A generalized iterative method and comparison results using projection techniques for solving linear systems

In [Y.-F. Jing, T.-Z. Huang, On a new iterative method for solving linear systems and comparison results, J. Comput. Appl. Math. 220 (2008) 74–84], Jing and Huang obtained a new iterative method for solving linear systems. This method can be considered as a projection method which uses a two-dimensional space at each step. In this paper, we generalize this method to a three-dimensional projection process. And a different approach is established, which is both theoretically and numerically proven to be better than (or at least the same as) [Jing and Huang’s (2008)].     

New iterative methods for solving linear systems

In this paper, we introduce some new iterative methods to solve linear systems \(Ax=b\}. We  show that these methods, comparing to the classical Jacobi or Gauss-Seidel method, can be applied to more systems and have faster convergence.     

关于解线性系统的一个有效的贝叶斯迭代方法

This paper concerns with the statistical methods for solving general linear systems. After a brief review of Bayesian perspective for inverse problems,a new and efficient iterative method for general linear systems from a Bayesian perspective is proposed.The convergence of this iterative method is proved,and the corresponding error analysis is studied.Finally, numerical experiments are given to support the efficiency of this iterative method,and some conclusions are obtained.     

A new implementation of the CMRH method for solving dense linear systems

The CMRH method [H. Sadok, Méthodes de projections pour les systèmes linéaires et non linéaires, Habilitation thesis, University of Lille1, Lille, France, 1994; H. Sadok, CMRH: A new method for solving nonsymmetric linear systems based on the Hessenberg reduction algorithm, Numer. Algorithms 20 (1999) 303–321] is an algorithm for solving nonsymmetric linear systems in which the Arnoldi component of GMRES is replaced by the Hessenberg process, which generates Krylov basis vectors which are orthogonal to standard unit basis vectors rather than mutually orthogonal. The iterate is formed from these vectors by solving a small least squares problem involving a Hessenberg matrix. Like GMRES, this method requires one matrix–vector product per iteration. However, it can be implemented to require half as much arithmetic work and less storage. Moreover, numerical experiments show that this method performs accurately and reduces the residual about as fast as GMRES. With this new implementation, we show that the CMRH method is the only method with long-term recurrence which requires not storing at the same time the entire Krylov vectors basis and the original matrix as in the GMRES algorithm. A comparison with Gaussian elimination is provided.     

On convergence of iterative methods for maximal correlation problems

Several iterative methods for maximal correlation problems (MCPs) have been proposed in the literature. This paper deals with the convergence of these iterations and contains three contributions. Firstly, a unified and concise proof of the monotone convergence of these iterative methods is presented. Secondly, a starting point strategy is analysed. Thirdly, some error estimates are presented to test the quality of a computed solution. Both theoretical results and numerical tests suggest that combining with this starting point strategy these methods converge rapidly and are more likely converging to a global maximizer of MCP. Copyright © 2016 John Wiley & Sons, Ltd.     

An EM-based iterative method for solving large sparse linear systems

ABSTRACT

We propose a novel iterative algorithm for solving a large sparse linear system. The method is based on the EM algorithm. If the system has a unique solution, the algorithm guarantees convergence with a geometric rate. Otherwise, convergence to a minimal Kullback–Leibler divergence point is guaranteed. The algorithm is easy to code and competitive with other iterative algorithms.     

A symmetric mixed finite element method for nearly incompressible elasticity based on biorthogonal systems

We present a symmetric version of the nonsymmetric mixed finite element method presented in (Lamichhane, ANZIAM J 50 (2008), C324–C338) for nearly incompressible elasticity. The displacement–pressure formulation of linear elasticity is discretized using a Petrov–Galerkin discretization for the pressure equation in (Lamichhane, ANZIAM J 50 (2008), C324–C338) leading to a non‐symmetric saddle point problem. A new three‐field formulation is introduced to obtain a symmetric saddle point problem which allows us to use a biorthogonal system. Working with a biorthogonal system, we can statically condense out all auxiliary variables from the saddle point problem arriving at a symmetric and positive‐definite system based only on the displacement. We also derive a residual based error estimator for the mixed formulation of the problem. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2012     

