An efficient parallel algorithm of variational nodal method for heterogeneous neutron transport problems

The heterogeneous variational nodal method(HVNM)has emerged as a potential approach for solving high-fidelity neutron transport problems.However,achieving accurate results with HVNM in large-scale problems using high-fidelity models has been challenging due to the prohibitive computational costs.This paper presents an efficient parallel algorithm tailored for HVNM based on the Message Passing Interface standard.The algorithm evenly distributes the response matrix sets among processors during the matrix formation process,thus enabling independent construction without communication.Once the formation tasks are completed,a collective operation merges and shares the matrix sets among the processors.For the solution process,the problem domain is decomposed into subdomains assigned to specific processors,and the red-black Gauss-Seidel iteration is employed within each subdomain to solve the response matrix equation.Point-to-point communication is conducted between adjacent subdomains to exchange data along the boundaries.The accuracy and efficiency of the parallel algorithm are verified using the KAIST and JRR-3 test cases.Numerical results obtained with multiple processors agree well with those obtained from Monte Carlo calculations.The parallelization of HVNM results in eigenvalue errors of 31 pcm/-90 pcm and fission rate RMS errors of 1.22%/0.66%,respectively,for the 3D KAIST problem and the 3D JRR-3 problem.In addition,the parallel algorithm significantly reduces computation time,with an efficiency of 68.51% using 36 processors in the KAIST problem and 77.14% using 144 processors in the JRR-3 problem.

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.

Variational regularization method of solving the Cauchy problem for Laplace's equation: Innovation of the Grad–Shafranov(GS) reconstruction

The simplified linear model of Grad-Shafranov （GS） reconstruction can be reformulated into an inverse boundary value problem of Laplace＇s equation. Therefore, in this paper we focus on the method of solving the inverse boundary value problem of Laplace＇s equation. In the first place, the variational regularization method is used to deal with the ill- posedness of the Cauchy problem for Laplace＇s equation. Then, the ＇L-Curve＇ principle is suggested to be adopted in choosing the optimal regularization parameter. Finally, a numerical experiment is implemented with a section of Neumann and Dirichlet boundary conditions with observation errors. The results well converge to the exact solution of the problem, which proves the efficiency and robustness of the proposed method. When the order of observation error δ is 10-1, the order of the approximate result error can reach 10-3.

A NEW ITERATIVE METHOD FOR FINDING COMMON SOLUTIONS OF GENERALIZED EQUILIBRIUM PROBLEM,FIXED POINT PROBLEM OF INFINITE k-STRICT PSEUDO-CONTRACTIVE MAPPINGS,AND QUASI-VARIATIONAL INCLUSION PROBLEM

In this article, we introduce a hybrid iterative scheme for finding a common element of the set of solutions for a generalized equilibrium problems, the set of common fixed point for a family of infinite k-strict pseudo-contractive mappings, and the set of solutions of the variational inclusion problem with multi-valued maximal monotone mappings and inverse-strongly monotone mappings in Hilbert space. Under suitable conditions, some strong convergence theorems are proved. Our results extends the recent results in G.L.Acedo and H.K.Xu [2], Zhang, Lee and Chan [8], Wakahashi and Toyoda [9], Takahashi and Takahashi [I0] and S. S. Chang, H. W. Joseph Lee and C. K. Chan [II], S.Takahashi and W.Takahashi [12]. Moreover, the method of proof adopted in this article is different from those of [4] and [12].

SEMI-INVERSE METHOD AND GENERALIZED VARIATIONAL PRINCIPLES WITH MULTI-VARIABLES IN ELASTICITY

Semi-inverse method, which is an integration and an extension of Hu's try-and-error method, Chien's veighted residual method and Liu's systematic method, is proposed to establish generalized variational principles with multi-variables without arty variational crisis phenomenon. The method is to construct an energy trial-functional with an unknown function F, which can be readily identified by making the trial-functional stationary and using known constraint equations. As a result generalized variational principles with two kinds of independent variables (such as well-known Hellinger-Reissner variational principle and Hu-Washizu principle) and generalized variational principles with three kinds of independent variables (such as Chien's generalized variational principles) in elasticity have been deduced without using Lagrange multiplier method. By semi-inverse method, the author has also proved that Hu-Washizu principle is actually a variational principle with only two kinds of independent variables, stress-strain relations are still its constraints.

A new two-step variational model for multiplicative noise removal with applications to texture images

Multiplicative noise removal problems have attracted much attention in recent years.Unlike additive noise,multiplicative noise destroys almost all information of the original image,especially for texture images.Motivated by the TV-Stokes model,we propose a new two-step variational model to denoise the texture images corrupted by multiplicative noise with a good geometry explanation in this paper.In the first step,we convert the multiplicative denoising problem into an additive one by the logarithm transform and propagate the isophote directions in the tangential field smoothing.Once the isophote directions are constructed,an image is restored to fit the constructed directions in the second step.The existence and uniqueness of the solution to the variational problems are proved.In these two steps,we use the gradient descent method and construct finite difference schemes to solve the problems.Especially,the augmented Lagrangian method and the fast Fourier transform are adopted to accelerate the calculation.Experimental results show that the proposed model can remove the multiplicative noise efficiently and protect the texture well.

