Fixed-point Iteration Method Research Articles

A Fixed-Point Iterative Method for Discrete Tomography Reconstruction Based on Intelligent Optimization

Discrete Tomography (DT) is a technology that uses image projection to reconstruct images. Its reconstruction problem, especially the binary image (0–1 matrix) has attracted strong attention. In this study, a fixed point iterative method of integer programming based on intelligent optimization is proposed to optimize the reconstructed model. The solution process can be divided into two procedures. First, the DT problem is reformulated into a polyhedron judgment problem based on lattice basis reduction. Second, the fixed-point iterative method of Dang and Ye is used to judge whether an integer point exists in the polyhedron of the previous program. All the programs involved in this study are written in MATLAB. The final experimental data show that this method is obviously better than the branch and bound method in terms of computational efficiency, especially in the case of high dimension. The branch and bound method requires more branch operations and takes a long time. It also needs to store a large number of leaf node boundaries and the corresponding consumption matrix, which occupies a large memory space.

Inexact fixed-point iteration method for nonlinear complementarity problems

Based on the modulus decomposition, the structured nonlinear complementarity problem is reformulated as an implicit fixed-point system of nonlinear equations. Distinguishing from some existing modulus-based matrix splitting methods, we present a flexible modulus-based inexact fixed-point iteration method for the resulting system, in which the subproblem can be solved approximately by a linear system-solver. The global convergence of the proposed method is established by assuming that the system matrix is positive definite. Some numerical comparisons are reported to illustrate the applicability of the proposed method, especially for large-scale problems.

Aircraft route recovery based on distributed integer programming method.

In order to further promote the application and development of unmanned aviation in the manned field, and reduce the difficulty that airlines cannot avoid due to unexpected factors such as bad weather, aircraft failure, and so on, the problem of restoring aircraft routes has been studied. To reduce the economic losses caused by flight interruption, this paper divides the repair problem of aircraft operation plans into two sub problems, namely, the generation of flight routes and the reallocation of aircraft. Firstly, the existing fixed-point iteration method proposed by Dang is used to solve the feasible route generation model based on integer programming. To calculate quickly and efficiently, a segmentation method that divides the solution space into mutually independent segments is proposed as the premise of distributed computing. The feasible route is then allocated to the available aircraft to repair the flight plan. The experimental results of two examples of aircraft fault grounding and airport closure show that the method proposed in this paper is effective for aircraft route restoration.

A fixed point iterative method for tensor complementarity problems with the implicit Z-tensors

A fixed point iterative method for tensor complementarity problems with the implicit Z-tensors

Revisiting graph neural networks from hybrid regularized graph signal reconstruction

Graph neural networks (GNNs) have shown strong graph-structured data processing capabilities. However, most of them are generated based on the message-passing mechanism and lack of the systematic approach to guide their developments. Meanwhile, a unified point of view is hard to explain the design concepts of different GNN models. This paper presents a unified optimization framework from hybrid regularized graph signal reconstruction to establish the connection between the aggregation operations of different GNNs, showing that exploring the optimal solution is the process of GNN information aggregation. We use this new framework to mathematically explain several classic GNN models and summarizes their commonalities and differences from a macro perspective. The proposed framework not only provides convenience to understand GNNs, but also has a guiding significance for the proposal of new GNNs. Moreover, we design a model-driven fixed-point iteration method and a data-driven dictionary learning network according to the corresponding optimization objective and sparse representation. Then the new model, GNN based on model-driven and data-driven (GNN-MD), is established by using alternating iteration methods. We also theoretically analyze its convergence. Numerous node classification experiments on multiple datasets illustrate that the proposed GNN-MD has excellent performance and outperforms all baselines on high-feature-dimension datasets.

