Two-step Iteration Research Articles

Nonlinear Meshfree Collocation Method for Solving Double-Diffusive Natural Convection in a Porous Enclosure

A meshfree collocation method formulated by using reproducing kernel shape functions with Newton–Raphson method is presented with application to analyzing a three-phase coupling system in a porous enclosure. To tackle such a highly nonlinear phenomenon caused by double-diffusive natural convection, direct collocation using local approximation in conjunction with a two-step iteration solver is employed to stably and efficiently solve this problem. It has been shown that a determined and well-conditioned system is established under the collocation framework with reproducing kernel approximation. By conducting the parametric study, the numerical results have demonstrated the robustness of the proposed method in analyzing the porous enclosure by considering various geometry, boundary conditions, and governing parameters.

A fast impact force identification method via constructing a dynamic reduced dictionary

Impact force identification is an important aspect of structural health monitoring for online assessment of the integrity of engineering structures. Increasing the number of monitoring positions is undoubtedly beneficial to accurate force localization, however, it may increase sharply the dimension of the transfer matrix and thus lead to a large-scale inverse problem. In this case, traditional methods, such as the Iterative Shrinkage Threshold algorithm (IST) and the Two-Step Iterative Shrinkage Thresholding algorithm (TwIST), suffer from low solving efficiency. In this paper, the dynamic reduced dictionary is firstly introduced according to the sparsity of impact force in both time and spatial domains so as to lower significantly the dimension of the transfer matrix. Then, a new reduced unconstrained optimization problem is established via the dynamic reduced dictionary for impact force identification. By integrating the reduced unconstrained optimization problem into each iteration step of IST/TwIST, two novel algorithms, which are the reduced Newton iteration shrinkage threshold algorithm (rNIST) and the reduced Newton two-step iteration shrinkage threshold algorithm (rNTwIST), are proposed to solve quickly the large-scale impact force identification problem using ℓ1-norm sparse regularization. Meanwhile, the adaptive setting of regularization parameters strategy is used to construct force more fast and accurately. The proposed algorithms are numerically and experimentally demonstrated through identifying impact forces at 16-point and 81-point distributions of a plate using one sensor response. The results show that rNIST/rNTwIST can more quickly and accurately identify impact forces under numerous monitoring points compared with other ℓ1-norm regularization methods and nonconvex sparse regularization methods. Moreover, the more the number of monitoring points, the more obvious the efficiency improvement of the impact force identification by rNIST/rNTwIST method.

A two-step iteration method for solving vertical nonlinear complementarity problems

A two-step iteration method for solving vertical nonlinear complementarity problems

Sparse prior-net: A sparse prior-based deep network for seismic data interpolation

Seismic data interpolation plays a crucial role in obtaining dense and regularly sampled data, contributing to improving the quality of seismic data in seismic exploration. Sparsity-promoting methods use a two-step iteration to gradually recover missing traces, by exploiting the sparsity representation of seismic data in transform domains, such as Fourier, wavelet, and curvelet transform, within the framework of the projection onto convex sets (POCS). In the first step, the missing traces are restored by applying the thresholding shrinkage to the transform coefficients. In the second step, the observed data are inserted into the updated result. However, this method relies on a preselected transform and lacks the capability to adaptively capture sparse representations. In addition, determining the optimal threshold parameters can pose difficulties. These limitations yield unsatisfactory reconstruction results. To address this issue, we propose a novel approach called sparse prior-based seismic interpolation network (SP-net) that combines the sparsity-promoting method with a deep neural network. Unlike traditional end-to-end networks, our proposed neural network integrates the widely used POCS method into its architecture, enabling automatic learning of the sparse transform, and threshold parameters from the training data set. By combining the merits of the sparsity-promoting techniques and data-driven deep-learning approaches, SP-net achieves enhanced adaptability and more accurate interpolation results. Through experiments conducted on synthetic and field seismic data, we demonstrate the effectiveness of our proposed method.

