Seidel Method Research Articles

Power Flow Analysis in DC Grids: Two Alternative Numerical Methods

This express brief proposes two new iterative approaches for solving the power flow problem in direct current networks as efficient alternatives to the classical Gauss–Seidel and Newton–Raphson methods. The first approach works with the set of nonlinear equations by rearranging them into a conventional fixed point form, generating a successive approximation methodology. The second approach is based on Taylors series expansion method by using a set of decoupling equations to linearize the problem around the desired operating point; these linearized equations are recursively solved until reach the solution of the power flow problem with minimum error. These two approaches are comparable to the classical Gauss–Seidel method and the classical Newton–Raphson method, respectively. Simulation results show that the proposed approaches have a better performance in terms of solution precision and computational requirements. All the simulations were conducted via MATLAB software by using its programming interface.

Worst-case complexity of cyclic coordinate descent: $$O(n^2)$$ gap with randomized version

This paper concerns the worst-case complexity of cyclic coordinate descent (C-CD) for minimizing a convex quadratic function, which is equivalent to Gauss–Seidel method, Kaczmarz method and projection onto convex sets (POCS) in this simple setting. We observe that the known provable complexity of C-CD can be $$\mathcal {O}(n^2)$$ times slower than randomized coordinate descent (R-CD), but no example was proven to exhibit such a large gap. In this paper we show that the gap indeed exists. We prove that there exists an example for which C-CD takes at least $$\mathcal {O}(n^4 \kappa _{\text {CD}} \log \frac{1}{\epsilon })$$ operations, where $$\kappa _{\text {CD}}$$ is related to Demmel’s condition number and it determines the convergence rate of R-CD. It implies that in the worst case C-CD can indeed be $$\mathcal {O}(n^2)$$ times slower than R-CD, which has complexity $$\mathcal {O}( n^2 \kappa _{\text {CD}} \log \frac{1}{\epsilon })$$ . Note that for this example, the gap exists for any fixed update order, not just a particular order. An immediate consequence is that for Gauss–Seidel method, Kaczmarz method and POCS, there is also an $$\mathcal {O}(n^2) $$ gap between the cyclic versions and randomized versions (for solving linear systems). One difficulty with the analysis is that the spectral radius of a non-symmetric iteration matrix does not necessarily constitute a lower bound for the convergence rate. Finally, we design some numerical experiments to show that the size of the off-diagonal entries is an important indicator of the practical performance of C-CD.

Quadratically Regularized Optimal Transport

We investigate the problem of optimal transport in the so-called Kantorovich form, i.e. given two Radon measures on two compact sets, we seek an optimal transport plan which is another Radon measure on the product of the sets that has these two measures as marginals and minimizes a certain cost function. We consider quadratic regularization of the problem, which forces the optimal transport plan to be a square integrable function rather than a Radon measure. We derive the dual problem and show strong duality and existence of primal and dual solutions to the regularized problem. Then we derive two algorithms to solve the dual problem of the regularized problem: A Gauss–Seidel method and a semismooth quasi-Newton method and investigate both methods numerically. Our experiments show that the methods perform well even for small regularization parameters. Quadratic regularization is of interest since the resulting optimal transport plans are sparse, i.e. they have a small support (which is not the case for the often used entropic regularization where the optimal transport plan always has full measure).

Iterative Methods for Solving a System of Linear Equations in a Bipolar Fuzzy Environment

We develop the solution procedures to solve the bipolar fuzzy linear system of equations (BFLSEs) with some iterative methods namely Richardson method, extrapolated Richardson (ER) method, Jacobi method, Jacobi over-relaxation (JOR) method, Gauss–Seidel (GS) method, extrapolated Gauss-Seidel (EGS) method and successive over-relaxation (SOR) method. Moreover, we discuss the properties of convergence of these iterative methods. By showing the validity of these methods, an example having exact solution is described. The numerical computation shows that the SOR method with ω = 1 . 25 is more accurate as compared to the other iterative methods.

