This paper considers Stokes and Newton iterations to solve stationary Navier- Stokes equations based on the finite element discretization. We obtain new sufficient conditions of stability and convergence for the two i...This paper considers Stokes and Newton iterations to solve stationary Navier- Stokes equations based on the finite element discretization. We obtain new sufficient conditions of stability and convergence for the two iterations. Specifically, when 0 〈 σ =N｜｜f｜｜-1/v2≤1/√2＋1 , the Stokes iteration is stable and convergent, where N is defined in the paper. When 0 〈 σ ≤5/11, the Newton iteration is stable and convergent. This work gives a more accurate admissible range of data for stability and convergence of the two schemes, which improves the previous results. A numerical test is given to verify the theory.展开更多
In this paper, based on the implicit Runge-Kutta(IRK) methods, we derive a class of parallel scheme that can be implemented on the parallel computers with Ns(N is a positive even number) processors efficiently, and di...In this paper, based on the implicit Runge-Kutta(IRK) methods, we derive a class of parallel scheme that can be implemented on the parallel computers with Ns(N is a positive even number) processors efficiently, and discuss the iteratively B-convergence of the Newton iterative process for solving the algebraic equations of the scheme, secondly we present a strategy providing initial values parallelly for the iterative process. Finally, some numerical results show that our parallel scheme is higher efficient as N is not so large.展开更多
Newton’s method is used to find the roots of a system of equations <span style="white-space:nowrap;"><em>f</em> (x) = 0</span>. It is one of the most important procedures in numerica...Newton’s method is used to find the roots of a system of equations <span style="white-space:nowrap;"><em>f</em> (x) = 0</span>. It is one of the most important procedures in numerical analysis, and its applicability extends to differential equations and integral equations. Analysis of the method shows a quadratic convergence under certain assumptions. For several years, researchers have improved the method by proposing modified Newton methods with salutary efforts. A modification of the Newton’s method was proposed by McDougall and Wotherspoon <a href="#ref1">[1]</a> with an order of convergence of <span style="white-space:nowrap;">1+ <span style="white-space:nowrap;">√2</span></span>. On a new type of methods with cubic convergence was proposed by H. H. H. Homeier <a href="#ref2">[2]</a>. In this article, we present a new modification of Newton method based on secant method. Analysis of convergence shows that the new method is cubically convergent. Our method requires an evaluation of the function and one of its derivatives.展开更多
Independent component analysis （ICA） is the primary statistical method for solving the problems of blind source separation. The fast ICA is a famous and excellent algorithm and its contrast function is optimized by ...Independent component analysis （ICA） is the primary statistical method for solving the problems of blind source separation. The fast ICA is a famous and excellent algorithm and its contrast function is optimized by the quadratic convergence of Newton iteration method. In order to improve the convergence speed and the separation precision of the fast ICA, an improved fast ICA algorithm is presented. The algorithm introduces an efficient Newton＇s iterative method with fifth-order convergence for optimizing the contrast function and gives the detail derivation process and the corresponding condition. The experimental results demonstrate that the convergence speed and the separation precision of the improved algorithm are better than that of the fast ICA.展开更多
Using a predictor-corrector tactic, this paper derives new iteration schemes for unconstrained optimization. It yields a point (predictor) by some line search from the current point;then with the two points it constru...Using a predictor-corrector tactic, this paper derives new iteration schemes for unconstrained optimization. It yields a point (predictor) by some line search from the current point;then with the two points it constructs a quadratic interpolation curve to approximate some ODE trajectory;it finally determines a new point (corrector) by searching along the quadratic curve. In particular, this paper gives a global convergence analysis for schemes associated with the quasi-Newton updates. In our computational experiments, the new schemes using DFP and BFGS updates outperformed their conventional counterparts on a set of standard test problems.展开更多
基金supported by the National Natural Science Foundation of China(No.11271298)
文摘This paper considers Stokes and Newton iterations to solve stationary Navier- Stokes equations based on the finite element discretization. We obtain new sufficient conditions of stability and convergence for the two iterations. Specifically, when 0 〈 σ =N｜｜f｜｜-1/v2≤1/√2＋1 , the Stokes iteration is stable and convergent, where N is defined in the paper. When 0 〈 σ ≤5/11, the Newton iteration is stable and convergent. This work gives a more accurate admissible range of data for stability and convergence of the two schemes, which improves the previous results. A numerical test is given to verify the theory.
基金national natural science foundation natural science foundation of Gansu province.
文摘In this paper, based on the implicit Runge-Kutta(IRK) methods, we derive a class of parallel scheme that can be implemented on the parallel computers with Ns(N is a positive even number) processors efficiently, and discuss the iteratively B-convergence of the Newton iterative process for solving the algebraic equations of the scheme, secondly we present a strategy providing initial values parallelly for the iterative process. Finally, some numerical results show that our parallel scheme is higher efficient as N is not so large.
文摘Newton’s method is used to find the roots of a system of equations <span style="white-space:nowrap;"><em>f</em> (x) = 0</span>. It is one of the most important procedures in numerical analysis, and its applicability extends to differential equations and integral equations. Analysis of the method shows a quadratic convergence under certain assumptions. For several years, researchers have improved the method by proposing modified Newton methods with salutary efforts. A modification of the Newton’s method was proposed by McDougall and Wotherspoon <a href="#ref1">[1]</a> with an order of convergence of <span style="white-space:nowrap;">1+ <span style="white-space:nowrap;">√2</span></span>. On a new type of methods with cubic convergence was proposed by H. H. H. Homeier <a href="#ref2">[2]</a>. In this article, we present a new modification of Newton method based on secant method. Analysis of convergence shows that the new method is cubically convergent. Our method requires an evaluation of the function and one of its derivatives.
文摘Independent component analysis （ICA） is the primary statistical method for solving the problems of blind source separation. The fast ICA is a famous and excellent algorithm and its contrast function is optimized by the quadratic convergence of Newton iteration method. In order to improve the convergence speed and the separation precision of the fast ICA, an improved fast ICA algorithm is presented. The algorithm introduces an efficient Newton＇s iterative method with fifth-order convergence for optimizing the contrast function and gives the detail derivation process and the corresponding condition. The experimental results demonstrate that the convergence speed and the separation precision of the improved algorithm are better than that of the fast ICA.
文摘Using a predictor-corrector tactic, this paper derives new iteration schemes for unconstrained optimization. It yields a point (predictor) by some line search from the current point;then with the two points it constructs a quadratic interpolation curve to approximate some ODE trajectory;it finally determines a new point (corrector) by searching along the quadratic curve. In particular, this paper gives a global convergence analysis for schemes associated with the quasi-Newton updates. In our computational experiments, the new schemes using DFP and BFGS updates outperformed their conventional counterparts on a set of standard test problems.
