Successive Overrelaxation Research Articles

Generalized Second-Order Value Iteration in Markov Decision Processes

Value iteration is a fixed point iteration technique utilized to obtain the optimal value function and policy in a discounted reward Markov decision process (MDP). Here, a contraction operator is constructed and applied repeatedly to arrive at the optimal solution. Value iteration is a first-order method and, therefore, it may take a large number of iterations to converge to the optimal solution. Successive relaxation is a popular technique that can be applied to solve a fixed point equation. It has been shown in the literature that under a special structure of the MDP, successive overrelaxation technique computes the optimal value function faster than standard value iteration. In this article, we propose a second-order value iteration procedure that is obtained by applying the Newton–Raphson method to the successive relaxation value iteration scheme. We prove the global convergence of our algorithm to the optimal solution asymptotically and show the second-order convergence. Through experiments, we demonstrate the effectiveness of our proposed approach.

Numerical Calculation of Fluid Film Force on Journal Bearings Based on a Biconjugate Gradient-Stabilized Algorithm

Abstract The aim of this work is to present a technical numerical method to improve the efficiency of solving the Reynolds equation for hydrodynamic bearings. Hydrodynamic bearings are largely employed in the shafting of high-speed rotating machinery to provide adequate support. The distribution of hydrodynamic pressure inside the bearing can be obtained by solving the Reynolds equation. Improving the efficiency of solving the Reynolds equation plays an essential role in the design and analysis of journal bearings. A numerical model has been developed to support the analysis and future design of hydrodynamic bearings. The primary objective of the model is to improve the efficiency of solving the Reynold equation during the steady-state and dynamic analysis. The developed method effectively combines the bi-conjugate gradient stabilized (Bi-CGSTAB) algorithm with the Reynolds boundary conditions, resulting in an effective methodology to characterize the pressure distribution within the lubricant for hydrodynamic bearings. A c++ program was implemented, and the difference between the BI-CGSTAB method and the successive over relaxation (SOR) method was evaluated against accuracy, convergence, and computational consumption. The Bi-CGSTAB algorithm has shown promising application in steady-state and dynamic analyses of hydrodynamic bearings. Validation of the results has been made with reference and analytical solutions.

Modeling of Three-Phase Written-Pole Motor and Performance Predetermination using Numerical Methods

Modeling has attracted increasing attention from researchers. A physical system transformation to a mathematical base is important to analyse the behaviour of the system. The solution of various engineering problems depends on appropriate mathematical models. Differential equations are the basic tool for the mathematical modeling of electric machines. A model that describes the machine performance is used to analyse the dynamic behaviour of a 3-phase written-pole motor.
 Mathematical modeling of a 3-phase written-pole motor will help us to do the numerical analysis. It provides insight and useful guidance for originating the applications. A method for studying the behaviour of a 3-phase written-pole motor under steady state condition is described. The numerical tools such as Gauss-Seidel method and Successive Over-Relaxation method are used. This study forms the foundation for further research and development.

Hybrid Precoding Algorithm for Millimeter-Wave Massive MIMO-NOMA Systems

In this paper, the performance of the millimeter-wave (mmWave) massive multiple-input multiple-output (mMIMO) non-orthogonal multiple access (NOMA) systems is investigated under multiple user scenarios. The performance of the system has been analyzed in terms of spectral efficiency (SE), energy efficiency (EE), and computational complexity. In the case of the mMIMO system, the linear precoder with matrix inversion becomes less efficient due to its high computational complexity. Therefore, the design of a low-complex hybrid precoder (HP) is the main aim of this paper. Here, the authors have proposed a symmetric successive over-relaxation (SSOR) complex regularized zero-forcing (CRZF) linear precoder. Through simulation, this paper demonstrates that the proposed SSOR-CRZF-HP performs better than the conventional linear precoder with reduced complexity.

