Delayed Over-Relaxation for iterative methods

Numerische Mathematik, 1990

We consider the solution of algebraic linear systems of the form Ax = b, where A is a matrix and x and b are vectors. Relaxation methods for solving this problem are de ned by splitting the matrix A as A = M ? N, and solving the system Mx k+1 = b + Nx k at each step k, from some initial guess x 0 . Generally M is choosen to be easily invertible. We will consider the situation where M is not necessarily easy to invert. This situation might arise in block Gauss-Seidel where the diagonal blocks become very large. In this case we will consider solving the Mv = g system by an iterative method. We call this a nested iterative method.

TURKISH JOURNAL OF MATHEMATICS, 2017

In this paper we introduce two strategies for picking relaxation parameters to control the semiconvergence behavior of a sequential block-iterative method. A convergence analysis is presented. We also demonstrate the performance of our strategies by examples taken from tomographic imaging.

2012

The objective of this dissertation is the design and analysis of iterative methods for the numerical solution of large, sparse linear systems. This type of systems emerges from the discretization of Partial Differential Equations. Two special types of linear systems are studied. The first type deals with systems whose coefficient matrix is two cyclic whereas the second type studies the augmented linear systems. Initially, the Preconditioned Simultaneous Displacement (PSD) method, which is a generalized version of the Symmetric SOR (SSOR) method, is studied when the Jacobi iteration matrix is weakly cyclic and its eigenvalues are all real “real case” or all imaginary “imaginary case”. The first result is that the PSD method has better convergence rate than the SSOR method. In particular, in the “imaginary case” its convergence is increased by an order of magnitude compared to the SSOR method. In an attempt to further increase the convergence rate of the PSD method, more parameters we...

Journal of Computational and Applied Mathematics, 1997

An iterative method for the numerical solution of singularly perturbed second-order linear elliptic problems is presented. It is a defect correction iteration in which the approximate operator is the product of two first-order operators, which is readily inverted numerically. The approximate operator is generated by formal asymptotic factorization of the original operator. Hence this is a Quasi Analytic Defect correction iteration (QUAD). Both its continuous and discrete versions are analyzed in one dimension. The scheme is extended to a variety of two dimensional operators and it is analyzed for a model advection-diffusion equation. Numerical calculations show the effectiveness of the scheme over a wide range of values of the small parameter.

Journal of Inverse and Ill-posed Problems, 2017

We study error minimizing relaxation (EMR) strategies for use in Landweber and Kaczmarz type iterations applied to linear systems with or without convex constraints. Convergence results based on operator theory are given, assuming exact data. The advantages and disadvantages of these relaxation strategies on a noisy and ill-posed problem are illustrated using examples taken from the field of image reconstruction from projections. We also consider combining EMR with penalization.

Applied Mathematics and Computation, 2000

In the recent paper of Bai and Su a class of parallel decomposition-type accelerated overrelaxation (PDAOR) methods suitable to the SIMD-systems is established and convergence conditions are concluded when the coecient matrices of the linear systems are L-matrices, H-matrices or symmetric positive-de®nite matrices. In the case of Hmatrices we improve the convergence area for relaxation parameters.

Foundations

Iterative methods which have high convergence order are crucial in computational mathematics since the iterates produce sequences converging to the root of a non-linear equation. A plethora of applications in chemistry and physics require the solution of non-linear equations in abstract spaces iteratively. The derivation of the order of the iterative methods requires expansions using Taylor series formula and higher-order derivatives not present in the method. Thus, these results cannot prove the convergence of the iterative method in these cases when such higher-order derivatives are non-existent. However, these methods may still converge. Our motivation originates from the need to handle these problems. No error estimates are given that are controlled by constants. The process introduced in this paper discusses both the local and the semi-local convergence analysis of two step fifth and multi-step 5+3r order iterative methods obtained using only information from the operators on t...

Solving a system of equations by = , where A is a × matrix and b and × 1 vector, can sometime be a daunting task because solving for x can be difficult. If you were given an algorithm that was efficient, that's great! What if you could make it solve the problem even faster? That's even better. We will first take a look at establishing the basics of the successive over-relaxation method (SOR for short), then we'll look at a real-world problem we applied the SOR method to, solving the heat-equation when a constant boundary temperature is applied to a flat plate.

International Journal for Numerical Methods in Engineering, 1993

Ia this paper we examine the dramatic influence that a severe stretching of finite difference grids can have on the convergence behaviour of iterative methods. For the most important classes of iterative methods this phenomenon is considered for a simple model problem with various boundary conditions and an exponential grid. It is shown that grid compression near a Neumann boundary or in the centre can make the convergence of some methods extremely slow, whereas grid compression near a Dirichlet boundary can be very advantageous. More theoretical insight is obtained by analysing the spectrum of the Jacobi matrix for one-and two-dimensional problems. Several bounds on dominant eigenvalues of this matrix are given. The final conclusions are also applicable to problems with a variable diffusion coefficient and convection-diffusion equations solved by central difference schemes.

Computers & Mathematics With Applications, 1997

In multigrid methods, it is preferred to employ smoothing techniques which are convergent. In practice, the standard Jacobi and Gauss-Seidel methods are the choices for “smoothers” However, it is known that if the spectral radius condition is violated, then convergence is not guaranteed for these methods. In this paper we develop a simple two-step Jacobi-type method which has better convergence properties and which can be employed as a convergent “smoother” wherever the standard iterative methods fail. We provide the convergence proofs and demonstrate the applicability of the method on a variety of problems.