Iterative Approximation Method Research Articles

Distributed least-squares progressive iterative approximation for blending curves and surfaces

Least-squares progressive iterative approximation (LSPIA) is an effective tool for fitting data points with curves and surfaces in CAD/CAM, due to its intuitive geometric meaning and its suitability for handling mass data. However, the classic LSPIA method for blending curves and surfaces has a slow convergence rate and takes a long CPU execution time, since the spectral radius of its iteration matrix is near to 1. To achieve a reduction in CPU execution time, this paper presents a distributed least-squares progressive iterative approximation (DLSPIA) method by dividing the collocation matrix into some blocks, which are called processors. The proposed method distributes the computation of the control points progressively in a way that each processor is responsible for a block of the whole point set. Combining the information obtained from the previous processors with that of the current processor, the DLSPIA method can progressively and quickly approximate the least-squares fitting result of the original data set block by block via distributed computation. And the algorithm converges within a finite number of iterations. Furthermore, the iterative formulae for blending surface fitting are expressed in matrix form, which can avoid the computation of the matrix Kronecker product to reduce the CPU execution time. Several numerical examples are presented to demonstrate the superiority of the proposed method compared with the previous methods.

Iterative Chebyshev approximation method for optimal control problems

We present a novel numerical approach for solving nonlinear constrained optimal control problems (NCOCPs). Instead of directly solving the NCOCPs, we start by linearizing the constraints and dynamic system, which results in a sequence of sub-problems. For each sub-problem, we use finite number of Chebyshev polynomials to estimate the control and state vectors. To eliminate the errors at non-collocation points caused by conventional collocation methods, we additionally estimate the coefficient functions involved in the linear constraints and dynamic system by Chebyshev polynomials. By leveraging the characteristics of Chebyshev polynomials, the approximate sub-problem is changed into an equivalent nonlinear optimization problem with linear equality constraints. Consequently, any feasible point of the approximate sub-problem will satisfy the constraints and dynamic system throughout the entire time scale. To validate the efficacy of the new method, we solve three examples and assess the accuracy of the method through the computation of its approximation error. Numerical results obtained show that our approach achieves lower approximation error when compared to the Chebyshev pseudo-spectral method. The proposed method is particularly suitable for scenarios that require high-precision approximation, such as aerospace and precision instrument production.

A cabin assembly optimization method based on force and position coordination analysis

In existing automated docking assembly solutions for aerospace products, segmented docking surfaces suffer from machining errors and deformation, making assembly difficult. The aim of this study was to reduce the internal stresses in cabin assembly and maintain sufficient assembly accuracy. A cabin assembly optimization method was developed based on force–position coordination analysis. To address the interference problem of the cabin flange surface, a penalty-function-based reweighting iterative approximation method was constructed based on the square-hole characteristics of the docking. In the assembly verification, mechanism calibration and measurement field iterative compensation optimized the mechanism pose, and the accuracy of the pose adjustment component and the cabin assembly stress were tested. The experimental results show that the method proposed can reduce the interference problem during the assembly of a flange face and optimize the assembly stress while achieving assembly accuracy. The results for accuracy and interference force after flange face assembly were obtained; however, no feedback on the analytical method was obtained through the experimental results. Therefore, it is necessary to test further the propositions investigated. The preassembly analysis method for large cabin assemblies proposed optimizes the cabin assembly accuracy and internal stress under the deformation state of the assembly surface.

GLS‐PIA: n‐Dimensional Spherical B‐Spline Curve Fitting based on Geodesic Least Square with Adaptive Knot Placement

AbstractDue to the widespread applications of curves on n‐dimensional spheres, fitting curves on n‐dimensional spheres has received increasing attention in recent years. However, due to the non‐Euclidean nature of spheres, curve fitting methods on n‐dimensional spheres often struggle to balance fitting accuracy and curve fairness. In this paper, we propose a new fitting framework, GLS‐PIA, for parameterized point sets on n‐dimensional spheres to address the challenge. Meanwhile, we provide the proof of the method. Firstly, we propose a progressive iterative approximation method based on geodesic least squares which can directly optimize the geodesic least squares loss on the n‐sphere, improving the accuracy of the fitting. Additionally, we use an error allocation method based on contribution coefficients to ensure the fairness of the fitting curve. Secondly, we propose an adaptive knot placement method based on geodesic difference to estimate a more reasonable distribution of control points in the parameter domain, placing more control points in areas with greater detail. This enables B‐spline curves to capture more details with a limited number of control points. Experimental results demonstrate that our framework achieves outstanding performance, especially in handling imbalanced data points. (In this paper, “sphere” refers to n‐sphere (n≥ 2) unless otherwise specified.)

