Optimization of Random Feature Method in the High-Precision Regime

Machine learning has been widely used for solving partial differential equations(PDEs)in recent years,among which the random feature method(RFM)exhibits spectral accuracy and can compete with traditional solvers in terms of both accuracy and efficiency.Potentially,the optimization problem in the RFM is more difficult to solve than those that arise in traditional methods.Unlike the broader machine-learning research,which frequently targets tasks within the low-precision regime,our study focuses on the high-precision regime crucial for solving PDEs.In this work,we study this problem from the following aspects:(i)we analyze the coeffcient matrix that arises in the RFM by studying the distribution of singular values;(ii)we investigate whether the continuous training causes the overfitting issue;(ii)we test direct and iterative methods as well as randomized methods for solving the optimization problem.Based on these results,we find that direct methods are superior to other methods if memory is not an issue,while iterative methods typically have low accuracy and can be improved by preconditioning to some extent.

Low-complexity signal detection for massive MIMO systems via trace iterative method

Linear minimum mean square error(MMSE)detection has been shown to achieve near-optimal performance for massive multiple-input multiple-output(MIMO)systems but inevitably involves complicated matrix inversion,which entails high complexity.To avoid the exact matrix inversion,a considerable number of implicit and explicit approximate matrix inversion based detection methods is proposed.By combining the advantages of both the explicit and the implicit matrix inversion,this paper introduces a new low-complexity signal detection algorithm.Firstly,the relationship between implicit and explicit techniques is analyzed.Then,an enhanced Newton iteration method is introduced to realize an approximate MMSE detection for massive MIMO uplink systems.The proposed improved Newton iteration significantly reduces the complexity of conventional Newton iteration.However,its complexity is still high for higher iterations.Thus,it is applied only for first two iterations.For subsequent iterations,we propose a novel trace iterative method(TIM)based low-complexity algorithm,which has significantly lower complexity than higher Newton iterations.Convergence guarantees of the proposed detector are also provided.Numerical simulations verify that the proposed detector exhibits significant performance enhancement over recently reported iterative detectors and achieves close-to-MMSE performance while retaining the low-complexity advantage for systems with hundreds of antennas.

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.

The Alternating Group Explicit Iterative Method for the Regularized Long-Wave Equation

An Alternating Group Explicit (AGE) iterative method with intrinsic parallelism is constructed based on an implicit scheme for the Regularized Long-Wave (RLW) equation. The method can be used for the iteration solution of a general tridiagonal system of equations with diagonal dominance. It is not only easy to implement, but also can directly carry out parallel computation. Convergence results are obtained by analysing the linear system. Numerical experiments show that the theory is accurate and the scheme is valid and reliable.

Variational Iteration Method for Solving Time Fractional Burgers Equation Using Maple

The Time Fractional Burger equation was solved in this study using the Mabel software and the Variational Iteration approach. where a number of instances of the Time Fractional Burger Equation were handled using this technique. Tables and images were used to present the collected numerical results. The difference between the exact and numerical solutions demonstrates the effectiveness of the Mabel program’s solution, as well as the accuracy and closeness of the results this method produced. It also demonstrates the Mabel program’s ability to quickly and effectively produce the numerical solution.

Nonlinear Algebraic Equations Solved by an Optimal Splitting-Linearizing Iterative Method

How to accelerate the convergence speed and avoid computing the inversion of a Jacobian matrix is important in the solution of nonlinear algebraic equations(NAEs).This paper develops an approach with a splitting-linearizing technique based on the nonlinear term to reduce the effect of the nonlinear terms.We decompose the nonlinear terms in the NAEs through a splitting parameter and then linearize the NAEs around the values at the previous step to a linear system.Through the maximal orthogonal projection concept,to minimize a merit function within a selected interval of splitting parameters,the optimal parameters can be quickly determined.In each step,a linear system is solved by the Gaussian elimination method,and the whole iteration procedure is convergent very fast.Several numerical tests show the high performance of the optimal split-linearization iterative method(OSLIM).

Ostrowski’s Method for Solving Nonlinear Equations and Systems

The dynamic characteristics and the efficiency of the Ostrowski’s method allow it to be crowned as an excellent tool for solving nonlinear problems.This article shows different versions of the classic method that allow it to be applied to a wide range of engineering problems.Among them stands out the derivative-free definition applying divided differences,the introduction of memory and its extension to the resolution of nonlinear systems of equations.All of these versions are compared in a numerical simulations section where the results obtained are compared with other classic methods.

