Fast and Accurate Predictor-Corrector Methods Using Feedback-Accelerated Picard Iteration for Strongly Nonlinear Problems

Although predictor-corrector methods have been extensively applied,they might not meet the requirements of practical applications and engineering tasks,particularly when high accuracy and efficiency are necessary.A novel class of correctors based on feedback-accelerated Picard iteration(FAPI)is proposed to further enhance computational performance.With optimal feedback terms that do not require inversion of matrices,significantly faster convergence speed and higher numerical accuracy are achieved by these correctors compared with their counterparts;however,the computational complexities are comparably low.These advantages enable nonlinear engineering problems to be solved quickly and accurately,even with rough initial guesses from elementary predictors.The proposed method offers flexibility,enabling the use of the generated correctors for either bulk processing of collocation nodes in a domain or successive corrections of a single node in a finite difference approach.In our method,the functional formulas of FAPI are discretized into numerical forms using the collocation approach.These collocated iteration formulas can directly solve nonlinear problems,but they may require significant computational resources because of the manipulation of high-dimensionalmatrices.To address this,the collocated iteration formulas are further converted into finite difference forms,enabling the design of lightweight predictor-corrector algorithms for real-time computation.The generality of the proposed method is illustrated by deriving new correctors for three commonly employed finite-difference approaches:the modified Euler approach,the Adams-Bashforth-Moulton approach,and the implicit Runge-Kutta approach.Subsequently,the updated approaches are tested in solving strongly nonlinear problems,including the Matthieu equation,the Duffing equation,and the low-earth-orbit tracking problem.The numerical findings confirm the computational accuracy and efficiency of the derived predictor-corrector algorithms.

Calculation of Mass Concrete Temperature Containing Cooling Water Pipe Based on Substructure and Iteration Algorithm

Mathematical physics equations are often utilized to describe physical phenomena in various fields of science and engineering.One such equation is the Fourier equation,which is a commonly used and effective method for evaluating the effectiveness of temperature control measures for mass concrete.One important measure for temperature control in mass concrete is the use of cooling water pipes.However,the mismatch of grids between large-scale concrete models and small-scale cooling pipe models can result in a significant waste of calculation time when using the finite element method.Moreover,the temperature of the water in the cooling pipe needs to be iteratively calculated during the thermal transfer process.The substructure method can effectively solve this problem,and it has been validated by scholars.The Abaqus/Python secondary development technology provides engineers with enough flexibility to combine the substructure method with an iteration algorithm,which enables the creation of a parametric modeling calculation for cooling water pipes.This paper proposes such a method,which involves iterating the water pipe boundary and establishing the water pipe unit substructure to numerically simulate the concrete temperature field that contains a cooling water pipe.To verify the feasibility and accuracy of the proposed method,two classic numerical examples were analyzed.The results showed that this method has good applicability in cooling pipe calculations.When the value of the iteration parameterαis 0.4,the boundary temperature of the cooling water pipes can meet the accuracy requirements after 4∼5 iterations,effectively improving the computational efficiency.Overall,this approach provides a useful tool for engineers to analyze the temperature control measures accurately and efficiently for mass concrete,such as cooling water pipes,using Abaqus/Python secondary development.

Value Iteration-Based Cooperative Adaptive Optimal Control for Multi-Player Differential Games With Incomplete Information

This paper presents a novel cooperative value iteration(VI)-based adaptive dynamic programming method for multi-player differential game models with a convergence proof.The players are divided into two groups in the learning process and adapt their policies sequentially.Our method removes the dependence of admissible initial policies,which is one of the main drawbacks of the PI-based frameworks.Furthermore,this algorithm enables the players to adapt their control policies without full knowledge of others’ system parameters or control laws.The efficacy of our method is illustrated by three examples.

Another SSOR Iteration Method

Kellogg gave a version of the Peaceman-Radford method. In this paper, we introduce a SSOR iteration method which uses Kellogg’s method. The new algorithm has some advantages over the traditional SSOR algorithm. A Cyclic Reduction algorithm is introduced via a decoupling in Kellogg’s method.