Asynchronous progressive iterative approximation method for least squares fitting

For large data fitting, the least squares progressive iterative approximation (LSPIA) methods have been proposed by Lin and Zhang (2013) and Deng and Lin (2014), in which a constant step size is used. In this paper, we further accelerate the LSPIA method in terms of a Chebyshev semi-iterative scheme and present an asynchronous LSPIA (denoted by ALSPIA) method. The control points in ALSPIA are updated by using an extrapolated variant in which an adaptive step size is chosen according to the roots of Chebyshev polynomial. Our convergence analysis shows that ALSPIA is faster than the original LSPIA method in both singular and non-singular least squares fitting cases. Numerical examples show that the proposed algorithm is feasible and effective.

An Efficiency Boost for Genetic Algorithms: Initializing the GA with the Iterative Approximate Method for Optimizing the Traveling Salesman Problem—Experimental Insights

The genetic algorithm (GA) is a well-known metaheuristic approach for dealing with complex problems with a wide search space. In genetic algorithms (GAs), the quality of individuals in the initial population is important in determining the final optimal solution. The classic GA using the random population seeding technique is effective and straightforward, but the generated population may contain individuals with low fitness, delaying convergence to the best solution. On the other side, heuristic population seeding strategies provide the advantages of producing individuals with high fitness and encouraging rapid convergence to the optimal solution. Using background information on the problem being solved, researchers have developed several population seeding approaches. In this paper, to enhance the genetic algorithm efficiency, we propose a new method for the initial population seeding based on a greedy approach. The proposed method starts by adding four extreme cities to the route, creating a loop, and then adding each city to the route through a greedy strategy that calculates the cost of adding every city to different locations along the route. This method identifies the best position to place the city as well as the best city to add. Employing local constant permutations improves the resultant route even more. Together with the suggested approach, we examine the GA’s effectiveness while employing conventional population seeding methods like nearest-neighbor, regression-based, and random seeding. Utilizing some of the well-known Traveling Salesman Problem (TSP) examples from the TSPLIB, the standard library for TSPs, tests were conducted. In terms of the error rate, average convergence, and time, the experimental results demonstrate that the GA that employs the suggested population seeding technique performs better than other GAs that use conventional population seeding strategies.

A Collaboratively Iterative Sequential Approximate Optimization Method for Aerodynamic Optimization

A Collaboratively Iterative Sequential Approximate Optimization Method for Aerodynamic Optimization

Study on the Relaxation Time Characteristics of Brain Tissue Based on Multi-Parametric Quantitative Magnetic Resonance Imaging

Traditional magnetic resonance imaging (MRI) is qualitative imaging, and doctors need to rely on experience to diagnose diseases, which cannot meet the current needs of precision medicine. As a new quantitative magnetic resonance imaging technology, magnetic resonance fingerprint imaging can obtain a variety of human tissue parameters at the same time through a data acquisition, which greatly improves the imaging speed and improves the impact of noise on image quality. Several pattern matching algorithms are compared, including direct matching method, Bloch response iterative projection method, covering tree and approximate nearest neighbor search method, and improved methods. Absolute error image, mean absolute error (MAE), normalized root means square error (RMSE) and running time are counted in the experimental results. The results show that the improved method is better than the traditional method, which can greatly improve the quality of MR fingerprint multi-parameter images (T1, T2, B0, PD), and make the running time within an acceptable range. In addition, the improved algorithm is insensitive to random additive noise.