Thermophoresis phenomenon in radiative flow about vertical movement of a rotating disk in porous region

This study examines the time-dependent, three-dimensional, laminar, incompressible, and inclined hydromagnetic flow of viscous fluid above a rotating disk. The disk is saturated in a porous space and moves vertically along axial direction. Heat transfer process is analyzed by the thermal radiation and convective boundary condition. Thermophoresis process is illustrated in the flow stream by mass equation. The similarity functions are exerted to transform the system of partial differential equations into a one independent variable expressions. The finite differences scheme is applied to the resultant dimensionalized system and then solved in iterative manner using Successive over Relaxation (SOR) method. The physical quantities are illustrated graphically in terms of velocity, thermal and concentration fields. Two-dimensional contour phenomenon and three-dimensional flow visualizations for some parameters are also sketched. The skin-friction co-efficient, frictional torque, Nusselt number and Sherwood number against selected parameters are discussed in tabular format.

Predetermination of Performance Parameters of 3-phase Induction Motor using Numerical Technique Tools

The Circle Diagram was used in past for the performance evaluation of 3-phase induction motor under different load conditions. But, now it is not preferred due to availability of programs for detailed analysis. The per-phase equivalent circuit is also not that suitable to estimate the dynamic performance. In this paper, a method is suggested to determine the performance using dq-model. The objective of this paper is to describe modelling in terms of dq variables. The analysis under steady state condition is done using numerical technique tools. The numerical tools such as Gauss-Seidel and Successive Relaxation methods are used and the results are compared with that obtained from the experiment conducted in the laboratory. The comparison of results confirms the validity as well as the accuracy of the proposed methods, as yet another approach to predetermine the performance.

A Direct Relaxation-Iteration Technique for the Troesch Problem

In the work reported herein, we propose a direct relaxation-iteration technique for the solution of a highly unstable parameter-sensitive nonlinear boundary value differential equation known as the Troesch problem. It arises in the confinement of plasma column by radiation pressure. The proposed technique is based on an SOR successive relaxation method and guarantees that convergence is achieved straightforwardly. Numerical experiments conducted to investigate the overall influence of sensitivity parameter on the solution profiles confirm the relative advantage of the proposed numerical technique over previous methods.

Deep solution operators for variational inequalities via proximal neural networks

Following Bauschke and Combettes (Convex analysis and monotone operator theory in Hilbert spaces, Springer, Cham, 2017), we introduce ProxNet, a collection of deep neural networks with ReLU activation which emulate numerical solution operators of variational inequalities (VIs). We analyze the expression rates of ProxNets in emulating solution operators for variational inequality problems posed on closed, convex cones in real, separable Hilbert spaces, covering the classical contact problems in mechanics, and early exercise problems as arise, e.g., in valuation of American-style contracts in Black–Scholes financial market models. In the finite-dimensional setting, the VIs reduce to matrix VIs in Euclidean space, and ProxNets emulate classical projected matrix iterations, such as projected Jacobi and projected SOR methods.

Relaxation parameters and composite refinement techniques

A composite refinement technique for two stationary iterative methods, one of them contains a relaxation parameter, is introduced. Four new techniques, Jacobi successive over relaxation (SOR) composite refinement (RJSOR), SOR Jacobi composite refinement (RSORJ), Gauss–Seidel (GS) SOR composite refinement (RGSSOR) and SOR with GS composite refinement (RSORGS) are compared with their classical forms. The efficient performance of the new forms is well established and confirmed through numerical example. The computational costs and the speed of convergence are considered. The decrease in the required number of iteration is established through the calculation of the spectral radius of the iteration matrices. It is illustrated that the convergence of Jacobi and Gauss–Seidel methods engage the divergence and extend the domain of convergence in the SOR method in the refinement technique. The calculations and graphs are performed by computer algebra system, Mathematica.

Numerical Modeling of Fracture Height Propagation in Multilayer Formations Considering the Plastic Zone and Induced Stress.

Hydraulic fracturing and acid fracturing are very effective stimulation technologies and are widely used in unconventional reservoir development. Fracture height, as an essential parameter to describe the geometric size of a fracture, is not only the input parameter of two-dimensional fracturing models but also the output parameter of three-dimensional fracturing models. Accurate prediction of fracture height growth can effectively avoid some risks. For example, petroleum reservoirs produce a large amount of formation water because wrong fracture height prediction leads to the connection between the oil or gas reservoir and the water layer. Although some fracture height prediction models were developed, few models considered the effects of the plastic zone, induced stress, and heterogeneous multilayer formation and its interaction. Therefore, considering the influence of many factors, an improved fracture-equilibrium-height model was developed in this study. The successive over-relaxation iteration method and the displacement discontinuity method were used to solve the model. We investigated the effects of the geological and engineering factors on fracture height growth by using the model, and some important conclusions were obtained. The higher the fracture height, the larger the plastic zone size, and the more obvious its influence on fracture height propagation. High overlying or underlying in situ stress and fracture toughness and low fluid density played a positive role in limiting the growth of the fracture height. Induced stress caused by fracture 1 could not only inhibit the height growth of fracture 2 but also promote its growth. The model established in this paper could be coupled to a fracturing simulator to provide a more reliable fracture height prediction.

