately captures both the convergence rate of the algorithm and the eventual error floor in the finite-sample regime, and is distinct from the commonly used “population” sequence that results from taking the infinite-sample limit. We apply our general framework to derive several concrete consequences for parameter estimation in popular statistical models including phase retrieval and mixtures of regressions. Provided the sample size scales near linearly in the dimension, we show sharp global convergence rates for both higher-order algorithms based on alternating updates and first-order algorithms based on subgradient descent. These corollaries, in turn, reveal multiple nonstandard phenomena that are then corroborated by extensive numerical experiments.

Algebra of L-Banded Matrices

Convergence is a crucial issue in iterative algorithms. Damping is commonly employed to ensure the convergence of iterative algorithms. The conventional ways of damping are scalar-wise, and either heuristic or empirical. Recently, an analytically optimized vector damping was proposed for memory message-passing (iterative) algorithms. As a result, it yields a special class of covariance matrices called L-banded matrices. In this paper, we show these matrices have broad algebraic properties arising from their L-banded structure. In particular, compact analytic expressions for the LDL decomposition, the Cholesky decomposition, the determinant after a column substitution, minors, and cofactors are derived. Furthermore, necessary and sufficient conditions for an L-banded matrix to be definite, a recurrence to obtain the characteristic polynomial, and some other properties are given. In addition, we give new derivations of the determinant and the inverse.

Convergence rate in structural equation models – analysis of estimation methods and implications in the number of observations

Structural Equation Modeling (SEM) is used to analyze the causal relationships between observable and unobservable variables. Among the assumptions considered, but not essential, for the application of the SEM are the presence of multivariate normality between the data, and the need for a large number of observations, in order to obtain the variances and covariances between the variables. It is not always possible to have access to a sufficiently large number of observations to enable the calculation of parameters, and the convergence of the iterative algorithm is one of the problems in obtaining the results. This work investigates the convergence of iterative algorithms, which minimize the variation of parameters, through a stipulated convergence rate, using the Maximum Likelihood (ML) and Generalized Least Squares (GLS) estimation methods on structural equation models using confirmatory factor analysis (CFA) and regression models. Convergences were evaluated in relation to the number of observations, in order to obtain a minimum quantity sufficient for a convergence rate above 50%. The calculations were performed in the statistical environment R® version 3.4.4, and the results obtained showed a convergence rate above 50% for models estimated by GLS, even with the data showing lack of multivariate normality, kurtosis and accentuated asymmetry. Thus, it was possible to define a minimum number of observations necessary for an adequate convergence of the iterative algorithms in obtaining the necessary parameters.

Convergence of Iterative Algorithms in Cat(0) Spaces

In this article, results have been shown via using a general quasi contraction multi-valued mapping in Cat(0) space. These results are used to prove the convergence of two iteration algorithms to a fixed point and the equivalence of convergence. We also demonstrate an appropriate conditions to ensure that one is faster than others.

A Simple proof for the algorithms of relaxed $(u, v)$-cocoercive mappings and $alpha$-inverse strongly monotone mappings

In this paper, a simple proof is presented for the Convergence of the algorithms for the class of relaxed $(u, v)$-cocoercive mappings and $alpha$-inverse strongly monotone mappings. Based on $alpha$-expansive maps, for example, a simple proof of the convergence of the recent iterative algorithms by relaxed $(u, v)$-cocoercive mappings due to Kumam-Jaiboon is provided. Also a simple proof for the convergence of the iterative algorithms by inverse-strongly monotone mappings due to Iiduka-Takahashi in a special case is provided. These results are an improvement as well as a refinement of previously known results.

High-resolution multi-planar coherent diffraction imaging with multimode fiber source

Coherent diffractive imaging (CDI) is a lensless, high-resolution imaging method. In order to overcome the problems of poor-quality image and slow convergence of iterative algorithm in traditional CDI, an innovative setup for the multimode-based phase retrieval method is proposed. Here the multimode fiber source was selected as the light source to replace the traditional plane-wave incidence. Due to the nature of the multimode fiber, a speckle light field can be generated on the output fiber facet and then is used to illuminate the object. Thus the diffraction field can be rapidly changed at a very short distance so as to realize much higher resolution and consume less iteration time. Then a phase-only spatial light modulator (SLM) is used as a negative lens instead of a positive lens in a 4f imaging system for the advantage of larger detection space. This proposed setup has many advantages such as a simpler structure and static recording. The comparison experiments with plane waves confirm that our proposed method has higher reconstruction resolution and smaller iteration errors.