Curve fitting by GLSPIA

We develop the generalized B-splines least square progressive iterative approximation (GLSPIA) method to improve LSPIA method (Deng and Lin, 2014) from the perspective of its equivalent neural network framework. We replace the B-spline basis function in standard LSPIA method with the generalized B-spline basis function (Juhász and Róth, 2013) in the GLSPIA method. By adjusting the control points as well as the parameter in the core function, higher precision fitting is achieved. Moreover, in order to solve the over-fitting problem of GLSPIA method when dealing with noise data, we propose three kinds of regularizations using the geometric properties of control polygon. Numerical examples are presented to show the effectiveness of our method.

A family of hybrid iterative approximation methods for fitting blending curves

A family of hybrid iterative approximation methods for fitting blending curves

Robust Reconstruction of Closed Parametric Curves by Topological Understanding with Persistent Homology

Curve reconstruction is a fundamental problem in reverse engineering, which has intrigued researchers for decades. In this paper, we propose a topological understanding based method for reconstructing parametric curves robustly from unorganized point clouds. Given a point cloud, we firstly understand the number of closed curves which need to be reconstructed using persistent homology. Then, by calculating the persistent 1-cycles of the point cloud, the initial shapes of the reconstructed parametric curves are generated. Finally, the closed parametric curves are reconstructed with the weighted least-squares progressive iterative approximation (W-LSPIA) method. Due to the topological understanding, the reconstructed parametric curves are faithful to the salient topological structure of the point cloud. Moreover, the developed reconstruction method is robust, and the reconstructed curve is much less affected by noise points and outliers, compared with the conventional parametric curve reconstruction algorithms. Experimental results demonstrated in this paper show the effectiveness of the developed curve reconstruction method.

Iterative sequential approximate solutions method to Hyers–Ulam stability of time-varying delayed fractional-order neural networks

There is a key issue in the field of stability analysis for fractional-order neural networks (FONNs) concerning whether the exact solution of the FONNs exists under the condition that a series of successive approximate solutions of the models can be obtained. If so, can the state differences between the approximate solutions and the exact solution be estimated? Hyers–Ulam stability, as one of the powerful tools to solve this problem, has attracted the attention of scholars, and various related research achievements have been reported. In view of this, this paper aims to investigate the Hyers–Ulam stability of FONNs with time-varying delays. Using the successive approximation method and the modified retarded Henry–Gronwall integral inequality, several conditions guaranteeing the existence and uniqueness of the exact solution of FONNs with time-varying delays are derived. Moreover, the corresponding results on the Hyers–Ulam–Rassias stability of the addressed models are given. Finally, a numerical example is designed to illustrate the efficiency and rationality of the theoretical results.

Dynamics of the Three-Compartment Financial Bubble Model under the Mittag-Leffler Operator

This paper presents an investigation into the dynamics of the emerging three-compartment financial bubble problem using a new non-singular kernel Atangana–Baleanu derivative operator. The problem is tested for at least one solution, and a unique root is determined using an iterative Newton approximation method, providing a globally stable fractional analysis technique. Curve sketches of the globalized model are provided, considering integers and other conformable orders. Sensitivities of the fractional order and other model parameters are examined, offering insights into their impact on the system dynamics. This research contributes to understanding financial bubbles and lays the groundwork for future studies in this field.

Improving geometric iterative approximation methods using local approximations

In this study, we propose improvements to Geometric Iterative Methods (GIM) for curve and surface approximation. Our key contributions include introducing an error vector function and deriving new error vectors, which enhance the approximation quality compared to the Least Squared Progressive and Iterative Approximation (LSPIA) methods. Additionally, we integrate parameter correction and control points update based on local approximations, achieving comparable accuracy to Geometric Approximation (GA) methods while being more computationally efficient. Extensive experiments demonstrate the superior accuracy and computational efficiency of our method.

