Richardson Iteration Method Research Articles

An Optimal Property of B-Bases for the Modified Richardson Method

A space with a normalized totally positive basis has a unique normalized B-basis. In computer-aided geometric design, normalized B-bases present optimal shape-preserving properties. More optimal properties of normalized B-bases were proved previously. This paper provides a new optimal property concerning the modified Richardson iterative method when applied to collocation matrices of normalized B-bases. Moreover, a similar optimal property is proved for the tensor product of normalized B-bases.

Richardson Iterative Method for Solving Multi-Linear System with M-Tensor

In this paper, Richardson iterative method is employed to solve M-Equation. In order to guarantee the solution can be found, convergence theorems are established and confirmed numerically. The optimal α, which is a parameter of Richardson iterative method that can provide the best convergence rate, is also determined theoretically and numerically. Furthermore, a theorem establishing the range of initial vector for general splitting methods is extended from the range in past study. To further accelerate the convergence rate, Anderson accelerator and three preconditioners are incorporated into Richardson iterative method. Numerical results reveal that by including these accelerators, the convergence rates are enhanced. Finally, we show that Richardson iterative methods with optimal α perform better than the SOR type methods in past studies in terms of number of iterative steps and CPU time.

Iterative photonic processor for fast complex-valued matrix inversion

AnN×Niterative photonic processor is proposed for the first time, we believe, for fast computation of complex-valued matrix inversion, a fundamental but computationally expensive linear algebra operation. Compared to traditional digital electronic processing, optical signal processing has a few unparalleled features that could enable higher representational efficiency and faster computing speed. The proposed processor is based on photonic integration platforms–the inclusion of III-V gain blocks offers net neutral loss in the phase-sensitive loops. This is essential for the Richardson iteration method that is adopted in this paper for complex linear systems. Wavelength multiplexing can be used to significantly improve the processing efficiency, allowing the computation of multiple columns of the inverse matrix using a single processor core. Performances of the key building blocks are modeled and simulated, followed by a system-level analysis, which serves as a guideline for designing anN×NRichardson iteration processor. An inversion accuracy of>98%can be predicted for a64×64photonic processor with a>80times faster inversion rate than electronic processors. Including the power consumed by both active components and electronic circuits, the power efficiency of the proposed processor is estimated to be over an order of magnitude more energy-efficient than electronic processors. The proposed iterative photonic integrated processor provides a promising solution for future optical signal processing systems.

Hyperpower least squares progressive iterative approximation

Geometric iterative methods (GIMs) generate curves or surfaces that fit (interpolate or approximate) given sets of data points. Standard GIMs use the Richardson iteration or gradient descent method, so they converge relatively slowly. This paper investigates the GIMs acceleration method, which is based on the use of more efficient methods for the matrix inverse approximation. That is, the possibility to use hyperpower iterative methods is being explored. As a result, a novel GIM named hyperpower least squares progressive iterative approximation (HPLSPIA) is proposed. The convergence of the HPLSPIA method is analyzed, and the computational complexity is briefly discussed. The method is then applied to tensor product B-spline surface fitting. Several HPLSPIA methods utilizing different hyperpower methods for the matrix inverse approximation are experimentally compared. It is shown that polynomial factorizations used in the hyperpower iterative methods significantly affect the efficiency of the HPLSPIA methods.

A Novel Numerical Method for Theoretical Tire Model Simulation

Theoretical tire models are often used in tire dynamics analysis and tire design. In the past, scholars have carried out a lot of research on theoretical model modeling; however, little progress has been made on its solution. This paper focuses on the numerical solution of the theoretical model. New force and moment calculation matrix equations are constructed, and different iterative methods are compared. The results show that the modified Richardson iteration method proposed in this paper has the best convergence-stability in the steady and unsteady state calculation, which mathematically solves the problem of nonconvergence of discrete theoretical models in the published reference. A novel discrete method for solving the total deformation of tires is established based on the Euler method. The unsteady characteristics of tire models are only related to the path frequency without changing its parameters, so the unsteady state ability of the tire model can be judged based on this condition. It shows that the method in the references have significant differences at different speeds with the same path frequency under turn slip or load variations input, but the method proposed in this paper has good results.

Accelerating the HS-type Richardson Iteration Method with Anderson Mixing

Accelerating the HS-type Richardson Iteration Method with Anderson Mixing

Joint steepest descent and non‐stationary Richardson method for low‐complexity detection in massive MIMO systems

AbstractSignal detection is a major challenge in massive multiple‐input multiple‐output (MIMO) wireless systems due to array of hundreds of antennas. Linear minimum mean square error (MMSE) enables near‐optimal detection performance in massive MIMO but suffers from unbearable computational complexity due to complicated matrix inversions. To address this problem, we propose a novel low‐complexity signal detector based on joint steepest descent (SD) and non‐stationary Richardson (NSR) iteration method. The SD is applied to get an efficient searching direction for the following NSR method to enhance the performance. The key idea of the proposed algorithm is to utilize a combination of the scaled‐diagonal initialization and the system‐ and iteration‐dependent acceleration mechanism, so that the convergence can be significantly speeded up. An antenna‐ and eigenvalue‐based scaling parameter is introduced for the proposed detector to further improve the error‐rate performance. We also provide convergence guarantees for the proposed technique. Numerical results demonstrate that the proposed joint detection approach attains superior performance compared to existing iterative approaches and provides a lower computational complexity than the conventional MMSE detector.