Modified Lagrange Multiplier Method and Generalized Variational Principle in Fluid Mechanics

The Lagrange multiplier method plays an important role in establishing generalized variational principles notonly in tluid mechallics. but also in elasticity. Sometimes, however, one may come across variational crisis(somemultipliers vanish identically). failing to achieve his aim. The crisis is caused by the fact that the Inultipliers are treatedas independent variables in the process of variatioll. but after identification they become functions of the originalindependent variables. To overcome it, a Inodified Lagrange multiplier method or semi-inverse method has beenproposed to deduce generalized varistional principles. Some e-camples are given to illustrate its convenience andeffectiveness of the novel method.

AN ITERATIVE METHOD FOR THE DISCRETE PROBLEMS OF A CLASS OF ELLIPTICAL VARIATIONAL INEQUALITIES

Based on the nonlinear characiers of the discrete problems of some ellipticalvariational inequalities, this paper presents a numerical iterative method, the schemesof which are pithy and converge rapidly The new method possesses a high efficiency. insolving such applied engineering problems as obstacle problems and .free boundary.problems arising in fluid lubrications.

Doubly Periodic Wave Solutions of Jaulent-Miodek Equations Using Variational Iteration Method Combined with Jacobian-function Method

One of the advantages of the variational iteration method is the free choice of initial guess. In this paper we use the basic idea of the Jacobian-function method to construct a generalized trial function with some unknown parameters. The Jaulent-Miodek equations are used to illustrate effectiveness and convenience of this method, some new explicit exact travelling wave solutions have been obtained, which include bell-type soliton solution, kink-type soliton solutions, solitary wave solutions, and doubly periodic wave solutions.

First-Arrival Picking Method for Active Source Data with Ocean Bottom Seismometers Based on Spatial Waveform Variation Characteristics

The precision and reliability of first-arrival picking are crucial for determining the accuracy of geological structure inversion using active source ocean bottom seismometer(OBS)refraction data.Traditional methods for first-arrival picking based on sample points are characterized by theoretical errors,especially in low-sampling-frequency OBS data because the travel time of seismic waves is not an integer multiple of the sampling interval.In this paper,a first-arrival picking method that utilizes the spatial waveform variation characteristics of active source OBS data is presented.First,the distribution law of theoretical error is examined;adjacent traces exhibit variation characteristics in their waveforms.Second,a label cross-correlation superposition method for extracting highfrequency signals is presented to enhance the first-arrival picking precision.Results from synthetic and field data verify that the proposed approach is robust,successfully overcomes the limitations of low sampling frequency,and achieves precise outcomes that are comparable with those of high-sampling-frequency data.

A RELAXED INERTIAL FACTOR OF THE MODIFIED SUBGRADIENT EXTRAGRADIENT METHOD FOR SOLVING PSEUDO MONOTONE VARIATIONAL INEQUALITIES IN HILBERT SPACES

In this paper,we investigate pseudomonotone and Lipschitz continuous variational inequalities in real Hilbert spaces.For solving this problem,we propose a new method that combines the advantages of the subgradient extragradient method and the projection contraction method.Some very recent papers have considered different inertial algorithms which allowed the inertial factor is chosen in[0;1].The purpose of this work is to continue working in this direction,we propose another inertial subgradient extragradient method that the inertial factor can be chosen in a special case to be 1.Under suitable mild conditions,we establish the weak convergence of the proposed algorithm.Moreover,linear convergence is obtained under strong pseudomonotonicity and Lipschitz continuity assumptions.Finally,some numerical illustrations are given to confirm the theoretical analysis.

GENERALIZED VARIATIONAL DATA ASSIMILATION METHOD AND NUMERICAL EXPERIMENT FOR NON-DIFFERENTIAL SYSTEM

The generalized variational data assimilation for non-differential dynamical systems is studied.There is no tangent linear model for non-differential systems and thus the general adjoint model can not be derived in the traditional way.The weak form of the original system was introduced, and then the generalized adjoint model was derived. The generalized variational data assimilation methods were developed for non-differential low dimensional system and non-differential high dimensional system with global and local observations. Furthermore, ideas in inverse problems are introduced to 4DVAR (Four-dimensional variational) of non-differential partial differential system with local observations.