Convergence improvement of the simultaneous relaxation method used in the finite element analysis of incompressible fluid flows

Purpose This paper aims to present an improved finite element method used for achieving faster convergence in simulations of incompressible fluid flows. For stable computations of incompressible fluid flows, it is important to ensure that the flow field satisfies the equation of continuity in each element of a generally distorted mesh. The study aims to develop a numerical approach that satisfies this requirement based on the highly simplified marker-and-cell (HSMAC) method and increases computational speed by introducing a new algorithm into the simultaneous relaxation of velocity and pressure. Design/methodology/approach First, the paper shows that the classical HSMAC method is equivalent to a Jacobi-type method in terms of the simultaneous relaxation of velocity and pressure. Then, a Gauss–Seidel or successive over-relaxation (SOR)-type method is introduced in the Newton–Raphson iterations to take into account all the derivative terms in the first-order Taylor series expansion of a nodal-averaged error explicitly. Here, the nine-node quadrilateral (Q2–Q1) elements are used. Findings The new finite element approach based on the improved HSMAC algorithm is tested on fluid flow problems including the lid-driven square cavity flow and the flow past a circular cylinder. The results show significant improvement of the convergence property with the accuracy of the numerical solutions kept unchanged even on a highly distorted mesh. Originality/value To the best of the author’s knowledge, the idea of using the Gauss–Seidel or SOR method in the simultaneous relaxation procedure of the HSMAC method has not been proposed elsewhere.

A Proof of the Correctness of an Algorithm Improving the Estimate of the Rate of Convergence of the Seidel Method

A Proof of the Correctness of an Algorithm Improving the Estimate of the Rate of Convergence of the Seidel Method

Techniques for iterative static analysis of a quad-orifice electrohydraulic steering actuator of rocket stages

The paper contains results of development and study of iterative techniques for static anal-ysis of quad-orifice electrohydraulic steering engine, namely, techniques for calculating its static characteristics (force and velocity characteristics) taking into account parameters of local hydraulic resistance parameters, making it possible to do analysis at various values of power supply voltage and temperature. The proposed techniques are based on solving systems of non-linear algebraic and transcendental equations of math models of the steering actuator describing its static operational modes. Taken as a basis for development of techniques for static analysis of the steering actuator are methods of integrated simulation of physical properties of working fluids of steering actuators and hydraulic drives, iterative methods for calculating parameters of working fluids flow in connecting lines, channels, flow-through elements and valves, results of studies of operating processes for steering actuator constituent elements, as well as a modification of the Seidel method for solving a system of non-linear algebraic and transcendental equations. The paper provides the results of testing the developed static analysis iterative techniques of such a steering actuator. Key words: static analysis, electrohydraulic steering actuator, non-linear algebraic and transcendent equations

Modelling and Analysis of Rails on Viscoelastic Foundation Under a Moving Load

The proposed analysis predicts the time-dependent behaviour of rails resting on geocell-improved soft clay subgrade subjected to moving load. Rail and geocell (with infill soil) have been modelled as Euler–Bernoulli beams of infinite length with finite bending stiffness. Upper granular mat has been characterized as Pasternak shear layer sandwiched between two beams resting over soft clay subgrade idealized by the four-parameter Burger model. Developed governing differential equations have been non-dimensionalised using appropriate dimensionless parameters. Numerical solution of these equations has been obtained by a finite difference scheme in conjunction with the Gauss–Seidel method in an iterative manner using appropriate boundary conditions. Detailed parametric study has been conducted to understand the influence of Burger model parameters, velocity and magnitude of applied load, flexural rigidity and location of geocell on the flexural response of rails. Critical velocity has been determined, and its time dependency has also been discussed. The present study also reports variation in degree of consolidation with elapsed time and procedure to obtain consolidation characteristics for such a double beam system representing geocell-reinforced railway tracks.

Fast corona discharge solver for precipitators using multi‐grid methods on fine grids

A novel approach for modelling the corona problem in the wire-duct precipitators using the finite difference technique integrated with multi-grid methods is adopted in this study. The multi-grid method is applied as a fast convergent iterative solution for the finite difference technique to solve Poisson equation especially on finer grids. Two schemes of the multi-grid method, mainly the V-cycle and the full multi-grid method, are adopted in the present study. Compared with Gauss–Seidel method, the above-mentioned schemes are successfully transcendent owing to timing performance. By using finer grids, the proposed algorithm allows to get a more accurate picture about the performance of the precipitators in the design stage without suffering from the excessive computational time. Accurate results for the potential and current density computations, closed to the previous published experimental measurements, are obtained in comparison with earlier numerical techniques for several design parameters of the precipitators. Finally, the effect of changing the effective ion mobility and the surface roughness factor on the voltage–current density is considered.

