Overrelaxation Algorithm Research Articles

Higher order phase-field modeling of brittle fracture via isogeometric analysis

AbstractThe evolution of brittle fracture in a material can be conveniently investigated by means of the phase-field technique introducing a smooth crack density functional. Following Borden et al. (2014), two distinct types of phase-field functional are considered: (i) a second-order model and (ii) a fourth-order one. The latter approach involves the bi-Laplacian of the phase field and therefore the resulting Galerkin form requires continuously differentiable basis functions: a condition we easily fulfill via Isogeometric Analysis. In this work, we provide an extensive comparison of the considered formulations performing several tests that progressively increase the complexity of the crack patterns. To measure the fracture length necessary in our accuracy evaluations, we propose an image-based algorithm that features an automatic skeletonization technique able to track complex fracture patterns. In all numerical results, damage irreversibility is handled in a straightforward and rigorous manner using the Projected Successive Over-Relaxation algorithm that is suitable to be adopted for both phase-field formulations since it can be used in combination with higher continuity isogeometric discretizations. Based on our results, the fourth-order approach provides higher rates of convergence and a greater accuracy. Moreover, we observe that fourth- and second-order models exhibit a comparable accuracy when the former methods employ a mesh-size approximately two times larger, entailing a substantial reduction of the computational effort.

Robust Fisher-Regularized Twin Extreme Learning Machine with Capped L1-Norm for Classification

Twin extreme learning machine (TELM) is a classical and high-efficiency classifier. However, it neglects the statistical knowledge hidden inside the data. In this paper, in order to make full use of statistical information from sample data, we first come up with a Fisher-regularized twin extreme learning machine (FTELM) by applying Fisher regularization into TELM learning framework. This strategy not only inherits the advantages of TELM, but also minimizes the within-class divergence of samples. Further, in an effort to further boost the anti-noise ability of FTELM method, we propose a new capped L1-norm FTELM (CL1-FTELM) by introducing capped L1-norm in FTELM to dwindle the influence of abnormal points, and CL1-FTELM improves the robust performance of our FTELM. Then, for the proposed FTELM method, we utilize an efficient successive overrelaxation algorithm to solve the corresponding optimization problem. For the proposed CL1-FTELM, an iterative method is designed to solve the corresponding optimization based on re-weighted technique. Meanwhile, the convergence and local optimality of CL1-FTELM are proved theoretically. Finally, numerical experiments on manual and UCI datasets show that the proposed methods achieve better classification effects than the state-of-the-art methods in most cases, which demonstrates the effectiveness and stability of the proposed methods.

Optical waveguide analysis using alternative direction implicit (ADI) method in combination with successive over-relaxation (SOR) algorithm

Optical waveguide analysis using alternative direction implicit (ADI) method in combination with successive over-relaxation (SOR) algorithm

A novel high-dimensional and multi-physics modeling approach of proton exchange membrane fuel cell for real-time simulation

Fuel cell model provides state observation and quantitative analysis for Hardware-In-the-Loop (HIL) test platform. For a high-dimensional multi-physics model, the observed distribution of physical properties is meaningful for fuel cell stack optimization, manufacturing process and model-based control development. However, traditional modeling approach is difficult to consider both the calculation accuracy and efficiency. To solve this problem, this paper proposes a high-dimensional multi-physics real-time modeling approach for fuel cells based on partial differential equations (PDEs)-tridiagonal matrix equation transformation (TMET) and symmetric successive overrelaxation (SSOR) algorithm. In the proposed method, a generalized framework of TMET and a novel staggered grid technique is developed. In this case, both the convection and diffusion phenomena can be accurately described. For different fuel cells or parameters change, supplementary model modification or development is not required, only the global variables need to be re-assigned. Moreover, SSOR with bi-directional iteration mechanism is used to increase the probability of computational convergence and thus improve the calculation efficiency. Compared with commercial COMSOL model, by using the proposed modeling approach, the mean absolute percentage error (MAPE) of the simulated polarization output characteristics is 1.77%, the MAPE of different distribution characteristics can remain within 10%, and the calculation time can be shortened from hours even days to seconds.