A new Chinese text clustering algorithm based on WRD and improved K-means

Text clustering has been widely used in data mining, document management, search engines, and other fields. The K-means algorithm is a representative algorithm of text clustering. However, traditional K-means algorithm often uses Euclidean distance or cosine distance to measure the similarity between texts, which is not effective in face of high-dimensional data and cannot retain enough semantic information. In response to the above problems, we combine word rotator’s distance with the K-means algorithm, and propose the WRDK-means algorithm, which use word rotator’s distance to calculate the similarity between texts and preserve more text features. Furthermore, we define a new cluster center initialization method that improves cluster instability during random initial cluster center selection. And, to solve the problem of inconsistent length between texts, we propose a new iterative approximation method of cluster centers. We selected three suitable datasets and five evaluation indicators to verify the feasibility of the proposed algorithm. Among them, the RI value of our algorithm exceeds 90%. And for Marco_F1, our scheme was about 37.77%, 23.2%, 13.06% and 20.12% better than other four methods, respectively.

A switching MPC-based power management strategy for DC microgrids with SOC consistency and energy loss

The DC microgrids equipped with photovoltaic and energy storage systems have increasingly been applied to unmanned vehicles, and energy management is attracted much more attention. In this work, a power management strategy based on switching model predictive control is proposed to tackle energy loss and inconsistency of energy storage systems. The dynamic characteristic of the battery are fully addressed, and the non-affine equation constraints are dealt with by the iterative approximation method. The proposed strategy balances energy loss and state-of-charge consistency of the energy storage system and can operate when future environmental information is unavailable. Finally, comparative examples with existing schemes verify the effectiveness of the proposed strategy.

A radial basis function approximation method for conservative Allen–Cahn equations on surfaces

In this paper, we present a meshless radial basis function method to solve conservative Allen–Cahn equation on smooth compact surfaces embedded in R3, which can inherits the mass conservation property. The proposed method is established on the operator splitting scheme. We approximate the surface Laplace–Beltrami operator by an iterative radial basis function approximation method and discretize the diffusion equation in time by the Euler method. The reaction equation containing the nonlinear function is solved analytically. Moreover, to make the mass conservation, we employ a kernel-based quadrature formula to approximate the Lagrange multiplier. The salient feature of the meshless conservative scheme is that it is explicit and more efficient than narrow band methods since few scattered nodes on the surface are adopted in spatial approximation. Several numerical experiments are performed to illustrate the accuracy and the conservation property of the scheme on spheres and other surfaces.

Improved Least-Squares Progressive Iterative Approximation for Tensor Product Surfaces

Geometric iterative methods, including progressive iterative approximation and geometric interpolation methods, are efficient for fitting a given data set. With the development of big data technology, the number of fitting data points has become massive, and the progressive iterative approximation for least-squares fitting (LSPIA) is generally applied to fit mass data. Combining the Schulz iterative method for calculating the Moore–Penrose generalized inverse matrix with the traditional LSPIA method, this paper presents an accelerated LSPIA method for tensor product surfaces and shows that the corresponding iterative surface sequence converged to the least-squares fitting surface of the given data set. The iterative format is that of a non-stationary iterative method, and the convergence rate increased rapidly as the iteration number increased. Some numerical examples are provided to illustrate that the proposed method has a faster convergence rate.

IAM-TSP: Iterative Approximate Methods for Solving the Travelling Salesman Problem

IAM-TSP: Iterative Approximate Methods for Solving the Travelling Salesman Problem

DINAMIC FORCES IN THE ECCENTRICALLY COMPRESSED MEMBERS OF REINFORCED CONCRETE FRAMES UNDER ACCIDENTAL IMPACTS

Interest to solving scientific problems related to the evaluation of facilities' resistance and its protection against progressive collapse increases and attracts more and more attention of specialists in the field of structural analysis and design. Therefore, the article presents the results of a computational analysis of dynamic forces in eccentrically compressed reinforced concrete members of structures under accidental impact such as sudden removal of a load bearing member. Using relations for specific deformation energy and integrating it through the cross-section area, the analytical expressions for dynamic strains and curvatures have been obtained for physically and structurally nonlinear RC frame members under eccentric compression. These expressions in some cases allow symbolic solution, for example, in MathCAD software. In contrary, it can be solved with the approximate iterative method. To assess the reliability and effectiveness of the proposed quasi-static method, the analysis of the cast-in-situ reinforced concrete frame resistance to progressive collapse has been performed. The article also provides comparison of the simulation results of the nonlinear quasi-static analysis and the nonlinear dynamic time-history analysis.