Sparse Microwave Imaging Algorithm Based on L1/2 Threshold Iteration and an Approximate Observation Operator

Aiming at the problems of synthetic-aperture radar (SAR), such as high sampling rate and vulnerability to noise interference in imaging, a sparse reconstruction algorithm based on approximate observation and L1/2 threshold iteration is proposed in this paper. To solve the problem of the large dimension and high computational complexity of the measurement matrix, a sparse reconstruction model based on approximate observation is constructed, and a threshold iterative algorithm to rapidly solve the L1/2 regularization problem is introduced, which can realize sparse signal reconstruction with fewer sampling data and quickly solve the problem. The simulation results of point targets and the measured data of spaceborne SAR show that compared with the traditional matched filtering algorithm and L1 threshold algorithm based on a two-step iteration, the proposed sparse reconstruction algorithm has a faster iteration speed and improves the image quality.

GraphHop: An Enhanced Label Propagation Method for Node Classification.

A scalable semisupervised node classification method on graph-structured data, called GraphHop, is proposed in this work. The graph contains all nodes' attributes and link connections but labels of only a subset of nodes. Graph convolutional networks (GCNs) have provided superior performance in node label classification over the traditional label propagation (LP) methods for this problem. Nevertheless, current GCN algorithms suffer from a considerable amount of labels for training because of high model complexity or cannot be easily generalized to large-scale graphs due to the expensive cost of loading the entire graph and node embeddings. Besides, nonlinearity makes the optimization process a mystery. To this end, an enhanced LP method, called GraphHop, is proposed to tackle these problems. GraphHop can be viewed as a smoothening LP algorithm, in which each propagation alternates between two steps: label aggregation and label update. In the label aggregation step, multihop neighbor embeddings are aggregated to the center node. In the label update step, new embeddings are learned and predicted for each node based on aggregated results from the previous step. The two-step iteration improves the graph signal smoothening capacity. Furthermore, to encode attributes, links, and labels on graphs effectively under one framework, we adopt a two-stage training process, i.e., the initialization stage and the iteration stage. Thus, the smooth attribute information extracted from the initialization stage is consistently imposed in the propagation process in the iteration stage. Experimental results show that GraphHop outperforms state-of-the-art graph learning methods on a wide range of tasks in graphs of various sizes (e.g., multilabel and multiclass classification on citation networks, social graphs, and commodity consumption graphs).

On the parameterized two-step iteration method for solving the matrix equation AXB = C

In this paper, based on the iteration framework [10], by introducing two tuning parameters α,β in the splittings of the matrices A and B, a parameterized two-step iteration (PTSI) method is presented for solving the matrix equation AXB=C. In the sequel, the convergence property and choices of the parameters α,β are analyzed in detail. For some special cases of the matrices A and B, the corresponding PTSI methods are also investigated, and the optimal parameters α,β can be obtained for the symmetric positive definite (SPD) matrices A and B. In addition, some comparison results of the PTSI method are given for the M-matrices A,B compared with the TSI method [10]. Finally, several numerical examples are performed to verify the efficiencies of the PTSI method and choices of the optimal parameters.

Numerical solution of Bratu’s boundary value problem based on Green’s function and a novel iterative scheme

We compute the numerical solution of the Bratu’s boundary value problem (BVP) on a Banach space setting. To do this, we embed a Green’s function into a new two-step iteration scheme. After this, under some assumptions, we show that this new iterative scheme converges to a sought solution of the one-dimensional non-linear Bratu’s BVP. Furthermore, we show that the suggested new iterative scheme is essentially weak w^{2}-stable in this setting. We perform some numerical computations and compare our findings with some other iterative schemes of the literature. Numerical results show that our new approach is numerically highly accurate and stable with respect to different set of parameters.

A Generalized Series Expansion of the Arctangent Function Based on the Enhanced Midpoint Integration

In this work, we derive a generalized series expansion of the acrtangent function by using the enhanced midpoint integration (EMI). Algorithmic implementation of the generalized series expansion utilizes a two-step iteration without surd or complex numbers. The computational test we performed reveals that such a generalization improves the accuracy in computation of the arctangent function by many orders of magnitude with increasing integer M, associated with subintervals in the EMI formula. The generalized series expansion may be promising for practical applications. It may be particularly useful in practical tasks, where extensive computations with arbitrary precision floating points are needed. The algorithmic implementation of the generalized series expansion of the arctangent function shows a rapid convergence rate in the computation of digits of π in the Machin-like formulas.