Convergence analysis for an iterative method for solving nonlinear parabolic systems

In a recent work, we have proposed a new iterative method based on the eigenfunction expansion to integrate nonlinear parabolic systems sequentially. In this paper, we prove that the method is convergent and give analytical rate for its convergence. Moreover, we determine the number of iterations needed to obtain a solution with a pre-determined level of accuracy. We then illustrate the convergence analysis with a problem in combustion theory. It is expected that the convergence analysis can be used for similar systems with time dependence.     

A fast Petrov–Galerkin method for solving the generalized airfoil equation

In this paper we develop a fast Petrov–Galerkin method for solving the generalized airfoil equation using the Chebyshev polynomials. The conventional method for solving this equation leads to a linear system with a dense coefficient matrix. When the order of the linear system is large, the computational complexity for solving the corresponding linear system is huge. For this we propose the matrix truncation strategy, which compresses the dense coefficient matrix into a sparse matrix. We prove that the truncated method preserves the optimal order of the approximate solution for the conventional method. Moreover, we solve the truncated equation using the multilevel augmentation method. The computational complexity for solving this truncated linear system is estimated to be linear up to a logarithmic factor.     

A new method for a class of linear variational inequalities

In this paper we introduce a new iterative scheme for the numerical solution of a class of linear variational inequalities. Each iteration of the method consists essentially only of a projection to a closed convex set and two matrix-vector multiplications. Both the method and the convergence proof are very simple.This work is supported by the National Natural Science Foundation of the P.R. China and NSF of Jiangsu.     

A new projection and contraction method for linear variational inequalities

In this paper, we presented a new projection and contraction method for linear variational inequalities, which can be regarded as an extension of He's method. The proposed method includes several new methods as special cases. We used a self-adaptive technique to adjust parameter β at each iteration. This method is simple, the global convergence is proved under the same assumptions as He's method. Some preliminary computational results are given to illustrate the efficiency of the proposed method.     

On a new exponential iterative method for solving nonsmooth equations

In this paper, we propose a new distinctive version of a generalized Newton method for solving nonsmooth equations. The iterative formula is not the classic Newton type, but an exponential one. Moreover, it uses matrices from B‐differential instead of generalized Jacobian. We prove local convergence of the method and we present some numerical examples.     

Multiscale discontinuous Petrov–Galerkin method for the multiscale elliptic problems

In this article, we present a new multiscale discontinuous Petrov–Galerkin method (MsDPGM) for multiscale elliptic problems. This method utilizes the classical oversampling multiscale basis in the framework of a Petrov–Galerkin version of the discontinuous Galerkin method, allowing us to better cope with multiscale features in the solution. MsDPGM takes advantage of the multiscale Petrov–Galerkin method (MsPGM) and the discontinuous Galerkin method (DGM). It can eliminate the resonance error completely and decrease the computational costs of assembling the stiffness matrix, thus, allowing for more efficient solution algorithms. On the basis of a new H² norm error estimate between the multiscale solution and the homogenized solution with the first‐order corrector, we give a detailed convergence analysis of the MsDPGM under the assumption of periodic oscillating coefficients. We also investigate a multiscale discontinuous Galerkin method (MsDGM) whose bilinear form is the same as that of the DGM but the approximation space is constructed from the classical oversampling multiscale basis functions. This method has not been analyzed theoretically or numerically in the literature yet. Numerical experiments are carried out on the multiscale elliptic problems with periodic and randomly generated log‐normal coefficients. Their results demonstrate the efficiency of the proposed method.     

A Petrov–Galerkin spectral method for fourth‐order problems

In this paper, we consider the Petrov–Galerkin spectral method for fourth‐order elliptic problems on rectangular domains subject to non‐homogeneous Dirichlet boundary conditions. We derive some sharp results on the orthogonal approximations in one and two dimensions, which play important roles in numerical solutions of higher‐order problems. By applying these results to a fourth‐order problem, we establish the H²‐error and L²‐error bounds of the Petrov–Galerkin spectral method. Numerical experiments are provided to illustrate the high accuracy of the proposed method and coincide well with the theoretical analysis. Copyright © 2013 John Wiley & Sons, Ltd.     