Numerical computation of magnetized bioconvection nanofluid flow with temperature-dependent viscosity and Arrhenius kinetic

The current study focused on the impact of activation energy and magnetic field on immiscible steady nanofluid moving over an elastic stretched surface containing motile gyrotactic microorganisms. We use the Reynolds exponential model to develop the mathematical modeling since the viscosity of the fluid is temperature-dependent. Under the presence of uniformly disseminated nanoparticles, the base fluid is electrically conductive. The Buongiorno model, which incorporates thermophoretic forces and Brownian motion, has been used. Using the spectral local linearization approach (SLLM), the looming nonlinear, coupled differential equations are numerically addressed. Because it is based on a smooth Uni-variant linearization of non-linear functions, the SLLM method is simple to construct and use. The numerical performance of SLLM is even better since it grows a collection of equations that are solved sequentially by operating the findings from one equation into the next. The successive over-relaxation technique was used to accelerate and enhance the convergence of the SLLM system. The reliability of the SLLM will be verified through the use of well-known techniques and comparisons with other data We have illustrated and analyzed in detail the graphical behavior of all emergent variables for temperature, velocity, and concentration distributions, and also the Nusselt number, Sherwood number, skin friction, and density number of motile microorganisms. It is significant to claim that the spectral local linearization approach has been demonstrated to be extremely stable and adaptive for the solution of non-linear differential equations.

On the Application of Successive Over-relaxation Algorithmic and Block Numerical Iterative Solutions for the Stationary Distribution in Markov Chain

The evolution of a system is represented by transitions from one state to the next, and the system's physical or mathematical behavior can also be depicted by defining all of the numerous states it can be in and demonstrating how it moves between them. In this study, the iterative solution methods for the stationary distribution of Markov chains were investigated, which start with an initial estimate of the solution vector and then alter it in such a way that it gets closer and closer to the genuine solution with each step or iteration., and also involved matrices operations such as multiplication with one or more vectors, which leaves the transition matrices unchanged and saves time. Our goal is to use Successive Overrelaxation Algorithmic and Block Numerical Iterative Solution Methods to compute the solutions. With the help of some existing Markov chain laws, theorems, and formulas, the normalization principle and matric operations such as lower, upper, and diagonal matrices are used. The stationary distribution vector’s π^((k+1) )={■(π_1^((k+1) )&π_2^((k+1) )&■(π_3^((k+1) )&π_4^((k+1) )&π_5^((k+1) ) )),k=0,1,2,…,n} are obtained for the illustrative examples, taken the initial stationary solution to be π^((0) )=(■(0.2&0.2&■(0.2&0.2&0.2)))^T and it was observed that all subsequent iterations yield exactly the same result as π^((1) ), and this shows that, the block iterative method requires only a single iteration to obtain the solution to full machine precision.

Half-Sweep Refinement of SOR Iterative Method via Linear Rational Finite Difference Approximation for Second-Order Linear Fredholm Integro-Differential Equations

