Overrelaxation Method Research Articles

A new SOR-Like method for the saddle point problems

In previous years, Golub, Wu and Yuan presented a generalized successive over-relaxation (SOR-Like) method for solving the saddle point problems. In this paper, we present a new SOR-Like (NSOR-Like) method which has three parameters. Our new method can be applied to the nonsingular saddle point problems as well as the singular cases. The characteristic of eigenvalues of the iteration matrix of this NSOR-Like method is analyzed. Then we give the convergence (semi-convergence) theorem of the new iterative method by giving the restrictions imposed on the parameter. Moreover, that convergence (semi-convergence) theorem is applied to some special cases to give the convergence region for the parameters. We can see that NSOR-Like method has a wider convergence (semi-convergence) region for ω and τ than the Parameterized Uzawa method which covers Preconditioned Uzawa method and Uzawa method. In addition, the optimal iteration parameters and the corresponding convergence (semi-convergence) factor for the Uzawa method are presented.

Exploration on a Fast EHL Computing Technology for Analyzing Journal Bearings with Engineered Surface Textures

Solving elastohydrodynamic lubrication (EHL) problems is a complex and time-consuming process due to the interactive solutions of the Reynolds equation and contact elasticity. Analyzing journal bearing EHL problems is even more difficult due to the scale difference in the structural and surface features, which may span four orders of magnitude. This article presents a fast EHL computing technology utilizing a parallel numerical iterative method (the red–black successive overrelaxation method) and multithreaded computing scheme conducted by OpenMP directives. The fast computational approaches allow the construction of high-density EHL meshes for effective descriptions of important texture features of journal bearing surfaces.

Call admission control and resource allocation of wireless networks

In the next-generation wireless communication, Orthogonal Frequency Division Multiplexing (OFDM) is a potential key technology. A CAC scheme is proposed for subcarrier allocation in OFDM-based wireless multi-service network. Call connection requests are divided into two classes: (i) narrowband calls and (ii) wideband calls. Traffic is characterised as batch arrival process for both classes of the calls, so that each call may request multiple subcarriers to satisfy its Quality of Service (QoS) requirement. There is a provision of buffer for narrowband calls, whereas the sub-rating scheme is used for wideband calls. The successive over-relaxation method is employed to solve the set of steady-state equations governing the traffic model. Various performance measures such as blocking probability of wide and narrowband calls, overall blocking probability of the calls and bandwidth utilisation are established. The sensitivity analysis has also been carried out to depict the effects of various system parameters on the performance measures.

A study of successive over-relaxation method parallelisation over modern HPC languages

Successive over-relaxation SOR is a computationally intensive, yet extremely important iterative solver for solving linear systems. Due to recent trends of exponential growth in the amount of data generated and increasing problem sizes, serial platforms have proved to be insufficient in providing the required computational power. In this paper, we present parallel implementations of red-black SOR method using three modern programming languages namely Chapel, D and Go. We employ SOR method for solving 2D steady-state heat conduction problem. We discuss the optimisations incorporated and the features of these languages which are crucial for improving the program performance. Experiments have been performed using two, four, and eight threads and performance results are compared with those obtained using serial execution. The analysis of results provides important insights into the working of SOR method.

An Efficient Matrix Splitting Method for the Second-Order Cone Complementarity Problem

Given a symmetric and positive (semi)definite $n$-by-$n$ matrix $M$ and a vector, in this paper, we consider the matrix splitting method for solving the second-order cone linear complementarity problem (SOCLCP). The matrix splitting method is among the most widely used approaches for large scale and sparse classical linear complementarity problems, and its linear convergence is proved by [Luo and Tseng, SIAM J. Control Optim., 30 (1992), pp. 408--425]. Our first contribution is to prove that, when the general matrix splitting algorithm is applied to SOCLCP with $M$ symmetric and positive definite, it also converges at least linearly. Numerically, our second contribution is to propose a special and efficient matrix splitting algorithm, the block successive overrelaxation method, for solving the SOCLCP. The algorithm makes good use of the underlying geometry of the SOCLCP and each iteration only involves solving triangular linear systems and requires $O(n^2)$ flops; moreover, the algorithm does not destroy the sparse structure of $M$ and is able to exploit the sparsity effectively. Our code BSOR_BN_L based on this method is tested against four other state-of-the-art methods on problems with dense, ill-conditioned symmetric and positive definite matrices, as well as on problems with large scale sparse, symmetric and positive (semi)definite matrices. The numerical results are quite encouraging and BSOR_BN_L exhibits a very good performance in terms of its speed, accuracy, and efficiency in exploiting the sparsity of $M$.