Numerical investigation of treated brain glioma model using a two-stage successive over-relaxation method

A brain tumor is a dynamic system in which cells develop rapidly and abnormally, as is the case with most cancers. Cancer develops in the brain or inside the skull when aberrant and odd cells proliferate in the brain. By depriving the healthy cells of leisure, nutrition, and oxygen, these aberrant cells eventually cause the healthy cells to perish. This article investigated the development of glioma cells in treating brain tumors. Mathematically, reaction-diffusion models have been developed for brain glioma growth to quantify the diffusion and proliferation of the tumor cells within brain tissues. This study presents the formulation the two-stage successive over-relaxation (TSSOR) algorithm based on the finite difference approximation for solving the treated brain glioma model to predict glioma cells in treating the brain tumor. Also, the performance of TSSOR method is compared to the Gauss-Seidel (GS) and two-stage Gauss-Seidel (TSGS) methods in terms of the number of iterations, the amount of time it takes to process the data, and the rate at which glioma cells grow the fastest. The implementation of the TSSOR, TSGS, and GS methods predicts the growth of tumor cells under the treatment protocol. The results show that the number of glioma cells decreased initially and then increased gradually by the next day. The computational complexity analysis is also used and concludes that the TSSOR method is faster compared to the TSGS and GS methods. According to the results of the treated glioma development model, the TSSOR approach reduced the number of iterations by between 8.0 and 71.95%. In terms of computational time, the TSSOR approach is around 1.18–76.34% faster than the TSGS and GS methods.

Any-Time Feasible Coordination for Multivehicle Systems

This article proposes a distributed control methodology for the coordination of multiple connected and automated vehicles (CAVs). Each vehicle computes its local trajectory based on a model predictive control (MPC) law and communicates the result with other relevant vehicles. In an iterative negotiation process, planned trajectories are optimized within a sampling time step. Inspired by the Jacobi over-relaxation (JOR) algorithm, we develop a distributed Jacobi over-relaxation algorithm (DJOR) for vehicle coordination. The modified algorithm exploits the structure of the distributed problem setting in which coupling occurs only in a bilateral way. Besides being able to guarantee any-time feasibility that implies collision-freeness, the algorithm scales well, unlike the standard JOR algorithm. The DJOR algorithm allows for significantly less conservatism in the choice of the update weightings. As a result, much faster convergence rates can be expected. Furthermore, the collision avoidance guarantee is extended for unforeseen scenarios such as emergency braking. Using an exact penalty function formulation ensures that the distributed optimization problem remains feasible even in previously unforeseen cases. Numerical simulations of an intersection crossing scenario illustrate the presented approach and show its benefits in comparison with standard traffic rules and a centralized computation.

An iterative technique for solving path planning in identified environments by using a skewed block accelerated algorithm

<abstract> <p>Currently, designing path-planning concepts for autonomous robot systems remains a topic of high interest. This work applies computational analysis through a numerical approach to deal with the path-planning problem with obstacle avoidance over a robot simulation. Based on the potential field produced by Laplace's equation, the formation of a potential function throughout the simulation configuration regions is obtained. This potential field is typically employed as a guide in the global approach of robot path-planning. An extended variant of the over-relaxation technique, namely the skewed block two-parameter over relaxation (SBTOR), otherwise known as the explicit decoupled group two-parameter over relaxation method, is presented to obtain the potential field that will be used for solving the path-planning problem. Experimental results with a robot simulator are presented to demonstrate the performance of the proposed approach on computing the harmonic potential for solving the path-planning problem. In addition to successfully validating pathways generated from various locations, it is also demonstrated that SBTOR outperforms existing over-relaxation algorithms in terms of the number of iterations, as well as the execution time.</p> </abstract>

Parallel Algorithm for Calculating the Dynamics of the Electron Beam Current Distribution during Pulsed Heating of a Metal Target

Speedup and efficiency of two parallel algorithms for calculating the dynamics of the current distribution when the surface of a tungsten sample is heated by an electron beam pulse are presented. The algorithms are implemented using OpenMP technology. A special case of axial symmetry without taking into account electric driving forces is considered. The temperature in the sample, calculated on the basis of the two-phase Stefan problem, is necessary for solving electrodynamic equations. To implement them, the methods of Jacobi and the successive over-relaxation methods were used. Pipelining is used to parallelize the successive over-relaxation algorithm. The current is considered as a possible source of rotation of the substance, which is observed in the experiment.