A stabilized finite element method for convection–diffusion problems

A stabilized finite element method (FEM) is presented for solving the convection–diffusion equation. We enrich the linear finite element space with local functions chosen according to the guidelines of the residual‐free bubble (RFB) FEM. In our approach, the bubble part of the solution (the microscales) is approximated via an adequate choice of discontinuous bubbles allowing static condensation. This leads to a streamline‐diffusion FEM with an explicit formula for the stability parameter τ_K that incorporates the flow direction, has the capability to deal with problems where there is substantial variation of the Péclet number, and gives the same limit as the RFB method. The method produces the same a priori error estimates that are typically obtained with streamline‐upwind Petrov/Galerkin and RFB. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2011     

On the gradient-projection method for solving the nonsymmetric linear complementarity problem

The Levitin-Poljak gradient-projection method is applied to solve the linear complementarity problem with a nonsymmetric matrixM, which is either a positive-semidefinite matrix or aP-matrix. Further-more, if the quadratic functionx ^T(Mx + q) is pseudoconvex on the feasible region {x R ⁿ |Mx + q 0,x0}, then the gradient-projection method generates a sequence converging to a solution, provided that the problem has a solution. For the case when the matrixM is aP-matrix and the solution is nondegenerate, the gradient-projection method is finite.This work is based on the author's PhD Dissertation, which was supported by NSF Grant No. MCS-79-01066 at the University of Wisconsin, Madison, Wisconsin.The author would like to thank Professor O. L. Mangasarian for his guidance of the dissertation.     

A Petrov‐Galerkin method with quadrature for semi‐linear second‐order hyperbolic problems

We propose and analyze a Crank–Nicolson quadrature Petrov–Galerkin (CNQPG) ‐spline method for solving semi‐linear second‐order hyperbolic initial‐boundary value problems. We prove second‐order convergence in time and optimal order H² norm convergence in space for the CNQPG scheme that requires only linear algebraic solvers. We demonstrate numerically optimal order H^k, k = 0,1,2, norm convergence of the scheme for some test problems with smooth and nonsmooth nonlinearities. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2006     

Linear sufficiency and linear admissibility in a continuous time Gauss–Markov model

This paper considers the problem of estimation in a linear model when a stochastic process instead of a random vector is observed. Estimators obtained as integrals of the observed process are studied. Characterizations of linear sufficiency and admissibility similar to those given in the classical linear model are obtained in this context. Moreover, a definition of generalized ridge estimators in continuous time is introduced and also a characterization of such estimators is given.     

On finite termination of an iterative method for linear complementarity problems

Based on a well-known reformulation of the linear complementarity problem (LCP) as a nondifferentiable system of nonlinear equations, a Newton-type method will be described for the solution of LCPs. Under certain assumptions, it will be shown that this method has a finite termination property, i.e., if an iterate is sufficiently close to a solution of LCP, the method finds this solution in one step. This result will be applied to a recently proposed algorithm by Harker and Pang in order to prove that their algorithm also has the finite termination property.     

A modified pseudospectral method for solving trajectory optimization problems with singular arc

This paper presents a direct method based on Legendre–Radau pseudospectral method for efficient and accurate solution of a class of singular optimal control problems. In this scheme, based on a priori knowledge of control, the problem is transformed to a multidomain formulation, in which the switching points appear as unknown parameters. Then, by utilizing Legendre‐Radau pseudospectral method, a nonlinear programming problem is derived which can be solved by the well‐developed parameter optimization algorithms. The main advantages of the present method are its superior accuracy and ability to capture the switching times. Accuracy and performance of the proposed method are examined by means of some numerical experiments. Copyright © 2016 John Wiley & Sons, Ltd.