Iterative Dichotomiser Posteriori Method Based Service Attack Detection in Cloud Computing

Cloud computing(CC)is an advanced technology that provides access to predictive resources and data sharing.The cloud environment represents the right type regarding cloud usage model ownership,size,and rights to access.It introduces the scope and nature of cloud computing.In recent times,all processes are fed into the system for which consumer data and cache size are required.One of the most security issues in the cloud environment is Distributed Denial of Ser-vice(DDoS)attacks,responsible for cloud server overloading.This proposed sys-tem ID3(Iterative Dichotomiser 3)Maximum Multifactor Dimensionality Posteriori Method(ID3-MMDP)is used to overcome the drawback and a rela-tively simple way to execute and for the detection of(DDoS)attack.First,the pro-posed ID3-MMDP method calls for the resources of the cloud platform and then implements the attack detection technology based on information entropy to detect DDoS attacks.Since because the entropy value can show the discrete or aggregated characteristics of the current data set,it can be used for the detection of abnormal dataflow,User-uploaded data,ID3-MMDP system checks and read risk measurement and processing,bug ratingfile size changes,orfile name changes and changes in the format design of the data size entropy value.Unique properties can be used whenever the program approaches any data error to detect abnormal data services.Finally,the experiment also verifies the DDoS attack detection capability algorithm.

Multi-scale seismic full waveform inversion in the frequency-domain with a multi-grid method

Although full waveform inversion in the frequency domain can overcome the local minima problem in the time direction, such problem still exists in the space direction because of the media subsurface complexity. Based on the optimal steep descent methods, we present an algorithm which combines the preconditioned bi-conjugated gradient stable method and the multi-grid method to compute the wave propagation and the gradient space. The multiple scale prosperity of the waveform inversion and the multi-grid method can overcome the inverse problems local minima defect and accelerate convergence. The local inhomogeneous three-hole model simulated results and the Marmousi model certify the algorithm effectiveness.

Two-level Bregmanized method for image interpolation with graph regularized sparse coding

A two-level Bregmanized method with graph regularized sparse coding （TBGSC） is presented for image interpolation. The outer-level Bregman iterative procedure enforces the observation data constraints, while the inner-level Bregmanized method devotes to dictionary updating and sparse represention of small overlapping image patches. The introduced constraint of graph regularized sparse coding can capture local image features effectively, and consequently enables accurate reconstruction from highly undersampled partial data. Furthermore, modified sparse coding and simple dictionary updating applied in the inner minimization make the proposed algorithm converge within a relatively small number of iterations. Experimental results demonstrate that the proposed algorithm can effectively reconstruct images and it outperforms the current state-of-the-art approaches in terms of visual comparisons and quantitative measures.

Combined method for fast 3-D finite element modeling of nondestructive testing signal

A combined method for the fast 3-D finite element modeling of defect responses in nondestructive testing of electromagnetics is presented. The method consists of three numerical techniques: zoom-in technique, difference field technique and iterative solution technique. Utilizing the zoom-in technique, the computational zone focuses on a relatively small domain around the defect. Employing the difference field technique, the axisymmetrical field solution corresponding to the case with no defect can be used to simplify the mesh generation and obtain the modeling results quickly. Using the iterative solution technique, the matrix equation system in the 3-D finite element modeling of nondestructive probe signals can easily be solved. The sample calculation shows that the presented method is highly effective and can consequently save significant computer resources.

Existence and numerical approximation of a solution to frictional contact problem for electro-elastic materials

In this paper,a frictional contact problem between an electro-elastic body and an electrically conductive foundation is studied.The contact is modeled by normal compliance with finite penetration and a version of Coulomb’s law of dry friction in which the coefficient of friction depends on the slip.In addition,the effects of the electrical conductivity of the foundation are taken into account.This model leads to a coupled system of the quasi-variational inequality of the elliptic type for the displacement and the nonlinear variational equation for the electric potential.The existence of a weak solution is proved by using an abstract result for elliptic variational inequalities and a fixed point argument.Then,a finite element approximation of the problem is presented.Under some regularity conditions,an optimal order error estimate of the approximate solution is derived.Finally,a successive iteration technique is used to solve the problem numerically and a convergence result is established.

Co-salient object detection with iterative purification and predictive optimization

Background Co-salient object detection(Co-SOD)aims to identify and segment commonly salient objects in a set of related images.However,most current Co-SOD methods encounter issues with the inclusion of irrelevant information in the co-representation.These issues hamper their ability to locate co-salient objects and significantly restrict the accuracy of detection.Methods To address this issue,this study introduces a novel Co-SOD method with iterative purification and predictive optimization(IPPO)comprising a common salient purification module(CSPM),predictive optimizing module(POM),and diminishing mixed enhancement block(DMEB).Results These components are designed to explore noise-free joint representations,assist the model in enhancing the quality of the final prediction results,and significantly improve the performance of the Co-SOD algorithm.Furthermore,through a comprehensive evaluation of IPPO and state-of-the-art algorithms focusing on the roles of CSPM,POM,and DMEB,our experiments confirmed that these components are pivotal in enhancing the performance of the model,substantiating the significant advancements of our method over existing benchmarks.Experiments on several challenging benchmark co-saliency datasets demonstrate that the proposed IPPO achieves state-of-the-art performance.