A Design Methodology for the Seismic Retrofitting of Existing Frame Structures Post-Earthquake Incident Using Nonlinear Control Systems

A structural design methodology for retrofitting weakened frame systems following earthquakes is developed and presented. The design procedure refers to frame systems in their degraded strength and stiffness states and restores their dynamic performance using nonlinear control systems. The control law associated with the employed systems regards the gains between the negative state feedback and the control force, which consists of linear, nonlinear, and hysteretic portions. Structural optimization is introduced in designing the nonlinear control systems, and the controller gains are optimized using the fixed-point iteration to improve the frame system’s dynamic performance. The fixed-point iteration method relates to first-order PDE equations; hence, a new state-space formulation for weakened inelastic frame systems is developed and presented using the frame system’s lateral force equilibrium equation. The design scheme and optimization strategy differ from designing passive control systems, given that the nonlinear control system’s force consists of linear, nonlinear, and hysteretic portions. The utilization of the fixed-point iteration in the structural design area is by itself a novel application due to its robustness in addressing the gains of any type of nonlinear control system. This paper’s nonlinear control system chosen to exhibit the application is Buckling Restrained Braces (BRBs) since force consists of linear and hysteretic portions. The implementation of hysteretic control force is rare in structural control applications. In the case of BRBs, the fixed-point iteration optimizes the cross-sectional areas. Two system optimization examples of 3-story and 15-story inelastic frames are provided and described. The examples demonstrate the fixed-point iteration’s applicability and robustness in optimizing control gains of nonlinear systems and regulating the dynamic response of weakened frame structures.

A Fixed-Point Iteration Method for High Frequency Helmholtz Equations

A Fixed-Point Iteration Method for High Frequency Helmholtz Equations

Constrained minimum fuzzy error entropy filtering for target tracking

To improve the accuracy of target tracking results in nonlinear systems under complex noise, the constrained minimum fuzzy error entropy Kalman filter (CMFEE-KF) is proposed. In this proposed filter, the double indices-induced fuzzy membership is introduced to represent the different effects of different error samples on the estimation results, solving the problem of the same weight in ordinary error entropy. And then, the minimum fuzzy error entropy criterion (MFEEC) is constructed and used to optimize the Kalman filter. In this proposed algorithm, error information is obtained by model reconstruction. Then the objective function is constructed based on MFEEC, and finally, the posterior state estimation is achieved by the fixed-point iteration method. In addition, soft constraints can be implemented by adding a regularization term into the loss function, deriving the CMFEE-KF. Simulations show that the proposed filter has strong stability and more accuracy in target tracking.

Shift-splitting fixed point iteration method for solving generalized absolute value equations

Shift-splitting fixed point iteration method for solving generalized absolute value equations

Fault section location for distribution network based on linear integer programming

In the power society, the fault location of the distribution network has always been the focus of scholars. When a fault occurs, it needs to be located quickly and accurately to avoid more significant losses. Due to the wide application of distributed generation in the distribution network, many traditional methods are no longer applicable. Thus, a fast and high fault tolerance method is urgently needed to solve the complex fault location problem of the distribution network. According to the logical and algebraic relationship of fault location, a linear integer programming model for fault section location of the distribution network is established. This paper adopts a new method for solving integer programming, called the fixed-point iterative method. This method has unique advantages in dealing with such large-scale problems and is easily realised through distributed computing, thereby saving time. Finally, through MATLAB experimental simulation, results show that the integer programming method can quickly locate the fault and has high fault tolerance.

Switching Gaussian-heavy-tailed distribution based robust Gaussian approximate filter for INS/GNSS integration

In inertial navigation system and global navigation satellite system (INS/GNSS) integration, the practical stochastic measurement noise may be non-stationary heavy-tailed distribution due to outlier measurements induced by multipath and/or non-line-of-sight receptions of the original GNSS signals. To address the problem, a new switching Gaussian-heavy-tailed (SGHT) distribution is presented, which models the measurement noise with the help of switching between the Gaussian and the an existing heavy-tailed distribution. Then, utilizing two auxiliary parameters satisfying categorical and Bernoulli distributions respectively, we construct the SGHT distribution as a hierarchical Gaussian presentation. Furthermore, applying variational Bayesian inference, a novel SGHT distribution based robust Gaussian approximate filter is derived. Meanwhile, to reduce the computational complexity of the filtering process, an improved fixed-point iteration method is designed. Finally, the simulation of integrated navigation for an aircraft illustrates effectiveness and superiority of the proposed filter as compared the existing robust filters.