Iteration and evaluation of shale oil development technology for continental rift lake basins

By benchmarking with the iteration of drilling technology,fracturing technology and well placement mode for shale oil and gas development in the United States and considering the geological characteristics and development difficulties of shale oil in the Jiyang continental rift lake basin,East China,the development technology system suitable for the geological characteristics of shale oil in continental rift lake basins has been primarily formed through innovation and iteration of the development,drilling and fracturing technologies.The technology system supports the rapid growth of shale oil production and reduces the development investment cost.By comparing it with the shale oil development technology in the United States,the prospect of the shale oil development technology iteration in continental rift lake basins is proposed.It is suggested to continuously strengthen the overall three-dimensional development,improve the precision level of engineering technology,upgrade the engineering technical indicator system,accelerate the intelligent optimization of engineering equipment,explore the application of complex structure wells,form a whole-process integrated quality management system from design to implementation,and constantly innovate the concept and technology of shale oil development,so as to promote the realization of extensive,beneficial and high-quality development of shale oil in continental rift lake basins.

Exploration of a Three-Stage Blended Teaching Model for the Medical Microbiology Course in the Context of Smart Education

Objective:This study explores the effectiveness of a three-stage blended teaching model in the Medical Microbiology course for Clinical Medicine students within the context of smart education.Methods:Clinical Medicine students from the 2021 and 2022 classes were studied,comparing the traditional and three-stage blended teaching models.Academic performance and survey feedback were analyzed.Results:The research group’s academic performance,including daily,final exam,and average scores,was significantly higher than the control group(P<0.05).Over 95%of students approved of the new teaching model,noting improvements in competencies.Conclusion:The three-stage blended teaching model effectively boosted student engagement,interaction,academic performance,and overall skills,showing potential for broader application in medical education.

Accelerated inexact Newton-Landweber iteration method for EIT image reconstruction

The image reconstruction of electrical impedance tomography(EIT)is a nonlinear and ill-posed inverse problem and the imaging results are easily affected by measurement noise,which needs to be solved by using regularization methods.The iterative regularization method has become a focus of the research due to its ease of implementation.To deal with the ill-posed and ill-conditional problems in image reconstruction,the inexact Newton-Landweber iterative method is considered and the Nesterov’s acceleration strategy is introduced.One Nesterov-type accelerated version of the inexact Newton-Landweber iteration is presented to determine the conductivity distributions inside an object from electrical measurements made on the surface.In order to further optimize the acceleration,both the steepest descent step-length and the minimal error step-length are exploited during the iterative image reconstruction process.Landweber iteration and its accelerated version are also implemented for comparison.All algorithms are terminated by the discrepancy principle.Finally,the performance is tested by reporting numerical simulations to verify the remarkable acceleration efficiency of the proposed method.

Toroidal torques due to n=1 magnetic perturbations in ITER baseline scenario

Toroidal torques,generated by the resonant magnetic perturbation(RMP)and acting on the plasma column,are numerically systematically investigated for an ITER baseline scenario.The neoclassical toroidal viscosity(NTV),in particular the resonant portion,is found to provide the dominant contribution to the total toroidal torque under the slow plasma flow regime in ITER.While the electromagnetic torque always opposes the plasma flow,the toroidal torque associated with the Reynolds stress enhances the plasma flow independent of the flow direction.A peculiar double-peak structure for the net NTV torque is robustly computed for ITER,as the toroidal rotation frequency is scanned near the zero value.This structure is found to be ultimately due to a non-monotonic behavior of the wave-particle resonance integral(over the particle pitch angle)in the superbanana plateau NTV regime in ITER.These findings are qualitatively insensitive to variations of a range of factors including the wall resistivity,the plasma pedestal flow and the assumed frequency of the rotating RMP field.