Heat Transfer Analysis of Viscous Incompressible Fluid by Combined Natural Convection and Radiation in an Open Cavity

The effect of radiation on natural convection of Newtonian fluid contained in an open cavity is investigated in this study. The governing partial differential equations are solved numerically using the Alternate Direct Implicit method together with the Successive Overrelaxation method. The study is focused on studying the flow pattern and the convective and radiative heat transfer rates are studied for different values of radiation parameters, namely, the optical thickness of the fluid, scattering albedo, and the Planck number. It was found that, in the optically thin limit, an increase in the optical thickness of the fluid raises the temperature and radiation heat transfer of the fluid. However, a further increase in the optical thickness decreases the radiative heat transfer rate due to increase in the energy level of the fluid, which ultimately reduces the total heat transfer rate within the fluid.

Boundary Value Problems on Triangular Domains and MKSOR Methods

The performance of six variants of the successive overrelaxation methods (SOR) are considered for an algebraic system arising from a finite difference treatment of an elliptic equation of Partial Differential Equations (PDEs) on a triangular region. The consistency of the finite difference representation of the system is achieved. In the finite difference method one obtains an algebraic system corresponding to the boundary value problem (BVP). The block structure of the algebraic system corresponding to four different labeling (the natural, the red- black and green (RBG), the electronic and the spiral) of the grid points is considered. Also, algebraic systems obtained from BVP with mixed derivatives are well established. Determination of the optimal relaxation parameters on the bases of the graphical representation of the spectral radius of the iteration matrices for the SOR, the Modified Successive over relaxation (MSOR) and their new variants KSOR, MKSOR, MKSOR1 and MKSOR2 are considered. Application of the treatment to two numerical examples is considered.

Stochastic averaging of quasi-partially integrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations

A stochastic averaging method for predicting the response of quasi-partially integrable and non-resonant Hamiltonian systems to combined Gaussian and Poisson white noise excitations is proposed. For the case with r(1<r<n) independent first integrals which are in involution, an r-dimensional averaged generalized Fokker–Planck–Kolmogorov (GFPK) equation for the transition probability density of r independent first integrals is derived from the stochastic integro-differential equations (SIDEs) of the original quasi-partially integrable and non-resonant Hamiltonian systems by using the stochastic jump-diffusion chain rule and the stochastic averaging theorem. An example is given to illustrate the applications of the proposed stochastic averaging method, and a combination of the finite difference method and the successive over-relaxation method is used to solve the reduced GFPK equation to obtain the stationary probability density of the system. The results are well verified by a Monte Carlo simulation.

Wave Diffraction by Semi-Infinite Venetian Blind Type Grating

The H-polarized electromagnetic wave diffraction by semi-infinite periodic venetian blind-type grating is considered. The grating consists of strips placed in the parallel planes. The problem is reduced to a nonlinear operator equation with respect to the unknown reflection operator of a structure. For its solution, a complex approach which consists in using the successive over-relaxation method and the Gauss-Seidel method is suggested. The near and far fields scattered by the grating are calculated. The dependences of the reflection coefficient and diffraction patterns are analyzed for different incidence angles, width of strips and distance between them.

Some results on preconditioned modified accelerated overrelaxation method

Some results on preconditioned modified accelerated overrelaxation method

Some results on preconditioned modified accelerated overrelaxation method

Some results on preconditioned modified accelerated overrelaxation method

Some results on preconditioned modified accelerated overrelaxation method

Some results on preconditioned modified accelerated overrelaxation method

Some results on preconditioned modified accelerated overrelaxation method

Some results on preconditioned modified accelerated overrelaxation method

Some results on preconditioned modified accelerated overrelaxation method

Some results on preconditioned modified accelerated overrelaxation method

Some results on preconditioned modified accelerated overrelaxation method

Some results on preconditioned modified accelerated overrelaxation method

A spectral relaxation method for thermal dispersion and radiation effects in a nanofluid flow

