Progressive Iterative Approximation Research Articles

Particle swarm optimization-based load balance-aware energy-efficient scheduling for mobile crowd computing

Due to their increasing computational power and energy-efficient hardware, today’s smart mobile devices (SMDs) are replacing desktops and laptops as casual computing devices. Moreover, a cluster of such powerful SMDs can garner substantial high-performance computing (HPC). Such an HPC is achieved by utilizing publicly owned SMDs in mobile crowd computing (MCC). Here, a large computing-intensive task is divided and scheduled for the available SMDs for execution, and the results are recollected. This approach provides an economical and sustainable HPC. However, battery-powered constrained energy is a great hindrance to achieving this goal. Therefore, in the MCC, it is crucial to minimize the overall energy consumption to complete the task. This can be achieved to some extent by optimizing task scheduling to the appropriate SMDs. However, considering only energy efficiency might lead to an enormous load imbalance among SMDs, i.e., the most energy-efficient SMDs would be overloaded most of the time. Considering this, in this paper, we present a modified particle swarm optimization (PSO)-based scheduling algorithm to minimize the overall energy consumption among a set of SMDs designated to execute a set of MCC tasks while maintaining a satisfactory load balance level. Extensive simulations with both synthetic and real data sets are carried out to analyze and validate the proposed method. The work was compared with popular heuristic (minimum completion time (MCT), MinMin, MaxMin, and preconditioned progressive iterative approximation (PPIA)) and metaheuristic (genetic algorithm (GA)) optimization algorithms, which yields significant improvements over others in terms of the considered objectives. In addition, an analysis of variance (ANOVA) test is conducted to provide further evidence regarding the distinctiveness of the proposed algorithm.

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.

Two novel iterative approaches for improved LSPIA convergence

This paper introduces two improved variants of the least squares progressive-iterative approximation (LSPIA) by leveraging momentum techniques. Specifically, based on the Polyak's and Nesterov's momentum techniques, the proposed methods utilize the previous iteration information to update the control points. We name these two methods PmLSPIA and NmLSPIA, respectively. The introduction of momentum enhances the determination of the search directions, leading to a significant improvement in convergence rate. The geometric interpretations of PmLSPIA and NmLSPIA are elucidated, providing insights into the underlying principles of these accelerated algorithms. Rigorous convergence analyses are conducted, revealing that both PmLSPIA and NmLSPIA exhibit faster convergence than LSPIA. Numerical results further validate the efficacy of the proposed algorithms.

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.

Randomized progressive iterative approximation for B-spline curve and surface fittings

For large-scale data fitting, the least-squares progressive iterative approximation is a widely used method in many applied domains because of its intuitive geometric meaning and efficiency. In this work, we present a randomized progressive iterative approximation (RPIA) for the B-spline curve and surface fittings. In each iteration, RPIA locally adjusts the control points according to a random criterion of index selections. The difference for each control point is computed concerning the randomized block coordinate descent method. From geometric and algebraic aspects, the illustrations of RPIA are provided. We prove that RPIA constructs a series of fitting curves (resp., surfaces), whose limit curve (resp., surface) can converge in expectation to the least-squares fitting result of the given data points. Numerical experiments are given to confirm our results and show the benefits of RPIA.

Periodic implicit representation, design and optimization of porous structures using periodic B-splines

Porous structures are intricate solid materials with numerous small pores, extensively used in fields like medicine, chemical engineering, and aerospace. However, the design of such structures using computer-aided tools is a time-consuming and tedious process. In this study, we propose a novel representation method and design approach for porous units that can be infinitely spliced to form a porous structure. We use periodic B-spline functions to represent periodic or symmetric porous units. Starting from a voxel representation of a porous sample, the discrete distance field is computed. To fit the discrete distance field with a periodic B-spline, we introduce the constrained least squares progressive-iterative approximation algorithm, which results in an implicit porous unit. This unit can be subject to optimization to enhance connectivity and utilized for topology optimization, thereby improving the model’s stiffness while maintaining periodicity or symmetry. The experimental results demonstrate the potential of the designed complex porous units in enhancing the mechanical performance of the model. Consequently, this study has the potential to incorporate remarkable structures derived from artificial design or nature into the design of high-performing models, showing the promise for biomimetic applications.

HSS-progressive interpolation for Loop and Catmull–Clark Subdivision Surfaces

Loop and Catmull–Clark subdivision schemes are the most popular and commonly employed approximation subdivision schemes. However, some features of the given mesh are lost in their refinement process. Thus, the subdivision surfaces of Loop and Catmull–Clark schemes does not interpolate the vertices of the given mesh and shrink in some cases. This paper proposed new progressive iterative approximation (PIA) formats based on Hermitian and skew-Hermitian splitting iteration technique (HSS). The proposed method named H-PIA and its weighted version called WH-PIA force the limit surface of Loop and Catmull–Clark subdivision schemes to interpolate the vertices of the given mesh. The approximate optimal weight of WH-PIA is given, and the convergence of H-PIA and WH-PIA are proved. Various test examples are provided to illustrate the efficiency and effectiveness of the proposed H-PIA and WH-PIA. Experimental results demonstrate that the rate of convergence of H-PIA and WH-PIA are faster than that of the PIA and weighted PIA (W-PIA), respectively.