Improving an Estimate of the Convergence Rate of the Seidel Method

The Seidel method for solving a system of linear algebraic equations and an estimate of the rate of its convergence are considered in this paper. It is proposed to construct an equivalent system for which the Seidel method also converges but yields a better rate of convergence. An equivalent system is constructed by a separate iterative process, where each step requires O(n) operations. The stability of this process is proved. Results of numerical experiments are presented that show an improvement in the estimate of the convergence rate.

Estimate of time-scale for the current relaxation of percolative Random Resistor cum Tunnelling Network model

The Random Resistor cum Tunnelling Network (RRTN) model was proposed from our group by considering an extra phenomenological (semi-classical) tunnelling process into a classical RRN bond percolation model. We earlier reported about early-stage two inverse power-laws, followed by large time purely exponential tail in some of the RRTN macroscopic current relaxations. In this paper, we investigate on the broader perspective of current relaxation. We present here an analytical argument behind the strong convergence (irrespective of initial voltage configuration) of the bulk current towards its steady-state, mapping the problem into a special kind of Gauss–Seidel method. We find two phenomenological time-scales (referred as τt and τs), those emerge from the variation of macroscopic quantities during current dynamics. We show that not both, only one of them is independent. Thus there exists a single scale in time which controls the entire dynamics.

�������� ����'�������� �����I �I�I��� ��� �I������ ����������� �� ��������� �I�������� ���� �I��������� �������

Many physical processes are described by the Helmholtz equation. By adding the Dirichlet, Neumann, or Robin boundary conditions, the corresponding problems are obtained. An eective method for solving such boundary problems is the nite dierence method. In this paper, we consider a Dirichlet problem for the Helmholtz equation. A nite dierence approximate scheme of the higher-order was constructed. For rst- and second-order derivatives the fourth-order nite dierence approximations are used. By adding boundary conditions at the grid nodes, we obtain a system of dierence equations (a system of linear algebraic equations) with a symmetric matrix that has a diagonal advantage. Therefore, it is advisable to solve this system by iterative methods. In the paper, the simple iterations method and Seidel method are used. They give an approximate solution with a given accuracy by a small number of iterations. A test problem with a constant wave number and a known exact solution is considered. The dierence equation and the iterative formulas of the simple iterations method and Seidel method for this problem are given. The results of numerical experiments, which conrm the eciency of the method and the theoretical order of convergence, are presented. The calculations are performed under dierent values of the wave number. Iterative processes are compared by the number of iterations and the values of absolute errors.

Magnetohydrodynamic effects on finite and infinite slider bearings

This paper presents a numerical investigation of lubricating slider bearings with conducting couple stress fluids using externally applied magnetics fields. The modified two-dimensional magnetohydrodynamic couple stress Reynolds-type equation is obtained. This governing equation is resolved numerically by using finite difference scheme, which involves the Gauss–Seidel method to compute the bearing characteristics. Numerical results using different considered values of the couple stress and Hartman number are presented. These results demonstrate that the transverse magnetic field and couple stress effects are significant.

Proof of correctness of an algorithm that enhances the estimate of the rate of Seidel method convergence

Proof of correctness of an algorithm that enhances the estimate of the rate of Seidel method convergence

Enhansing an estimate of the rate of the Seidel method convergence

Enhansing an estimate of the rate of the Seidel method convergence

On the continuous analogue of the Seidel method

Continuous Seidel method for solving systems of linear and nonlinear algebraic equations is constructed in the article, and the convergence of this method is investigated. According to the method discussed, solving a system of algebraic equations is reduced to solving systems of ordinary differential equations with delay. This allows to use rich arsenal of numerical ODE solution methods while solving systems of algebraic equations. The main advantage of the continuous analogue of the Seidel method compared to the classical one is that it does not require all the elements of the diagonal matrix to be non-zero while solving linear algebraic equations’ systems. The continuous analogue has the similar advantage when solving systems of nonlinear equations.