Convergence analysis of Anderson‐type acceleration of Richardson's iteration

SummaryWe consider Anderson extrapolation to accelerate the (stationary) Richardson iterative method for sparse linear systems. Using an Anderson mixing at periodic intervals, we assess how this benefits convergence to a prescribed accuracy. The method, named alternating Anderson–Richardson, has appealing properties for high‐performance computing, such as the potential to reduce communication and storage in comparison to more conventional linear solvers. We establish sufficient conditions for convergence, and we evaluate the performance of this technique in combination with various preconditioners through numerical examples. Furthermore, we propose an augmented version of this technique.

Richardson iteration for linear equations and application in two-point boundary value problem

The main contribution of present work is to validate the performance of Richardson iteration method for linear equations with symmetric positive definite (SPD) matrix. Meanwhile, Richardson iteration method is applied to solve the two-point boundary value problem.

A numerical investigation of multi space reduced basis preconditioners for parametrized elliptic advection-diffusion equations

Abstract We analyze the numerical performance of a preconditioning technique recently proposed in [1] for the efficient solution of parametrized linear systems arising from the finite element (FE) discretization of parameterdependent elliptic partial differential equations (PDEs). In order to exploit the parametric dependence of the PDE, the proposed preconditioner takes advantage of the reduced basis (RB) method within the preconditioned iterative solver employed to solve the linear system, and combines a RB solver, playing the role of coarse component, with a traditional fine grid (such as Additive Schwarz or block Jacobi) preconditioner. A sequence of RB spaces is required to handle the approximation of the error-residual equation at each step of the iterative method at hand, whence the name of Multi Space Reduced Basis (MSRB) method. In this paper, a numerical investigation of the proposed technique is carried on in the case of a Richardson iterative method, and then extended to the flexible GMRES method, in order to solve parameterized advection-diffusion problems. Particular attention is payed to the impact of anisotropic diffusion coefficients and (possibly dominant) transport terms on the proposed preconditioner, by carrying out detailed comparisons with the current state of the art algebraic multigrid preconditioners.

An iterative method for solving general restricted linear equations

The Newton iterative method for computing outer inverses with prescribed range and null space is used in the non-stationary Richardson iterative method to develop an iterative method for solving general restricted linear equations. Starting with any suitably chosen initial iterate, our method generates a sequence of iterates converging to the solution. The necessary and sufficient conditions for the convergence along with the error bounds are established. The applications of the iterative method for solving some special linear equations are also discussed. A number of numerical examples are worked out. They include singular square, rectangular, randomly generated rank deficient matrices, full rank matrices and a set of singular matrices given in Matrix Computation Toolbox (mctoolbox) with the condition numbers ranging from order 1016 to 1050. The mean CPU time (MCT) and the error bounds are the performance measures used. Our results when compared with the results obtained by Chen (1997)leads to substantial improvement in terms of both computational speed and accuracy.

Using canonical dual frames in adaptive Richardson iterative method for solving operator equations

Using canonical dual frames in adaptive Richardson iterative method for solving operator equations

A Preconditioned Richardson Regularization for the Data Completion Problem and the Kozlov-Maz’ya-Fomin Method

Using a preconditioned Richardson iterative method as a regularization to the data completion problem is the aim of the contribution. The problem is known to be exponentially ill posed that makes its numerical treatment a hard task. The approach we present relies on the Steklov-Poincaré variational framework introduced in [Inverse Problems, vol. 21, 2005]. The resulting algorithm turns out to be equivalent to the Kozlov-Maz’ya-Fomin method in [Comp. Math. Phys., vol. 31, 1991]. We conduct a comprehensive analysis on the suitable stopping rules that provides some optimal estimates under the General Source Condition on the exact solution. Some numerical examples are finally discussed to highlight the performances of the method. L’objectif est d’utiliser une méthode itérative de Richardson préconditionnée comme une technique de régularisation pour le problème de complétion de données. Le problème est connu pour être sévèrement mal posé qui rend son traitement numérique ardu. L’approche adoptée est basée sur le cadre variationnel de Steklov-Poincaré introduit dans [Inverse Problems, vol. 21, 2005].L’algorithme obtenu s’avère être équivalent à celui de Kozlov-Maz’ya-Fomin parû dans [Comp. Math. Phys., vol. 31, 1991]. Nous menons une analyse complète pour le choix du critère d’arrêt, et établissons des estimations optimales sous les Conditions Générale de Source sur la solution exacte. Nous discutons, enfin, quelques exemples numériques qui confortent les pertinence de la méthode.