The numerical solutions of the second-order linear Fredholm integro-differential equations have been considered and discussed based on several discretization schemes. In this paper, the new schemes are developed derived on the hybrid of the three-point half-sweep linear rational finite difference (3HSLRFD) approaches with the half-sweep composite trapezoidal (HSCT) approach. The main advantage of the established schemes is that they discretize the differential terms and integral term of second-order linear Fredholm integro-differential equations into the algebraic equations and generate the corresponding linear system. Furthermore, the half-sweep (HS) concept is combined with the refinement of the successive over-relaxation (RSOR) iterative method to create the new half-sweep successive over-relaxation (HSRSOR) iterative method, which is implemented to get the numerical solution of a system of linear algebraic equations. Apart from that, the classical or full-sweep Gauss-Seidel (FSGS) and full-sweep successive over-relaxation iterative (FSSOR) methods are presented, which serve as the control method in this paper. In the end, we employed FSGS, FSRSOR and HSRSOR methods to obtain numerical solutions of three examples and make a detailed comparison from three aspects of the number of iterations, elapsed time and maximum absolute error. Numerical results demonstrate that FSRSOR and HSRSOR methods have lesser iterations, faster elapsed time, and are more accurate than FSGS. In addition, HSRSOR is the most effective of the three methods. To sum up, this paper has successfully proposed the applicability and superiority of the new HSRSOR method based on 3HSLRFD-HSCT schemes.

Hybrid Methods for Computing the Streamfunction and Velocity Potential for Complex Flow Fields over Mesoscale Domains

Three types of previously used numerical methods are revisited for computing the streamfunction ψ and velocity potential χ from the horizontal velocity v in limited domains. The first type, called the SOR-based method, uses a classical successive over-relaxation (SOR) scheme to compute ψ (or χ) first with an arbitrary boundary condition (BC) and then χ (or ψ) with the BC derived from ν. The second type, called the spectral method, uses spectral formulations to construct the inner part of (ψ, χ)—the inversion of (vorticity, divergence) with a homogeneous BC, and then the remaining harmonic part of (ψ, χ) with BCs from ν. The third type, called the integral method, uses integral formulas to compute the internally induced (ψ, χ)—the inversion of domain-internal (vorticity, divergence) using the free-space Green’s function without BCs and then the remaining harmonic ψ (or χ) with BCs from v minus the internally-induced part. Although these methods have previously been successfully applied to flows in large-scale and synoptic-scale domains, their accuracy is compromised when applied to complex flows over mesoscale domains, as shown in this paper. To resolve this problem, two hybrid approaches, the integral-SOR method and the integral-spectral method, are developed by combining the first step of the integral method with the second step adopted from the SOR-based and spectral methods, respectively. Upon testing these methods on real-case complex flows, the integral-SOR method is significantly more accurate than the integral-spectral method, noting that the latter is still generally more accurate than the three previously-used methods. The integral-SOR method is recommended for future applications and diagnostic studies of complex flows.

Intelligent conventional and proposed hybrid 5G detection techniques

In recent years, Multiple Inputs and Multiple Outputs (MIMO) have gained significant attention due to their characteristics such as spectral gain, high throughput, and energy efficient. It is seen as one of the integral parts and backbone of the Fifth Generation (5G) and beyond 5G (B5G). However, the use of a large number of antennas requires complex algorithms to detect the received signal. Though, several detection methods have been proposed which can efficiently enhance the Bit Error Rate (BER) gain of the framework, it also increases the computational complexity. The proposed article introduces a hybrid algorithm for different sizes of MIMO. The hybrid algorithm is designed by combining QR Decomposition M−algorithm−Maximum Likelihood Detection (QRM-MLD) and Beam Forming (BF). Further, we compare the performance of proposed hybrid algorithms with that of conventional algorithms, namely Zero Forcing (ZF), Minimum Mean Square Error (MMSE), Successive over Relaxation (SOR), Gauss Seidel Detector (GSD), Jacobi Scheme (JS), and Approximate Message Passing (AMP). In computer simulation, it is noted that the proposed algorithm outperforms the conventional detection algorithms with minimum computational complexity.

A New Improved Classical Iterative Algorithm for Solving System of Linear Equations

The fundamental problem of linear algebra is to solve the system of linear equations (SOLE’s). To solve SOLE’s, is one of the most crucial topics in iterative methods. The SOLE’s occurs throughout the natural sciences, social sciences, engineering, medicine and business. For the most part, iterative methods are used for solving sparse SOLE’s. In this research, an improved iterative scheme namely, ‘’a new improved classical iterative algorithm (NICA)’’ has been developed. The proposed iterative method is valid when the co-efficient matrix of SOLE’s is strictly diagonally dominant (SDD), irreducibly diagonally dominant (IDD), M-matrix, Symmetric positive definite with some conditions and H-matrix. Such types of SOLE’s does arise usually from ordinary differential equations (ODE’s) and partial differential equations (PDE’s). The proposed method reduces the number of iterations, decreases spectral radius and increases the rate of convergence. Some numerical examples are utilized to demonstrate the effectiveness of NICA over Jacobi (J), Gauss Siedel (GS), Successive Over Relaxation (SOR), Refinement of Jacobi (RJ), Second Refinement of Jacobi (SRJ), Generalized Jacobi (GJ) and Refinement of Generalized Jacobi (RGJ) methods.