Modeling 3-D Elastic Wave Propagation in TI Media Using Discontinuous Galerkin Method on Tetrahedral Meshes

Transversely isotropic (TI) medium is a widely studied anisotropic solid medium in seismology. The numerical simulation of seismic wave propagation in TI media is an effective tool to analyze the mechanism of seismic waves in complex anisotropic media. In this study, we introduce a double-weighted Runge-Kutta discontinuous Galerkin (RKDG) method for numerically solving wave propagation problems in 3D TI media with surface topography. This method incorporates the discontinuous Galerkin spatial discretization with an explicit double-weighted two-step iteration time discretization. The local Lax-Friedrichs flux is used as the numerical flux in the formulations. This method can solve the first-order velocity-stress seismic wave equations including TI media as well as more general anisotropic media. Due to the large scale of 3D problems, the parallel technology is adopted. Regions with irregular boundaries are discretized into unstructured tetrahedral meshes. Numerical experiments for various anisotropic media including the vertical and tilted TI media are presented. The results demonstrate the effectiveness of the double-weighted RKDG method in wavefield simulations in 3D complicated anisotropic media.

A Two-Step Iteration Method for Vertical Linear Complementarity Problems

In this paper, a two-step iteration method was established which can be viewed as a generalisation of the existing modulus-based methods for vertical linear complementarity problems. The convergence analysis of the proposed method is presented, which can enlarge the convergence domain of the parameter matrix compared to the recent results. Numerical examples show that the proposed method is efficient with the two-step technique and confirm the improvement of the theoretical results.

Adaptive parameter based matrix splitting iteration method for the large and sparse linear systems

For the large and sparse linear systems, we utilize the efficient splittings of the system matrix and introduce an intermediate variable. The main contribution of this paper is that the adaptive parameter based matrix splitting iteration method is constructed from the view of numerical optimization and hybrid procedure to solve the derived equation instead. The novel method adopts the prediction and correction two-step iteration and the global convergence results are established under some suitable conditions. Compared with some well-known methods, numerical experiments show the efficiency and effectiveness of the new proposal with application to the three-dimensional convection-diffusion equation and the image restoration problems.

Pion damping width and pion spectral function in hot pion gas

The temperature behavior of the pion width in the hadronic phase is investigated in the framework of the Nambu–Jona–Lasinio (NJL) model. The contribution to the width from the pion–pion collision is considered with a scalar sigma-meson as an intermediate state. It is shown that the pion width significantly broadens at [Formula: see text][Formula: see text]GeV. Using the two-step iteration method, suggested by Kadanoff and Baym, the pion spectral function in a hot pion gas is calculated at different temperatures.

Exterior computed tomography image reconstruction based on anisotropic relative total variation in polar coordinates.

In computed tomography (CT) image reconstruction problems, exterior CT is an important application in industrial non-destructive testing (NDT). Different from the limited-angle problem that misses part of the rotation angle, the rotation angle of the exterior problem is complete, but for each rotation angle, the projection data through the central region of the object cannot be collected, so that the exterior CT problem is ill-posed inverse problem. The results of traditional reconstruction methods like filtered back-projection (FBP) and simultaneous algebra reconstruction technique (SART) have artifacts along the radial direction edges for exterior CT reconstruction. In this study, we propose and test an anisotropic relative total variation in polar coordinates (P-ARTV) model for addressing the exterior CT problem. Since relative total variation (RTV) can effectively distinguish edges from noises, and P-ARTV with different weights in radial and tangential directions can effectively enhance radial edges, a two-step iteration algorithm was developed to solve the P-ARTV model in this study. The fidelity term and the regularization term are solved in Cartesian and polar coordinate systems, respectively. Numerical experiments show that our new model yields better performance than the existing state-of-the-art algorithms.