A note on a faster fixed point iterative method

Very recently, Piri et al. (Numer Algorithms 81:1129–1148, 2019) introduced an iterative process to approximate a fixed point of generalized \(\alpha\)-nonexpansive mappings and discussed its convergence analysis. In this paper, we introduce an iteration process to approximate a fixed point of a contractive self-mapping that is faster than all of Picard, Mann, Ishikawa, Noor, Agarwal, Abbas, and Thakur processes for contractive mappings including the recent one by Piri et al. We also obtain convergence and stability theorems of this iterative process for a contractive self-mapping. Numerical examples show that our iteration process for approximating a fixed point of a contractive self-mapping is faster than the method proposed by Piri et al. Based on this process, we finally present a new modified Newton–Raphson method for finding the roots of a function and generate some nice polynomiographs.

Solution Bounds and Numerical Methods of the Unified Algebraic Lyapunov Equation

In this paper, applying some properties of matrix inequality and Schur complement, we give new upper and lower bounds of the solution for the unified algebraic Lyapunov equation that generalize the forms of discrete and continuous Lyapunov matrix equations. We show that its positive definite solution exists and is unique under certain conditions. Meanwhile, we present three numerical algorithms, including fixed point iterative method, the acceleration fixed point method and the alternating direction implicit method, to solve the unified algebraic Lyapunov equation. The convergence analysis of these algorithms is discussed. Finally, some numerical examples are presented to verify the feasibility of the derived upper and lower bounds, and numerical algorithms.

On the probability of ventricular fibrillation due to electric shock

When exposed to a specified electrical shock, theprobability that a randomly chosen individual will undergofatal ventricular fibrillation can be regarded as a functionof random variation in the human population along twodimensions. The first dimension is the individual's bodyimpedance characteristic: for this, we introduce a newtwo-parameter model that improves on the simplerone-parameter model used in previous work. The seconddimension is the individual's current tolerance: we codifysome curves used in previous practice. We also considermethods of solving the resulting shock circuit and show thatthe fixed-point iteration method can give incorrect results.

Color Occlusion Face Recognition Method Based on Quaternion Non-Convex Sparse Constraint Mechanism.

As the acquisition and application of color images become more and more extensive, color face recognition technology has also been vigorously developed, especially the recognition methods based on convolutional neural network, which have excellent performance. However, with the increasing depth and complexity of network models, the number of calculated parameters also increases, which means the training of most high-performance models depends on large-scale samples and expensive equipment. Therefore, the key to the current research is to realize a lightweight model while ensuring the recognition accuracy. At present, PCANet, a typical lightweight framework for deep learning, has achieved good results in most of the image recognition tasks, but its recognition accuracy for color face images, especially under occlusion, still needs to be improved. Therefore, a color occlusion face recognition method based on quaternion non-convex sparse constraint mechanism is proposed in this paper. Firstly, a quaternion non-convex sparse principal component analysis network model was constructed based on Lp regularization of strong sparsity. Secondly, the fixed point iteration method and coordinate descent method were established to solve the non-convex optimization problem. Finally, the occlusion recognition performance of the proposed method was verified on Georgia Tech, Color FERET, AR, and LFW-A Color face datasets.

A finite difference scheme for non-Cartesian mesh: Applications to rarefied gas flows

A novel numerical scheme based on the finite-difference framework is developed, which allows us to model moderately rarefied gas flows in irregular geometries. The major hurdle in constructing numerical methods for rarefied gas flows is the prescription of the velocity-slip and temperature-jump boundary conditions as well as the discretization of an intricate set of partial differential equations. The proposed scheme is demonstrated to solve the non-linear coupled constitutive relations model along with the corresponding non-linear slip and jump boundary conditions. The computation of the discretized weights is proposed using two approaches: (i) polynomial shape functions and (ii) a generalized inverse distance approach. The non-linear terms are discretized using the fixed-point iteration method. The numerical method is validated for the Laplace equation over an annulus, and results are presented for a lid-driven curved cavity and a triangular lid-driven cavity, which delineates its performance on a skewed non-Cartesian grid. The results are validated with direct simulation Monte Carlo data from the literature, and a robust convergence for the solutions is demonstrated.