Full-LSPIA: A Least-Squares Progressive-Iterative Approximation Method with Optimization of Weights and Knots for NURBS Curves and Surfaces

The Least-Squares Progressive-Iterative Approximation (LSPIA) method offers a powerful and intuitive approach for data fitting. Non-Uniform Rational B-splines (NURBS) are a popular choice for approximation functions in data fitting, due to their robust capabilities in shape representation. However, a restriction of the traditional LSPIA application to NURBS is that it only iteratively adjusts control points to approximate the provided data, with weights and knots remaining static. To enhance fitting precision and overcome this constraint, we present Full-LSPIA, an innovative LSPIA method that jointly optimizes weights and knots alongside control points adjustments for superior NURBS curves and surfaces creation. We achieve this by constructing an objective function that incorporates control points, weights, and knots as variables, and solving the resultant optimization problem. Specifically, control points are adjusted using LSPIA, while weights and knots are optimized through the LBFGS method based on the analytical gradients of the objective function with respect to weights and knots. Additionally, we present a knot removal strategy known as Decremental Full-LSPIA. This strategy reduces the number of knots within a specified error tolerance, and determines optimal knot locations. The proposed Full-LSPIA and Decremental Full-LSPIA maximize the strengths of LSPIA, with numerical examples further highlighting the superior performance and effectiveness of these methods. Compared to the classical LSPIA, Full-LSPIA offers greater fitting accuracy for NURBS curves and surfaces while maintaining the same number of control points, and automatically determines suitable weights and knots. Moreover, Decremental Full-LSPIA yields fitting results with fewer knots while maintaining the same error tolerance.

Preconditioned geometric iterative methods for B-spline interpolation

The geometric iterative method (GIM) is widely used in data interpolation/fitting, but its slow convergence affects the computational efficiency. Recently, much work has been done to guarantee the acceleration of GIM in the literature. This work aims to accelerate the convergence rate by introducing a preconditioning technique. After constructing the preconditioner, we preprocess the progressive iterative approximation (PIA) and its variants, called the preconditioned GIMs. We show that the proposed preconditioned GIMs converge, and the extra computation cost of the preconditioning technique is negligible. Several numerical experiments are given to demonstrate that our preconditioner can accelerate the convergence rate of PIA and its variants.

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.

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.

Reparameterization of B-spline surface and its application in ship hull modeling

NURBS are widely used in the design of free curves and surfaces and have been the industrial standard for product design in CAD/CAM. Ship hull surfaces are complex doubly-curved surfaces and usually decomposed into patches based on feature curves. Due to the topologically rectangular limitations of tensor-product NURBS surfaces and the inevitable problem of surface merging caused by decomposition, the reconstruction of ship hull still remains a complicated problem. The theoretical research of merging NURBS surfaces is relatively comprehensive, but the difficulty lies in how to merge surfaces defined in different knot vectors at the common boundary curve since control meshes form T-junctions. This paper proposes a methodology of reparameterizing B-spline surfaces, using a fast progressive-iterative approximation (FPIA) method to refit section curves or isoparametric curves with constraints, thus making the topological structure of control meshes regular, uniform and orthogonal. Subsequently, a KCS container ship with bulbous bow and stern is utilized as an illustration for reconstructing with B-splines. The lighting effects and lines plan derived from the hull surface demonstrate that this approach can achieve fair, smooth and watertight surfaces. The approach exhibits strong robustness, versatility and generalization ability, and provides theoretical innovation and practical design methods for representing and reconstructing complex ship hull.

Free-form multi-level porous model design based on truncated hierarchical B-spline functions

Porous structures have been widely applied in aerospace, paramedical, transportation, and other fields owing to their high interconnectivity, low weight, and extensive internal surface areas. Multi-level porous structures with different hole shapes and distributions at varying scales have shown excellent biological and mechanical performances compared with single-level porous structures. However, designing multi-level porous models with complex topological and geometric structures is a challenge. In this work, we propose a multi-level porous model design framework. Given a model represented by a trivariate B-spline solid and porous structures represented by implicit functions, we design the porous model in the parametric domain. We use threshold distribution fields represented by truncated hierarchical B-spline (THB-spline) functions to control the pore shapes, locations, and number of levels. Furthermore, the proposed THB-spline least squares progressive-iterative approximation algorithm can flexibly adjust the porous model. The model meets the expected porosity through the proposed optimization algorithm. The porous structure in the parametric domain is finally mapped to the trivariate B-spline solid to generate the final model. Moreover, we define a new storage space-saving file format to store multi-level porous models. The experimental results demonstrate the effectiveness and efficiency of the method and the superior performance of the new storage format.