一类分数阶调和映射迭代算法的收敛性

一类分数阶调和映射迭代算法的收敛性

Quantum annealing for systems of polynomial equations

Numerous scientific and engineering applications require numerically solving systems of equations. Classically solving a general set of polynomial equations requires iterative solvers, while linear equations may be solved either by direct matrix inversion or iteratively with judicious preconditioning. However, the convergence of iterative algorithms is highly variable and depends, in part, on the condition number. We present a direct method for solving general systems of polynomial equations based on quantum annealing, and we validate this method using a system of second-order polynomial equations solved on a commercially available quantum annealer. We then demonstrate applications for linear regression, and discuss in more detail the scaling behavior for general systems of linear equations with respect to problem size, condition number, and search precision. Finally, we define an iterative annealing process and demonstrate its efficacy in solving a linear system to a tolerance of 10−8.

Dual-mode phase-shifting interferometry based on iterative algorithms

Iterative algorithms are useful for retrieving wavefront phase from vibration disturbed interferograms for phase-shifting interferometry (PSI). But the dependence of convergence on the initial value deviation from the exact value impairs their application in severe vibration. In this paper, performance investigation of a representative iterative algorithm indicates that a higher frame rate of the camera, with matched phase shift, helps in enhancing the success probability of convergence. However, the camera frame rate is limited by its transmission speed and could not be increased easily. To improve the convergence of iterative algorithms, a dual-mode (DM) PSI is proposed that utilises the binning function of cameras. In DMPSI, two modes, high-speed and high-resolution modes, work consecutively by switching the camera binning function and two series of interferograms are collected respectively. The wavefront phase could be accurately estimated from high-speed interferograms and then is input to the iterative calculation with high-resolution interferograms as the initial value, and a wavefront phase with high-resolution is achieved ultimately. DMPSI combines the advantages of two complementary modes of cameras to reconstruct an accurate and high-resolution wavefront. Both simulations and a practical measurement verify the enhancement of vibration resistance of iterative algorithms. DMPSI is easy to implement with low cost and good compatibility, predicating a reliable solution for optical wavefront measurement in the severe vibration.

RECONFIGURATION OF THE AIRCRAFT INTEGRATED CONTROL SYSTEM REGARDING CONTROL CONSTRAINTS UNDER ACTUATOR FAILURES

The aircraft integrated control system reconfiguration laws under failures of actuators, calculated disregarding physical constraints on control surfaces saturation, can lead to a complete loss of aircraft controllability and stability. Despite the large number of scientific publications in this field, practical systematic results have been obtained only for SISO (single input – single output) systems. Problems of the convergence of iterative algorithms restricting the set of admissible solutions and the conservatism of the reconfiguration laws designed using weight matrices do not allow solving this problem in general. For complex MIMO (multi input – multi output) systems there is still no widely accepted universal approach. In this work, control surfaces constraints are regarded in terms of the power of reconfiguration control. It is shown that by slight modification of pseudoinverse (optimal) solution it is possible to obtain approximate pseudoinverse (suboptimal) solutions with priory known minimum power (compensation matrix norm) and error (residual matrix norm) of the reconfiguration for a given degree of approximation. This allows for a multistep consistent reduction in power and increasing in error of reconfiguration, until an acceptable solution is obtained. By providing the greater reconfiguration error at each step we have additional freedom in reducing the reconfiguration power. This leads to a decrease in the amplitude of the deviations of the control surfaces, to which the signals from the failed control channels are redistributed. The simulation example of the aircraft integrated control system reconfiguration under the stabilizer’s actuator failure is presented. It is shown that the pseudoinverse reconfiguration problem solution leads to the significant ailerons’ constraints violation and the loss of aircraft controllability. Regarding control constraints solution reduces several times the deviation of the control surfaces and provides an effective problem solution in the permissible power and error reconfiguration range.