Milestones in the Development of Iterative Solution Methods

Iterative solution methods to solve linear systems of equations were originally formulated as basic iteration methods of defect‐correction type, commonly referred to as Richardson′s iteration method. These methods developed further into various versions of splitting methods, including the successive overrelaxation (SOR) method. Later, immensely important developments included convergence acceleration methods, such as the Chebyshev and conjugate gradient iteration methods and preconditioning methods of various forms. A major strive has been to find methods with a total computational complexity of optimal order, that is, proportional to the degrees of freedom involved in the equation. Methods that have turned out to have been particularly important for the further developments of linear equation solvers are surveyed. Some of them are presented in greater detail.

Dirichlet–Neumann alternating algorithm based on the natural boundary reduction for time-dependent problems over an unbounded domain

In this paper, a new iterative algorithm to solve a time-dependent problem over an unbounded domain is suggested. This method is based on the natural boundary reduction and is suitable for solving initial boundary value problems of time-dependent wave equation over an unbounded domain. Firstly, an circular artificial boundary is introduced. Then the original unbounded domain is decomposed into a bounded domain and an exterior unbounded domain outside the artificial boundary. The natural integral equation obtained by the natural boundary reduction is used as a boundary condition on the artificial boundary. Secondly, a Dirichlet–Neumann (D–N) alternating iterative algorithm is constructed. The algorithm is equivalent to preconditioned Richardson iteration method. Thirdly, numerical studies are performed by finite element methods, and the results demonstrate the effectiveness of this algorithm. Finally, some remarks are presented.

Variable Additive Preconditioning Procedures

Iterative methods with variable preconditioners of additive type are proposed. The scaling factors of each summand in the additive preconditioners are optimized within each iteration step. It is proved that the presented methods converge at least as fast as the Richardson's iterative method with the corresponding additive preconditioner with optimal scaling factors. In the presented numerical experiments the suggested methods need nearly the same number of iterations as the usual preconditioned conjugate gradient method with the corresponding additive preconditioner with numerically determined fixed optimal scaling factors.

Adaptive Richardson iteration based on Leja points

An adaptive Richardson iteration method is presented for the solution of large linear systems of equations with a sparse symmetric positive definite matrix. The relaxation parameters for Richardson iteration are chosen to be reciprocal values of Leja points for an interval [ a, b] on the positive real axis, and the endpoints a and b are determined adaptively by computing certain modified moments during the iterations. Computed examples show that the adaptive Richardson method can be competitive with the conjugate gradient algorithm.

An adaptive Richardson iteration method for indefinite linear systems

An adaptive Richardson iteration method is presented for the solution of large linear systems of equations with a sparse, symmetric, nonsingular, indefinite matrix. The relaxation parameters for Richardson iteration are chosen to be reciprocal values of Leja points for a compact setK:=[a,b]∪[c,d], where [a,b] is an interval on the negative real axis and [c, d] is an interval on the positive real axis. Endpoints of these intervals are determined adaptively by computing certain modified moments during the iterations. Computed examples show that this adaptive Richardson method can be competitive with the SYMMLQ and the conjugate residual methods, which are based on the Lanczos process.

Application of a block modified Chebyshev algorithm to the iterative solution of symmetric linear systems with multiple right hand side vectors

An adaptive Richardson iteration method is described for the solution of large sparse symmetric positive definite linear systems of equations with multiple right-hand side vectors. This scheme ``learns'' about the linear system to be solved by computing inner products of residual matrices during the iterations. These inner products are interpreted as block modified moments. A block version of the modified Chebyshev algorithm is presented which yields a block tridiagonal matrix from the block modified moments and the recursion coefficients of the residual polynomials. The eigenvalues of this block tridiagonal matrix define an interval, which determines the choice of relaxation parameters for Richardson iteration. Only minor modifications are necessary in order to obtain a scheme for the solution of symmetric indefinite linear systems with multiple right-hand side vectors. We outline the changes required.

A hybrid Arnoldi-Faber iterative method for nonsymmetric systems of linear equations

We present here a new hybrid method for the iterative solution of large sparse nonsymmetric systems of linear equations, say of the formAx=b, whereA ? ? N, N , withA nonsingular, andb ? ? N are given. This hybrid method begins with a limited number of steps of the Arnoldi method to obtain some information on the location of the spectrum ofA, and then switches to a Richardson iterative method based on Faber polynomials. For a polygonal domain, the Faber polynomials can be constructed recursively from the parameters in the Schwarz-Christoffel mapping function. In four specific numerical examples of non-normal matrices, we show that this hybrid algorithm converges quite well and is approximately as fast or faster than the hybrid GMRES or restarted versions of the GMRES algorithm. It is, however, sensitive (as other hybrid methods also are) to the amount of information on the spectrum ofA acquired during the first (Arnoldi) phase of this procedure.