An Improvement on a Class of Fixed Point Iterative Methods for Solving Absolute Value Equations

Abstract We consider a class of iterative methods based on block splitting (BBS) to solve absolute value equations A ⁢ x - | x | = b Ax-\lvert x\rvert=b . Recently, several works were devoted to deriving sufficient conditions for the convergence of iterative methods of this type under certain assumptions including ν := ∥ A - 1 ∥ < 1 \nu:=\lVert A^{-1}\rVert<1 . However, the BBS-type iterative methods tend to converge slowly when 𝜈 is very close to one (i.e., ν ≈ 1 \nu\approx 1 ). In this paper, using an auxiliary matrix, we develop a new approach by first rewriting the main problem into a new equivalent block system having shifted ( 1 , 1 ) (1,1) -block and then constructing a fixed point iteration. The exploited strategy can significantly improve the convergence speed of the BBS-type iterative methods when ν ≈ 1 \nu\approx 1 . Numerical experiments are reported to demonstrate the superiority of the new modified iterative scheme over the existing original form of BBS-type methods in the literature.

Energy quadratization Runge–Kutta scheme for the conservative Allen–Cahn equation with a nonlocal Lagrange multiplier

In this study, we present a high-order energy stable scheme for the conservative Allen–Cahn equation with a nonlocal Lagrange multiplier by combining the concept of energy quadratization and the Runge–Kutta method. Under the stability condition for the Runge–Kutta coefficients, we analytically demonstrate that the scheme is unconditionally stable with respect to the reformulated energy. Additionally, we develop a Newton-type fixed point iteration method to implement the scheme, enabling the achievement of a fast iterative convergence. Numerical experiments are presented to demonstrate the accuracy and energy stability of the proposed scheme.

An assessment of solvers for algebraically stabilized discretizations of convection–diffusion–reaction equations

Abstract We consider flux-corrected finite element discretizations of 3D convection-dominated transport problems and assess the computational efficiency of algorithms based on such approximations. The methods under investigation include flux-corrected transport schemes and monolithic limiters. We discretize in space using a continuous Galerkin method and ℙ1 or ℚ1 finite elements. Time integration is performed using the Crank–Nicolson method or an explicit strong stability preserving Runge–Kutta method. Nonlinear systems are solved using a fixed-point iteration method, which requires solution of large linear systems at each iteration or time step. The great variety of options in the choice of discretization methods and solver components calls for a dedicated comparative study of existing approaches. To perform such a study, we define new 3D test problems for time dependent and stationary convection–diffusion–reaction equations. The results of our numerical experiments illustrate how the limiting technique, time discretization and solver impact on the overall performance.

Multi-kernel correntropy based extended Kalman filtering for state-of-charge estimation

As a powerful tool for real-time battery management, the extended Kalman filter (EKF) can achieve an online estimation for state of charge (SOC). The EKF, however, may yield biased estimates since the measured system suffers from the abnormal operation conditions, i.e., sensor faults, sensor bias and sensor noise. Thus, this paper proposes a robust extended Kalman filter based on maximum multi-kernel correntropy (MMKC-EKF) for SOC estimate when the system is subjected to complex non-Gaussian disturbances. To derive MMKC-EKF, a batch-mode regression is formulated by integrating the uncertainties of process and measurement, which is solved by using maximum multi-kernel correntropy (MMKC) criterion to suppress the influences of abnormal conditions. An effective optimization method is introduced to determine the free parameters of MMKC, and a fixed-point iteration method gives the state estimation. Then, the posterior error covariance matrix is updated with the help of total influence function, which contributes to the robustness improvement. In addition, a novel filtering scheme is presented for reducing computational complexity, which is beneficial for solving battery pack state estimation in practice. Extensive simulations are carried out for SOC estimate to validate the accuracy and robustness of the proposed MMKC-EKF in the Gaussian and non-Gaussian distributed process and measurement noises.