Fairing-PIA: progressive-iterative approximation for fairing curve and surface generation

The fairing curves and surfaces are used extensively in geometric design, modeling, and industrial manufacturing. However, the majority of conventional fairing approaches, which lack sufficient parameters to improve fairness, are based on energy minimization problems. In this study, we develop a novel progressive-iterative approximation method for the fairing curve and surface generation (fairing-PIA). Fairing-PIA is an iteration method that can generate a series of curves (surfaces) by adjusting the control points of B-spline curves (surfaces). In fairing-PIA, each control point is endowed with an individual weight. Thus, the fairing-PIA has many parameters to optimize the shapes of curves and surfaces. Not only a fairing curve (surface) can be generated globally through fairing-PIA, but also the curve (surface) can be improved locally. Moreover, we prove the convergence of the developed fairing-PIA and show that the conventional energy minimization fairing model is a special case of fairing-PIA. Finally, numerical examples indicate that the proposed method is effective and efficient.

IG-LSPIA: Least Squares Progressive Iterative Approximation for Isogeometric Collocation Method

The isogeometric collocation method (IGA-C), which is a promising branch of isogeometric analysis (IGA), can be considered fitting the load function with the combination of the numerical solution and its derivatives. In this study, we develop an iterative method, isogeometric least-squares progressive-iterative approximation (IG-LSPIA), to solve the fitting problem in the collocation method. IG-LSPIA starts with an initial blending function, where the control coefficients are combined with the B-spline basis functions and their derivatives. A new blending function is generated by constructing the differences for collocation points (DCP) and control coefficients (DCC), and then adding the DCC to the corresponding control coefficients. The procedure is performed iteratively until the stop criterion is reached. We prove the convergence of IG-LSPIA and show that the computation complexity in each iteration of IG-LSPIA is related only to the number of collocation points and unrelated to the number of control coefficients. Moreover, an incremental algorithm is designed; it alternates with knot refinement until the desired precision is achieved. After each knot refinement, the result of the last round of IG-LSPIA iterations is used to generate the initial blending function of the new round of iteration, thereby saving great computation. Experiments show that the proposed method is stable and efficient. In the three-dimensional case, the total computation time is saved twice compared to the traditional method.

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.

Implicit Randomized Progressive-Iterative Approximation for Curve and Surface Reconstruction

In this paper, we propose the implicit randomized progressive-iterative approximation (IR-PIA) method for curve and surface reconstruction. By introducing the effective probability criterion, the IR-PIA method selects the working elements of the collocation matrix to adjust the control coefficients. It is proved that the sequence of curves and surfaces generated by the control coefficients converge to the least-norm results in expectation. The IR-PIA method reduces the computation complexity and speeds up the curve and surface reconstruction compared with the implicit progressive-iterative approximation (I-PIA) method (Hamza et al., 2020). Numerical examples show that the IR-PIA method can be more efficient than the I-PIA method (Hamza et al., 2020). • The IR-PIA method converge to the least-norm solution in expectation. • The IR-PIA method eliminates the extra zero-level sheets effectively. • The working elements are selected to update the control coefficients. • The larger entries of the difference vectors can be annihilated as far as possible.

Hyperpower least squares progressive iterative approximation

Geometric iterative methods (GIMs) generate curves or surfaces that fit (interpolate or approximate) given sets of data points. Standard GIMs use the Richardson iteration or gradient descent method, so they converge relatively slowly. This paper investigates the GIMs acceleration method, which is based on the use of more efficient methods for the matrix inverse approximation. That is, the possibility to use hyperpower iterative methods is being explored. As a result, a novel GIM named hyperpower least squares progressive iterative approximation (HPLSPIA) is proposed. The convergence of the HPLSPIA method is analyzed, and the computational complexity is briefly discussed. The method is then applied to tensor product B-spline surface fitting. Several HPLSPIA methods utilizing different hyperpower methods for the matrix inverse approximation are experimentally compared. It is shown that polynomial factorizations used in the hyperpower iterative methods significantly affect the efficiency of the HPLSPIA methods.

Progressive Iterative Approximation of Non-Uniform Cubic B-Spline Curves and Surfaces via Successive Over-Relaxation Iteration

Geometric iteration (GI) is one of the most efficient curve- or surface-fitting techniques in recent years, which is famous for its remarkable geometric significance. In essence, GI can be thought of as the sum of iterative methods for solving systems of linear equations, such as progressive iterative approximation (PIA) which relies on the theory of Richardson iteration. Thus, when the curve- or surface-fitting error is at a desired level, we want to have as few iterations as possible to improve efficiency when dealing with large data sets. Based on the idea of successive over-relaxation (SOR) iteration, we formulate a faster PIA curve and surface interpolation scheme using classical non-uniform cubic B-splines, named SOR-PIA. The genetic algorithm is utilized to estimate the best approximate relaxation factor of SOR-PIA. Similar to standard PIA, SOR-PIA can also be regarded as a process in which the control points move in one direction, but it can greatly reduce the number of iterations in the iterative process with the same fitting accuracy. By comparing with the standard PIA and WPIA algorithms, the effectiveness of the SOR-PIA iterative interpolation algorithm can be verified.