A Unification of the Concepts of the Variational Iteration, Adomian Decomposition and Picard Iteration Methods;and a Local Variational Iteration Method

This paper compares the variational iteration method(VIM),the Adomian decomposition method(ADM)and the Picard iteration method(PIM)for solving a system of first o rder n onlinear o rdinary d ifferential e quations(ODEs).A unification of the concepts underlying these three methods is attempted by considering a very general iterative algorithm for VIM.It is found that all the three methods can be regarded as special cases of using a very general matrix of Lagrange multipliers in the iterative algorithm of VIM.The global variational iteration method is briefly reviewed,and further recast into a Local VIM,which is much more convenient and capable of predicting long term complex dynamic responses of nonlinear systems even if they are chaotic.

A Smoothing Newton Method for the Box Constrained Variational Inequality Problems

The box constrained variational inequality problem can be reformulated as a nonsmooth equation by using median operator.In this paper,we present a smoothing Newton method for solving the box constrained variational inequality problem based on a new smoothing approximation function.The proposed algorithm is proved to be well defined and convergent globally under weaker conditions.

Variational Homotopy Perturbation Method for Solving Benjamin-Bona-Mahony Equation

In this article, the application of variational homotopy perturbation method is applied to solve Benjamin-Bona-Mahony equation. Then, we obtain the numerical solution of BBM equation using the initial condition. Comparison with Adomian’s decomposition method, homotopy perturbation method, and with the exact solution shows that VHPM is more effective and accurate than ADM and HPM, and is reliable and manageable for this type of equation.

Weak and Strong Convergence of Self Adaptive Inertial Subgradient Extragradient Algorithms for Solving Variational Inequality Problems

Many solutions of variational inequalities have been proposed,among which the subgradient extragradient method has obvious advantages.Two different algorithms are given for solving variational inequality problem in this paper.The problem we study is defined in a real Hilbert space and has L-Lipschitz and pseudomonotone condition.Two new algorithms adopt inertial technology and non-monotonic step size rule,and their convergence can still be proved when the value of L is not given in advance.Finally,some numerical results are designed to demonstrate the computational efficiency of our two new algorithms.

STRONG CONVERGENCE OF AN INERTIAL EXTRAGRADIENT METHOD WITH AN ADAPTIVE NONDECREASING STEP SIZE FOR SOLVING VARIATIONAL INEQUALITIES

In this work,we investigate a classical pseudomonotone and Lipschitz continuous variational inequality in the setting of Hilbert space,and present a projection-type approximation method for solving this problem.Our method requires only to compute one projection onto the feasible set per iteration and without any linesearch procedure or additional projections as well as does not need to the prior knowledge of the Lipschitz constant and the sequentially weakly continuity of the variational inequality mapping.A strong convergence is established for the proposed method to a solution of a variational inequality problem under certain mild assumptions.Finally,we give some numerical experiments illustrating the performance of the proposed method for variational inequality problems.

Three-Dimensional Wind Field Retrieved from Dual-Doppler Radar Based on a Variational Method:Refinement of Vertical Velocity Estimates

In this paper,a scheme of dual-Doppler radar wind analysis based on a three-dimensional variational method is proposed and performed in two steps.First,the horizontal wind field is simultaneously recovered through minimizing a cost function defined as a radial observation term with the standard conjugate gradient method,avoiding a weighting parameter specification step.Compared with conventional dual-Doppler wind synthesis approaches,this variational method minimizes errors caused by interpolation from radar observation to analysis grid in the iterative solution process,which is one of the main sources of errors.Then,through the accelerated Liebmann method,the vertical velocity is further reestimated as an extra step by solving the Poisson equation with impermeable conditions imposed at the ground and near the tropopause.The Poisson equation defined by the second derivative of the vertical velocity is derived from the mass continuity equation.Compared with the method proposed by O’Brien,this method is less sensitive to the uncertainty of the boundary conditions and has better stability and reliability.Furthermore,the method proposed in this paper is applied to Doppler radar observation of a squall line process.It is shown that the retrieved vertical wind profile agrees well with the vertical profile obtained with the velocity–azimuth display(VAD)method,and the retrieved radial velocity as well as the analyzed positive and negative velocity centers and horizontal wind shear of the squall line are in accord with radar observations.There is a good correspondence between the divergence field of the derived wind field and the vertical velocity.And,the horizontal and vertical circulations within and around the squall line,as well as strong updrafts,the associated downdrafts,and associated rear inflow of the bow echo,are analyzed well.It is worth mentioning that the variational method in this paper can be applied to simultaneously synthesize the three-dimensional wind field from multiple-Doppler radar observations.

New Improved Variational Homotopy Perturbation Method for Bratu-Type Problems

This research paper deals with the boundary and initial value problems for the Bratu-type model by using the New Improved Variational Homotopy Perturbation Method. The New Method does not require discritization, linearization or any restrictive assumption of any form in providing analytical or approximate solutions to linear and nonlinear equation without the integral related with nonlinear term. Theses virtues make it to be reliable and its efficiency is demonstrated with numerical examples.