A Markovian approach to modeling a periodic order‐up‐to‐level policy under stochastic discrete demand and lead time with lost sales

AbstractIn this paper, a periodic order‐up‐to‐level policy is developed by using a Markovian approach. Customer demand is modeled by using an empirical distribution. Unmet customer demand is considered as lost sales. The internal storage capacity is limited, and excessive units are stored in external storage space. Replenishment lead time is discrete and stochastic. Because of the characteristics of the demand and lead time, the demand during the lead time and the demand during the protection interval are empirical. The Markovian approach is used to estimate the system measures of performance, including the expected inventory on‐hand, expected lost sales, and expected over‐storage. The successive overrelaxation algorithm is implemented to overcome the dimensional issue of the Markov model. The optimal policy that minimizes the total cost is obtained by using a customized golden section search procedure. The proposed method is applied to a real dataset of consumable products. In addition, the effectiveness of the approach is demonstrated by a comparison with a simulation. The results indicate that the model from our approach can accurately describe the behavior of the periodic order‐up‐to‐level review system.

Monte Carlo simulation fused with target distribution modeling via deep reinforcement learning for automatic high-efficiency photon distribution estimation

Particle distribution estimation is an important issue in medical diagnosis. In particular, photon scattering in some medical devices extremely degrades image quality and causes measurement inaccuracy. The Monte Carlo (MC) algorithm is regarded as the most accurate particle estimation approach but is still time-consuming, even with graphic processing unit (GPU) acceleration. The goal of this work is to develop an automatic scatter estimation framework for high-efficiency photon distribution estimation. Specifically, a GPU-based MC simulation initially yields a raw scatter signal with a low photon number to hasten scatter generation. In the proposed method, assume that the scatter signal follows Poisson distribution, where an optimization objective function fused with sparse feature penalty is modeled. Then, an over-relaxation algorithm is deduced mathematically to solve this objective function. For optimizing the parameters in the over-relaxation algorithm, the deep Q -network in the deep reinforcement learning scheme is built to intelligently interact with the over-relaxation algorithm to accurately and rapidly estimate a scatter signal with the large range of photon numbers. Experimental results demonstrated that our proposed framework can achieve superior performance with structural similarity > 0.94 , peak signal-to-noise ratio > 26.55 dB , and relative absolute error < 5.62 % , and the lowest computation time for one scatter image generation can be within 2 s.

Fisher-regularized supervised and semi-supervised extreme learning machine

The structural information of data contains useful prior knowledge and thus is important for designing classifiers. Extreme learning machine (ELM) has been a potential technique in handling classification problems. However, it only simply considers the prior class-based structural information and ignores the prior knowledge from statistics and geometry of data. In this paper, to capture more structural information of the data, we first propose a Fisher-regularized extreme learning machine (called Fisher-ELM) by applying Fisher regularization into the ELM learning framework, the main goals of which is to build an optimal hyperplane such that the output weight and within-class scatter are minimized simultaneously. The proposed Fisher-ELM reflects both the global characteristics and local properties of samples. Intuitively, the Fisher-ELM can approximatively fulfill the Fisher criterion and can obtain good statistical separability. Then, we exploit graph structural formulation to obtain semi-supervised Fisher-ELM version (called Lap-FisherELM) by introducing manifold regularization that characterizes the geometric information of the marginal distribution embedded in unlabeled samples. An efficient successive overrelaxation algorithm is used to solve the proposed Fisher-ELM and Lap-FisherELM, which converges linearly to a solution, and can process very large datasets that need not reside in memory. The proposed Fisher-ELM and Lap-FisherELM do not need to deal with the extra matrix and burden the computations related to the variable switching, which makes them more suitable for relatively large-scale problems. Experiments on several datasets verify the effectiveness of the proposed methods.