ITERATED DESIGN OF NON EQUIRIPPLE LOW PASS FILTER

Non equiripple approximation of filter characteristics can be realized either odd order or even order in the symmetric load case.This paper presents a method of synthesizing non equiripple low pass filter based on iteration analysis,in which the rational fraction formed of Chebyshev polynomial is used as the filter characteristic function.This method is convenient for computer programming,because the attenuation zeros and poles of the filter can be determined easily and the synthesis procedure is simple,too.The given examples show that the method is of a practical value in filter design.

Static and Dynamic Analysis of Mooring Lines by Nonlinear Finite Element Method

This study has focused on developing numerical procedures for the static and dynamic nonlinear analysis of mooring lines. A geometrically nonlinear finite element method using isoparametric cable element with two nodes is briefly presented on the basis of the total Lagrangian formulation. The static and dynamic equilibrium equations of mooring lines are established. An incremental-iterative method is used to determine the initial static equilibrium state of cable systems under the action of self weights, buoyancy and current. Also the Newmark method is used for dynamic nonlinear analysis of ocean cables. Numerical examples are presented to validate the present numerical method, and examine the effect of various parameters.

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.

New Optimal Newton-Householder Methods for Solving Nonlinear Equations and Their Dynamics

The classical iterative methods for finding roots of nonlinear equations,like the Newton method,Halley method,and Chebyshev method,have been modified previously to achieve optimal convergence order.However,the Householder method has so far not been modified to become optimal.In this study,we shall develop two new optimal Newton-Householder methods without memory.The key idea in the development of the new methods is the avoidance of the need to evaluate the second derivative.The methods fulfill the Kung-Traub conjecture by achieving optimal convergence order four with three functional evaluations and order eight with four functional evaluations.The efficiency indices of the methods show that methods perform better than the classical Householder’s method.With the aid of convergence analysis and numerical analysis,the efficiency of the schemes formulated in this paper has been demonstrated.The dynamical analysis exhibits the stability of the schemes in solving nonlinear equations.Some comparisons with other optimal methods have been conducted to verify the effectiveness,convergence speed,and capability of the suggested methods.

3D elastic waveform modeling with an optimized equivalent staggered-grid finite-difference method

Equivalent staggered-grid(ESG) as a new family of schemes has been utilized in seismic modeling,imaging,and inversion.Traditionally,the Taylor series expansion is often applied to calculate finite-difference(FD) coefficients on spatial derivatives,but the simulation results suffer serious numerical dispersion on a large frequency zone.We develop an optimized equivalent staggered-grid(OESG) FD method that can simultaneously suppress temporal and spatial dispersion for solving the second-order system of the 3 D elastic wave equation.On the one hand,we consider the coupling relations between wave speeds and spatial derivatives in the elastic wave equation and give three sets of FD coefficients with respect to the P-wave,S-wave,and converted-wave(C-wave) terms.On the other hand,a novel plane wave solution for the 3 D elastic wave equation is derived from the matrix decomposition method to construct the time-space dispersion relations.FD coefficients of the OESG method can be acquired by solving the new dispersion equations based on the Newton iteration method.Finally,we construct a new objective function to analyze P-wave,S-wave,and C-wave dispersion concerning frequencies.The dispersion analyses show that the presented method produces less modeling errors than the traditional ESG method.The synthetic examples demonstrate the effectiveness and superiority of the presented method.

A Fourth-order Covergence Newton-type Method

A fourth-order convergence method of solving roots for nonlinear equation, which is a variant of Newton＇s method given. Its convergence properties is proved. It is at least fourth-order convergence near simple roots and one order convergence near multiple roots. In the end, numerical tests are given and compared with other known Newton and Newton-type methods. The results show that the proposed method has some more advantages than others. It enriches the methods to find the roots of non-linear equations and it is important in both theory and application.

A Family of Fifth-order Iterative Methods for Solving Nonlinear Equations

In this paper, we present and analyze a family of fifth-order iterative methods free from second derivative for solving nonlinear equations. It is established that the family of iterative methods has convergence order five. Numerical examples show that the new methods are comparable with the well known existing methods and give better results in many aspects.