Solving non-LTE problems in rotational transitions using the Gauss–Seidel method and its implementation in the Atmospheric Radiative Transfer Simulator

This article presents our implementation of a non-LTE solver in spherical symmetry for molecular rotational transition in static or expanding atmospheres. The new open-source code relies on the Gauss–Seidel Accelerated Lambda Iteration methodology that provides a rapid and accurate convergence of the non-LTE problems, which is now routinely used in astrophysical and planetary research. The non-LTE code is interfaced with the widely used package, the Atmospheric Radiative Transfer Simulator (ARTS), to facilitate spectral line simulations for various viewing geometries. In this paper we describe the numerical implementation, provide the first validation results for the populations against two other non-LTE codes, and then discuss the possible application. The quantitative comparisons are performed using an established ortho-water non-LTE model applied to cases of optical thick and thin conditions of Ganymede’s atmosphere. The differences in populations expressed as excitation temperatures show very good agreement in both cases. Finally, we also apply this model to a sample of data from the Microwave Instrument for the Rosetta Orbiter (MIRO) instrument. The new non-LTE package is demonstrated to be fast and accurate, and we hope that it will be a useful addition to the planetary community. In addition, being open source and part of the ARTS, it will be further improved and developed.

Gauss–Seidel Method for Multi-leader–follower Games

The multi-leader–follower game has many applications such as the bilevel structured market in which two or more enterprises, called leaders, have initiatives, and the other firms, called followers, observe the leaders’ decisions and then decide their own strategies. A special case of the game is the Stackelberg model, or the single-leader–follower game, which has been studied for many years. The Stackelberg game may be reformulated as a mathematical program with equilibrium constraints, which has also been studied extensively in recent years. On the other hand, the multi-leader–follower game may be formulated as an equilibrium problem with equilibrium constraints, in which each leader’s problem is an mathematical program with equilibrium constraints. However, finding an equilibrium point of an equilibrium problem with equilibrium constraints is much more difficult than solving a single mathematical program with equilibrium constraints, because each leader’s problem contains those variables which are common to other players’ problems. Moreover, the constraints of each leader’s problem depend on the other rival leaders’ strategies. In this paper, we propose a Gauss–Seidel type algorithm with a penalty technique for solving an equilibrium problem with equilibrium constraints associated with the multi-leader–follower game, and then suggest a refinement procedure to obtain more accurate solutions. We discuss convergence of the algorithm and report some numerical results to illustrate the behavior of the algorithm.

A new method for frequency constrained structural optimization of tall buildings under wind loads

SummaryTall buildings are usually sensitive to wind loads, and effective and accurate methods for the wind‐resistant structural optimal design of tall buildings have become a widely concerned problem. Conventionally, the frequency constraint for the structural optimal design of a tall building is formulated through the strain energy method, and with the objective functions and constraint functions being established, the structural optimization problem is mathematically modeled, and the corresponding solution is solved using the optimal criteria (OC) method. This procedure has two disadvantages. First, the strain energy usually does not consider the gradient of the structural internal forces to the design variables, and so the computed sensitivity of the frequency constraint is inaccurate. Second, in the OC method, the Lagrangian multipliers are computed by solving a set of linear equations using the Gauss–Seidel method, which may lead to inaccurate results. To overcome these disadvantages, a new method is proposed by using an eigenvalue approach for computing the sensitivity of the frequency constraint and a quadratic programming approach for computing the Lagrangian multipliers. It is shown that the proposed new method can perform accurate wind‐resistant structural optimal design for tall buildings and make the iterations of the optimal design converge faster and more stable.

Fuzzy linear systems via boundary value problem

Reduction in storage and number of operations are considered through avoiding the representation of zeros in storage as well as in the calculations. The importance of this approach has its effect in large problems that appear in numerical treatments of boundary value problems in general and becomes more effective when fuzzy concepts are considered. We introduce an extended embedding solution model named fuzzy compact storage Gauss–Seidel (FCGS) for solving linear systems of equations with a fuzzy-based right-hand side. The model starts by applying the embedding approach to the $$n \times n$$ fuzzy linear system, a compact storage technique is then applied to the resultant $$2n \times 2n$$ de-fuzzification matrix, and finally, a Gauss–Seidel method is applied to the system. The FCGS experimental results and algorithm are clarified on some numerical examples including a fuzzy boundary value problem (FBVP). The error improvements through Gauss–Seidel iterations of fuzzy solution computations are reported. The fuzzy solutions at $$\alpha $$ -cuts are shown and compared to the exact solutions. FCGS achieved a reduction of at least 50% of storage by using the compact storage concepts and consequently obtain a reduction in the mathematical operations and accordingly the running time especially in FBVP applications.