Investment analysis of power producers under reliability option scheme

Reliability Options (ROs) are one type of Capacity Remuneration Mechanisms (CRMS) to meet generation adequacy requirement, and it has been applied in some power markets. When a power producer sells a RO, he will receive an option fee but have to sell power at a price below the strike price which is set by the buying side. This mechanism may have some impacts on the investment decisions of power producers. In this paper, the investment timing of power producers is studied. A stochastic differential electricity price model and an investment decision model are built. Some characteristics, such as mean-reversion, multi-periodic and gradual growth are considered in the price model. Real option pricing model is used to make generation investment decision and it is equivalent to a linear complementarity problem. The numerical solution is obtained by the Successive Over Relaxation (SOR) method. The example results show that the different settings of strike price and option fee will lead to early or late investment.

The Rate of Convergence of the SOR Method in the Positive Semidefinite Case

In this paper, we derive upper bounds that characterize the rate of convergence of the SOR method for solving a linear system of the form G x = b , where G is a real symmetric positive semidefinite n × n matrix. The bounds are given in terms of the condition number of G , which is the ratio κ = α / β , where α is the largest eigenvalue of G and β is the smallest nonzero eigenvalue of G . Let H denote the related iteration matrix. Then, since G has a zero eigenvalue, the spectral radius of H equals 1, and the rate of convergence is determined by the size of η , the largest eigenvalue of H whose modulus differs from 1. The bound has the form η 2 ≤ 1 − 1 / κ c , where c = 2 + log 2 n . The main consequence from this bound is that small condition number forces fast convergence while large condition number allows slow convergence.

A generalized numerical framework for solving cocurrent and counter-current membrane models for gas separation

Membrane separation has become a panacea for various scientific and engineering problems, including water treatment, gas separation, purification, hemodialysis, and drug delivery. Modeling and simulation of such systems are necessary for the design, analysis, and optimization of membrane separation processes. Despite numerous studies, an efficient numerical solution of such systems is an open problem, especially when speed and reliability matter. In this study, a generalized numerical framework for solving cocurrent and counter-current membrane models is proposed, which hinges on a straightforward and reliable Gauss-Seidel method with successive over-relaxation. The results confirm the speed and reliability of the proposed algorithm, while it is validated by the experimental data for the separation of a mixture of CH4 and CO2, as well as a mixture of He, CO2, N2, and CH4. The permeate outlet pressure estimation error can be reduced to any value as low as ∼10−14%, while the computational time on a personal laptop is not more than 4.5 s. This algorithm can be readily implemented in various programming languages and commercial software applications.

Effectiveness of magnetize flow on nanofluid via unsteady natural convection inside an inclined U-shaped cavity with discrete heating

This analysis investigates the unsteady MHD free convection flow and heat transfer in an inclined U-shaped cavity filled with Cu-water nanofluid. Both the right and left walls of the cavity and the internal walls of the cavity are supposed to be adiabatic, while the upper horizontal walls of the cavity are kept at a lower temperature. A heat source is located on the bottom horizontal wall with its position that alters the left vertical wall. The dimensionless governing equations are solved using the Successive Under-Relaxation technique. The investigation is achieved by controlling the impact of a set of pertinent parameters, namely; the size and position of the heat element (B = 0.3–0.6, D = 0.3–0.6), Hartman number (Ha = 0–50), nanoparticle volume fraction (ϕ = 0.0–0.07), heat source generation/ absorption (Q = −4.0–2.0), and aspect ratio (AR = 0.2–0.6). The results show that the mean Nu declines with heat source length and Hartmann number, whereas it augments with dimensionless heat source location. Furthermore, the mean Nu rises in parallel with the solid volume fraction of nanoparticles and decreases with aspect ratio. In addition, the increasing of aspect ratio reduces the convection mode compared to conduction heat transfer.