Abstract In this study we use a new spectral relaxation method to investigate heat transfer in a nanofluid flow over an unsteady stretching sheet with thermal dispersion and radiation. Three water-based nanofluids containing copper oxide CuO, aluminium oxide Al2O3 and titanium dioxide TiO2 nanoparticles are considered in this study. The transformed governing system of nonlinear differential equations was solved numerically using the spectral relaxation method that has been proposed for the solution of nonlinear boundary layer equations. Results were obtained for the skin friction coefficient, the local Nusselt number as well as the velocity, temperature and nanoparticle fraction profiles for some values of the governing physical and fluid parameters. Validation of the results was achieved by comparison with limiting cases from previous studies in the literature. We show that the proposed technique is an efficient numerical algorithm with assured convergence that serves as an alternative to common numerical methods for solving nonlinear boundary value problems. We show that the convergence rate of the spectral relaxation method is significantly improved by using the method in conjunction with the successive over-relaxation method.

Radiation hydrodynamics integrated in the PLUTO code

The transport of energy through radiation is very important in many astrophysical phenomena. In dynamical problems the time-dependent equations of radiation hydrodynamics have to be solved. We present a newly developed radiation-hydrodynamics module specifically designed for the versatile MHD code PLUTO. The solver is based on the flux-limited diffusion approximation in the two-temperature approach. All equations are solved in the co-moving frame in the frequency independent (grey) approximation. The hydrodynamics is solved by the different Godunov schemes implemented in PLUTO, and for the radiation transport we use a fully implicit scheme. The resulting system of linear equations is solved either using the successive over-relaxation (SOR) method (for testing purposes), or matrix solvers that are available in the PETSc library. We state in detail the methodology and describe several test cases in order to verify the correctness of our implementation. The solver works in standard coordinate systems, such as Cartesian, cylindrical and spherical, and also for non-equidistant grids. We have presented a new radiation-hydrodynamics solver coupled to the MHD-code \PLUTO that is a modern, versatile and efficient new module for treating complex radiation hydrodynamical problems in astrophysics. As test cases, either purely radiative situations, or full radiation-hydrodynamical setups (including radiative shocks and convection in accretion discs) have been studied successfully. The new module scales very well on parallel computers using MPI. For problems in star or planet formation, we have added the possibility of irradiation by a central source.

Convergence of accelerated modulus-based matrix splitting iteration methods for linear complementarity problem with an [formula omitted]-matrix

The theoretical analysis of the accelerated modulus-based matrix splitting iteration methods for the solution of the large sparse linear complementarity problem is further studied. The convergence conditions are presented by fully utilizing the H+-matrix property of the system matrix, and the optimal iteration parameters in accelerated modulus-based accelerated overrelaxation method are determined by minimizing the spectral radius of the iteration matrix. Numerical experiments further confirm the theoretical discussion, and show that the proposed methods accelerate the convergence of the modulus-based matrix splitting methods with less iteration steps and CPU time.

Improved Ray Casting of Procedural Distance Bounds

In this article, we present performance and quality enhancements for ray casting procedural distance bounds using sphere tracing: First, we propose a safe overrelaxation method for accelerating the tracing process and evaluate different variants of the algorithm. Second, we introduce a technique for preventing self-intersections, which enables controlling the precision and rendering performance. In addition, we show a method for significant acceleration of the intersection test when sphere tracing convex objects. We further present a screen-space metric for retrieval of intersection point candidates, in case sphere tracing does not converge. Thereby, we allow for an increase of the rendering quality without sacrificing performance. Further, we reduce common discontinuity artifacts in sphere tracing using a fixed-point iteration algorithm. The algorithms depicted in this article are universally applicable beyond rendering complex procedurally generated scenes in real time. We demonstrate this by integratin...

MODULUS-BASED SUCCESSIVE OVERRELAXATION METHOD FOR PRICING AMERICAN OPTIONS

We consider the modulus-based successive overrelaxation met- hod for the linear complementarity problems from the discretization of Black-Scholes American options model. The H+-matrix property of the system matrix discretized from American option pricing which guaran- tees the convergence of the proposed method for the linear complementar- ity problem is analyzed. Numerical experiments confirm the theoretical analysis, and further show that the modulus-based successive overrelax- ation method is superior to the classical projected successive overrelaxation method with optimal parameter.