Complex seismic wavefi eld interpolation based on the Bregman iteration method in the sparse transform domain

In seismic prospecting, fi eld conditions and other factors hamper the recording of the complete seismic wavefi eld; thus, data interpolation is critical in seismic data processing. Especially, in complex conditions, prestack missing data affect the subsequent highprecision data processing workfl ow. Compressive sensing is an effective strategy for seismic data interpolation by optimally representing the complex seismic wavefi eld and using fast and accurate iterative algorithms. The seislet transform is a sparse multiscale transform well suited for representing the seismic wavefield, as it can effectively compress seismic events. Furthermore, the Bregman iterative algorithm is an efficient algorithm for sparse representation in compressive sensing. Seismic data interpolation methods can be developed by combining seismic dynamic prediction, image transform, and compressive sensing. In this study, we link seismic data interpolation and constrained optimization. We selected the OC-seislet sparse transform to represent complex wavefields and used the Bregman iteration method to solve the hybrid norm inverse problem under the compressed sensing framework. In addition, we used an H-curve method to choose the threshold parameter in the Bregman iteration method. Thus, we achieved fast and accurate reconstruction of the seismic wavefi eld. Model and fi eld data tests demonstrate that the Bregman iteration method based on the H-curve norm in the sparse transform domain can effectively reconstruct missing complex wavefi eld data.

Variational iteration method for solving the mechanism of the Equatorial Eastern Pacific El Nino-Southern Oscillation

A class of coupled system for the E1 Nifio-Southern Oscillation （ENSO） mechanism is studied. Using the method of variational iteration for perturbation theory, the asymptotic expansions of the solution for ENSO model are obtained and the asymptotic behaviour of solution for corresponding problem is considered.

Variational iteration solving method for El Nio phenomenon atmospheric physics of nonlinear model

A class of E1 Niйo atmospheric physics oscillation model is considered. The E1 Niйo atmospheric physics oscillation is an abnormal phenomenon involved in the tropical Pacific ocean-atmosphere interactions. The conceptual oscillator model should consider the variations of both the eastern and western Pacific anomaly patterns. An E1 Niйo atmospheric physics model is proposed using a method for the variational iteration theory. Using the variational iteration method, the approximate expansions of the solution of corresponding problem are constructed. That is, firstly, introducing a set of functional and accounting their variationals, the Lagrange multiplicators are counted, and then the variational iteration is defined, finally, the approximate solution is obtained. From approximate expansions of the solution, the zonal sea surface temperature anomaly in the equatorial eastern Pacific and the thermocline depth anomaly of the sea-air oscillation for E1 Niйo atmospheric physics model can be analyzed. E1 Niйo is a very complicated natural phenomenon. Hence basic models need to be reduced for the sea-air oscillator and are solved. The variational iteration is a simple and valid approximate method.

Existence and iteration of monotone positive solutions for a third-order two-point boundary value problem

The existence of nondecreasing positive solutions for the nonlinear third-order twopoint boundary value problem u′″（t） ＋ q（t）f（t,u（t）,u′（t）） = 0, 0 〈 t 〈 1, u（0） = u″（0） = u′（1） = 0 is studied. The iterative schemes for approximating the solutions are obtained by applying a monotone iterative method.

Three-dimensional gravity inversion based on sparse recovery iteration using approximate zero norm

This research proposes a novel three-dimensional gravity inversion based on sparse recovery in compress sensing. Zero norm is selected as the objective function, which is then iteratively solved by the approximate zero norm solution. The inversion approach mainly employs forward modeling; a depth weight function is introduced into the objective function of the zero norms. Sparse inversion results are obtained by the corresponding optimal mathematical method. To achieve the practical geophysical and geological significance of the results, penalty function is applied to constrain the density values. Results obtained by proposed provide clear boundary depth and density contrast distribution information. The method's accuracy, validity, and reliability are verified by comparing its results with those of synthetic models. To further explain its reliability, a practical gravity data is obtained for a region in Texas, USA is applied. Inversion results for this region are compared with those of previous studies, including a research of logging data in the same area. The depth of salt dome obtained by the inversion method is 4.2 km, which is in good agreement with the 4.4 km value from the logging data. From this, the practicality of the inversion method is also validated.

Generalized Variational Iteration Solution of Soliton for Disturbed KdV Equation

The corresponding solution for a class of disturbed KdV equation is considered using the analytic method. From the generalized variational iteration theory, the problem of solving soliton for the corresponding equation translates into the problem of variational iteration. And then the approximate solution of the soliton for the equation is obtained.