Numerical Solution of European and American Option with Dividends using Finite Difference Methods

Numerical methods form an important part of the pricing of financial derivatives where there is no closed form analytical formula. Black-Scholes equation is a well known partial differential equation in financial mathematics. In this paper, we have studied the numerical solutions of the Black-Scholes equation for European options (Call and Put) as well as American options with dividends. We have used different approximate to discretize the partial differential equation in space and explicit (Forward Euler’s), fully implicit with projected Successive Over-Relaxation (SOR) algorithm and Crank-Nicolson scheme for time stepping. We have implemented and tested the methods in MATLAB. Finally, some numerical results have been presented and the effects of dividend payments on option pricing have also been considered.

Fast clustering-based weighted twin support vector regression

Construction of an effective model for regression to fit data samples with noise or outlier is a challenging work. In this paper, in order to reduce the influence of noise or outlier on regression and further improve the prediction performance of standard twin support vector regression (TSVR), we proposed a fast clustering-based weighted TSVR, termed as FC-WTSVR. First, we use a fast clustering algorithm to quickly classify samples into different categories based on their similarities. Secondly, to reflect the prior structural information and distinguish contributions of samples located at different positions to regression, we introduce the covariance matrix and weighted diagonal matrix into the primal problems of FC-WTSVR, respectively. Finally, to shorten the training time, we adopt the successive over-relaxation algorithm to solve the quadratic programming problems. The results show that the proposed FC-WTSVR can obtain better prediction performance and anti-interference capability than some state-of-the-art algorithms.

Precursors of the spin glass transition in three dimensions

We study energy landscape and dynamics of the three-dimensional Heisenberg spin glass model in the paramagnetic phase, i.e. for temperature T larger than the critical temperature . The landscape is non-trivially related to the equilibrium states even in the high-temperature phase, and reveals an onset of non-trivial behavior at a temperature , which is also seen through the behavior of the thermoremanent magnetization. We also find a power-law growth of the relaxation times far from the spin-glass transition, indicating a dynamical crossover at a temperature , . The arising picture is reminiscent of the phenomenology of supercooled liquids, and poses questions on which mean-field models can describe qualitatively well the phenomenology in three dimensions. On the technical side, local energy minima are found with the Successive Overrelaxation algorithm, which reveals very efficient for energy minimization in this kind of models.

A parallel second-order unstructured finite volume method for 3D free-surface flows using a σ coordinate

In this paper, we introduce a second-order time- and space-accurate technique, developed to solve in parallel free-surface flows in arbitrary three-dimensional geometries. The discretization is based on a second-order finite-volume technique on prisms elements, consisting of triangular grids on the horizontal and bounded by a free surface and an irregular bottom on the vertical. The equations are transformed vertically to the σ-coordinate system in order to obtain an accurate representation of top and bottom topography. The reconstruction of three presure/velocity decoupling methods using a Crank-Nicolson scheme formulation is proposed. The Momentum Interpolation Method (MIM) is combined with Local Extremum Diminishing (LED) second-order upstream scheme for convective terms is developed. The parallelization is designed by a block domain decomposition technique. The discretization results in non-symmetric variable-coefficient linear systems which are solved using a parallel multi-color Successive Over-Relaxation algorithm. Several test cases of surface wave motion are used to demonstrate the capabilities, numerical stability and performance of the model.

Accelerated low rank matrix approximate algorithms for matrix completion

In this paper, we propose two accelerated algorithms for the low-rank approximate method in Wang et al. (0000) for matrix completion. The main idea is to use the successive over-relaxation technique. Based on the successive over-relaxation method for the feasible matrices or projection matrices, the low-rank matrix approximate method is modified and accelerated. Meanwhile, we discuss the convergence of the over-relaxation algorithm for the feasible matrix. Finally, the numerical experiments show them to be effective.