Analytical Study of Algorithms for Solving Inverse Kinematic Problems in Robot Motion Control

The given article covers the general formulations of inverse kinematic problems for robot motion control systems. We have discussed the difficulties how to solve such problems using analytical and numerical methods. We have also analyzed the convergence of iterative algorithms with the regularization on the trajectory with the points outside of the gripper reachability. The example of an iterative calculation of joint trajectories for a 3-link robot using the recursive algorithm for the Jacobi matrix calculation has been presented.

Strong Convergence of Iterative Algorithms for a Class of Generalized Variational Inequality Problems in Banach Spaces

Strong Convergence of Iterative Algorithms for a Class of Generalized Variational Inequality Problems in Banach Spaces

Solution of a Bivariate Regularized Problem

We derive the mapping that takes an observation vector to the minimizer of a bivariate cost consisting of the sum of a quadratic data fidelity term and an $\ell_1$ norm. The derived mapping is useful for accelerating convergence of iterative algorithms that aim to solve $\ell_1$ regularized problems. We discuss how to use the mapping in practice and demonstrate the improvement in convergence rate with numerical experiments.

A block coordinate variable metric forward–backward algorithm

A number of recent works have emphasized the prominent role played by the Kurdyka-Łojasiewicz inequality for proving the convergence of iterative algorithms solving possibly nonsmooth/nonconvex optimization problems. In this work, we consider the minimization of an objective function satisfying this property, which is a sum of two terms: (i) a differentiable, but not necessarily convex, function and (ii) a function that is not necessarily convex, nor necessarily differentiable. The latter function is expressed as a separable sum of functions of blocks of variables. Such an optimization problem can be addressed with the Forward–Backward algorithm which can be accelerated thanks to the use of variable metrics derived from the Majorize–Minimize principle. We propose to combine the latter acceleration technique with an alternating minimization strategy which relies upon a flexible update rule. We give conditions under which the sequence generated by the resulting Block Coordinate Variable Metric Forward–Backward algorithm converges to a critical point of the objective function. An application example to a nonconvex phase retrieval problem encountered in signal/image processing shows the efficiency of the proposed optimization method.

Convergence of iterative algorithms for continuous pseudocontractive mappings

In this paper, we prove strong convergence of a path for a convex combination of a pseudocontractive type of operators in a real reflexive Banach space having a weakly continuous duality mapping J? with gauge function ?. Using path convergency, we establish strong convergence of an implicit iterative algorithm for a pseudocontractive mapping combined with a strongly pseudocontractive mapping in the same Banach space.

Relaxed Linearized Algorithms for Faster X-Ray CT Image Reconstruction.

Statistical image reconstruction (SIR) methods are studied extensively for X-ray computed tomography (CT) due to the potential of acquiring CT scans with reduced X-ray dose while maintaining image quality. However, the longer reconstruction time of SIR methods hinders their use in X-ray CT in practice. To accelerate statistical methods, many optimization techniques have been investigated. Over-relaxation is a common technique to speed up convergence of iterative algorithms. For instance, using a relaxation parameter that is close to two in alternating direction method of multipliers (ADMM) has been shown to speed up convergence significantly. This paper proposes a relaxed linearized augmented Lagrangian (AL) method that shows theoretical faster convergence rate with over-relaxation and applies the proposed relaxed linearized AL method to X-ray CT image reconstruction problems. Experimental results with both simulated and real CT scan data show that the proposed relaxed algorithm (with ordered-subsets [OS] acceleration) is about twice as fast as the existing unrelaxed fast algorithms, with negligible computation and memory overhead.

Graph Convergence for the $$H(\cdot ,\cdot )$$ H ( · , · ) -Co-accretive Mapping with an Application

In this paper, we introduce a concept of graph convergence for the $$H(\cdot ,\cdot )$$ -co-accretive mapping in Banach spaces and prove an equivalence theorem between graph convergence and resolvent operator convergence for the $$H(\cdot ,\cdot )$$ -co-accretive mapping. Further, we consider a system of generalized variational inclusions involving $$H(\cdot ,\cdot )$$ -co-accretive mapping in real $$q$$ -uniformly smooth Banach spaces. Using resolvent operator technique, we prove the existence and uniqueness of solution and suggest an iterative algorithm for the system of generalized variational inclusions under some suitable conditions. Further, we discuss the convergence of iterative algorithm using the concept of graph convergence. Our results can be viewed as a refinement and generalization of some known results in the literature.