Estimating regional coal resource efficiency in China using three-stage DEA and bootstrap DEA models

Based on three-stage data envelopment analysis(DEA) model estimates of resource utilization efficiency of coal,we selected 29 provinces from China's 2012 input-output data and used the bootstrap DEA model to correct the bias.The results show that the mean overall technical efficiency,pure technical efficiency,and scale efficiency was 0.833,0.998,and 0.711 in 2012,respectively.Moreover,the comprehensive technical efficiency score indicates that the scale is invalid.Area utilization efficiencies for the eastern,central,and western regions were 0.917,0.731,and 0.629,respectively.The results prove that there are significant differences in the distribution of coal resources utilization efficiency across regions.

The Equivalence between the Mann and Ishikawa Iterations for Generalized Contraction Mappings in a Cone

In this paper, equivalence between the Mann and Ishikawa iterations for a generalized contraction mapping in cone subset of a real Banach space is discussed.

ON THE MONOTONE CONVERGENCE OF THE PROJECTED ITERATION METHODS FOR LINEAR COMPLEMENTARITY PROBLEMS

Under suitable conditions,the monotone convergence about the projected iteration method for solving linear complementarity problem is proved and the influence of the involved parameter matrix on the convergence rate of this method is investigated.

Low-Complexity Signal Detection Based on Relaxation Iteration Method in Massive MIMO Systems

Minimum mean square error(MMSE) detection algorithm can achieve nearly optimal performance when the number of antennas at the base station(BS) is large enough compared to the number of users. But the traditional MMSE involves complicated matrix inversion. In this paper, we propose a modified MMSE algorithm which exploits the channel characteristics occurring in massive multiple-input multipleoutput(MIMO) channels and the relaxation iteration(RI) method to avoid the matrix inversion. A proper initial solution is given to accelerate the convergence speed. In addition, we point out that the channel estimation scheme used in channel hardening-exploiting message passing(CHEMP) receiver is very appropriate for our proposed detection algorithm. Simulation results verify that the proposed algorithm can achieve very close performance of the traditional MMSE algorithm with a small number of iterations.

Higher Order Aitken Extrapolation with Application to Converging and Diverging Gauss-Seidel Iterations

In this paper, Aitken’s extrapolation normally applied to convergent fixed point iteration is extended to extrapolate the solution of a divergent iteration. In addition, higher order Aitken extrapolation is introduced that enables successive decomposition of high Eigen values of the iteration matrix to enable convergence. While extrapolation of a convergent fixed point iteration using a geometric series sum is a known form of Aitken acceleration, it is shown that in this paper, the same formula can be used to estimate the solution of sets of linear equations from diverging Gauss-Seidel iterations. In both convergent and divergent iterations, the ratios of differences among the consecutive values of iteration eventually form a convergent (divergent) series with a factor equal to the largest Eigen value of the iteration matrix. Higher order Aitken extrapolation is shown to eliminate the influence of dominant Eigen values of the iteration matrix in successive order until the iteration is determined by the lowest possible Eigen values. For the convergent part of the Gauss-Seidel iteration, further acceleration is made possible by coupling of the extrapolation technique with the successive over relaxation (SOR) method. Application examples from both convergent and divergent iterations have been provided. Coupling of the extrapolation with the SOR technique is also illustrated for a steady state two dimensional heat flow problem which was solved using MATLAB programming.

Investigation on seismic response of a three-stage soil slope supported by anchor frame structure

Based on a typical prototype of a soil slope in engineering practice, a numerical model of a three-stage soil slope supported by the anchor frame structure was established by means of FLAC3D code. The dynamic responses of three-stage soil slope and frame structure were studied by performing a series of bidirectional Wenchuan motions in terms of the failure mode of three-stage structure, the acceleration of soil slope, the displacement of frame structure, and the anchor stress of frame structure. The response accelerations in both horizontal and vertical directions are the most largely amplified at the slope top of each stage subjected to different shaking cases. The platforms among the stages reduce the amplification effect of response acceleration. The residual displacement of frame structure increases significantly as the intensity of shaking case increases. The frame structure at each stage presents a combined displacement mode consisting of a translation and a rotation around the vertex. The anchor stress of frame structure is mainly increased by the first intense pulse of Wenchuan seismic wave, and it is sensitive to the intensity of shaking case. The anchor stress of frame structure at the first stage is the most considerably enlarged by earthquake loading.