A novel projection nonparallel support vector machine for pattern classification

In this paper, we propose a novel nonparallel classifier termed as projection nonparallel support vector machine (PNPSVM), which is fully different from the existing nonparallel classifiers. The new classifier needs two steps to obtain the optimal proximal hyperplanes. The first step is to obtain two projection directions, which could achieve maximum class separability by minimizing the within-class distance and maximizing the between-class distance simultaneously, to be treated as the normal vectors of the optimal proximal hyperplanes, and the second step is to determine the specific locations of the optimal proximal hyperplanes based on an appropriate central sample. Furthermore, the improved successive overrelaxation (SOR) algorithm is applied to solve our PNPSVM. The incomparable advantages of this paper can be summarized as follows: (1) implementing the structural risk minimization principle in the primal problems; (2) utilizing the potential structural information of data by considering both the tightness between the similar patterns and the discrepancy between the dissimilar pairs; (3) the kernel trick can be applied directly since the dual problems have the similar elegant formulation as that of standard support vector machine; (4) SOR technique is introduced to solve our optimization problems. The comprehensive experimental results on an artificial dataset and twenty-four UCI datasets demonstrate the effectiveness of our method in classification accuracy.

Parallel simulations for a 2D x/z two-phase flow fluid-solid particle model

The importance of the two-phase flow model relies upon the correct simulation of specific problems in which the passive tracer model fails; however, the mathematical and numerical models need to be improved in order to reproduce fluid-solid particle interactions of greater complexity. Such is the case with regard to simulating erosion patterns suffered upon horizontal granular beds by means of a vertical water jet as shown here. Moreover, sequential platforms have proven to be insufficient in providing the required computational power needed to obtain fast and detailed simulations. In this paper, a fully parallel algorithm for a two-dimensional x/z two-phase Eulerian approach is presented and applied for the numerical solution of the erosion of sediment beds. The parallelization is designed by a row and column block domain decomposition technique using a distributed memory platform with Message Passing Interface (MPI). Arising from the numerical method, a Poisson problem for the pressure is solved at each time step. The discretization results in a non-symmetric variable-coefficient linear system which is solved using several parallel Successive Over-Relaxation (SOR) algorithms, including partitioning and coloring methods. The specification of the optimal relaxation parameter to achieve efficiency is found numerically. Results show that SOR methods achieve faster convergence rates and the simulation time achieves an order similar to that required for the typical, widely-used Bi-Conjugate Gradient Stabilized (Bi-CGSTAB) method. The performances of the algorithms are evaluated in terms of speedup and efficiency. The results indicate that the parallel code significantly improves the results of the sequential calculation in general.

Stochastic convergence analysis of recursive successive over-relaxation algorithm in adaptive filtering

A stochastic convergence analysis of the parameter vector estimation obtained by the recursive successive over-relaxation (RSOR) algorithm is performed in mean sense and mean-square sense. Also, excess of mean-square error and misadjustment analysis of the RSOR algorithm is presented. These results are verified by ensemble-averaged computer simulations. Furthermore, the performance of the RSOR algorithm is examined using a system identification example and compared with other widely used adaptive algorithms. Computer simulations show that the RSOR algorithm has better convergence rate than the widely used gradient-based algorithms and gives comparable results obtained by the recursive least-squares RLS algorithm.

MLTSVM: A novel twin support vector machine to multi-label learning

Multi-label learning paradigm, which aims at dealing with data associated with potential multiple labels, has attracted a great deal of attention in machine intelligent community. In this paper, we propose a novel multi-label twin support vector machine (MLTSVM) for multi-label classification. MLTSVM determines multiple nonparallel hyperplanes to capture the multi-label information embedded in data, which is a useful promotion of twin support vector machine (TWSVM) for multi-label classification. To speed up the training procedure, an efficient successive overrelaxation (SOR) algorithm is developed for solving the involved quadratic programming problems (QPPs) in MLTSVM. Extensive experimental results on both synthetic and real-world multi-label datasets confirm the feasibility and effectiveness of the proposed MLTSVM.