Hydrodynamic investigation of wet swimming platform problem for semi-planning motor yachts

One of the main optimization problems for the performance and comfort of yachts is the wet swimming platform problem, also known as the rooster-tail phenomena .And this research presents an investigation using Computational Fluid Dynamics (CFD) method for a real case of an 88 ft. semi-planning motor yacht hull form that encountered this phenomena. First of all, the original hull form was analyzed to define the problem at cruising and maximum speeds, 14 knots, and 24 knots respectively. To solve this problem, modifications to the stern form of the hull were made in a two-step iteration. Initially, the size of the current interceptor was increased with a tunnel wedge. Results of the CFD analysis show that this modification eliminated the wet platform problem and increased the top speed of the yacht. Since the applicability of the wedge to mount is complex and costly, a second analysis with only enlarged interceptors was performed. As a result, the wet platform problem is solved, however, the resistance reduction was slightly lower.

A new two-step iteration method for discrete ill-posed problems and image restoration

In this study, for the augmented linear system of discrete ill-posed problems we establish a new two-step (NTS) iteration method containing a parameter and a parameter matrix, which is based on the Hermitian and skew-Hermitian splitting (HSS) and the upper and lower triangular splitting (ULT) of the coefficient matrix. Then, we theoretically study its convergence properties and determine its optimal iteration parameters. It is seen that the NTS method converges faster when the parameters are chosen properly. Experimental examples are carried out to further validate the effectiveness and accuracy of the new method compared to the newly developed methods in terms of the numerical performance and image recovering quality.

Sparse reconstruction of magnetic resonance image combined with two-step iteration and adaptive shrinkage factor.

As an advanced technique, compressed sensing has been used for rapid magnetic resonance imaging in recent years, Two-step Iterative Shrinkage Thresholding Algorithm (TwIST) is a popular algorithm based on Iterative Thresholding Shrinkage Algorithm (ISTA) for fast MR image reconstruction. However TwIST algorithms cannot dynamically adjust shrinkage factor according to the degree of convergence. So it is difficult to balance speed and efficiency. In this paper, we proposed an algorithm which can dynamically adjust the shrinkage factor to rebalance the fidelity item and regular item during TwIST iterative process. The shrinkage factor adjusting is judged by the previous reconstructed results throughout the iteration cycle. It can greatly accelerate the iterative convergence while ensuring convergence accuracy. We used MR images with 2 body parts and different sampling rates to simulate, the results proved that the proposed algorithm have a faster convergence rate and better reconstruction performance. We also used 60 MR images of different body parts for further simulation, and the results proved the universal superiority of the proposed algorithm.

Convergence theorems and stability results of a two-step iteration scheme for pointwise asymptotically nonexpansive self mappings and pointwise asymptotically nonexpansive nonself mappings in uniformly convex Banach spaces

Convergence theorems and stability results of a two-step iteration scheme for pointwise asymptotically nonexpansive self mappings and pointwise asymptotically nonexpansive nonself mappings in uniformly convex Banach spaces

Multiple Kernel Subspace Learning for Clustering and Classification

In the face of high-dimensional and complex data, effective subspace can preserve specific statistical properties and provide an appropriate representation of data, which generally facilitates the underlying tasks such as clustering or classification. Meanwhile, multiple kernel learning is a technique to combine multiple kernels from different feature spaces effectively. Thus, by incorporating multiple kernels into the process of subspace learning, different feature spaces can be projected into a unified subspace. This paper proposes the Multiple Kernel Subspace Learning (MKSL) for embedding the original space into a unified subspace. Multiple kernels of different feature spaces are combined by MKSL in the process of learning, which can extend the suitability for various applications. Moreover, to generate the optimal combination kernel of subspace learning, we propose a two-step iteration strategy to learn the appropriate kernel weights and transformation matrix of projecting simultaneously. Furthermore, our proposed formulation of MKSL can introduce different prior knowledge such as class information and neighborhood relationships. Thus it is competent to the unsupervised learning, semi-supervised learning, and supervised learning. Extensive experiments are conducted on diverse datasets, and the performances are comprehensively evaluated on different tasks. The experimental results indicate that the proposed algorithm is outstanding in unsupervised clustering task and effective in supervised and semi-supervised classification tasks.

On the convergence of two-step modulus-based matrix splitting iteration method

Abstract In this paper, based on the relationship between the linear complementarity problem and its reformulated fixed-point equation, we discuss the conditions of the modulus-based type iteration methods. Moreover, we present some convergence results on the two-step modulus-based matrix splitting iteration method with an H + {H}_{+} -matrix. Finally, we give the numerical experiments.