#### DMCA

## Research Article Modi cations of Steepest Descent Method and Conjugate Gradient Method Against Noise for Ill-posed Linear Systems

### Citations

473 |
Introduction to Matrix Computations
- Stewart
- 1973
(Show Context)
Citation Context ...tion method [16], the relaxed steepest descent method [11], as well as the optimal iterative algorithm [14]. A measure of the ill-posedness of Eq. (1.1) can be performed by using the condition number =-=[22]-=-: cond(A) = ∥A∥∥A1∥, (1.2) where ∥A∥ is the Frobenius norm of A. For arbitrary ϵ > 0, there exists a matrix norm ∥A∥ such that ρ(A) ≤ ∥A∥ ≤ ρ(A) + ϵ, where ρ(A) is a radius of the spectrum of A. Ther... |

312 |
Two-point step size gradient methods
- Barzilai, Borwein
- 1988
(Show Context)
Citation Context ...with ill-conditioned A may appear in several fields. For example, finding an n-degree polynomial function p(x) = a0 + a1x+ . . .+ anx n to best match a continuous function f(x) in the interval of x ∈ =-=[0, 1]-=-: min deg(p)n ∫ 1 0 [f(x)− p(x)]2dx, (1.4) leads to a problem governed by Eq. (1.1); A is the (n+1)× (n+1) Hilbert matrix defined by Aij = 1 i+ j − 1 , (1.5) x is composed of the n+ 1 coefficients a0... |

119 |
The Barzilai and Borwein gradient method for the large scale unconstrained minimization problem
- Raydan
- 1997
(Show Context)
Citation Context ... [1] were able to produce a substantial improvement of the convergence speed for a certain test. The results of Barzilai and Borwein [1] have initiated many researches on the SDM, for example, Raydan =-=[18, 19]-=-, Friedlander et al. [6], Raydan and Svaiter [20], Dai et al. [3], Dai and Liao [2], Dai and Yuan [4], Fletcher [5], and Yuan [26]. In this paper we will approach this problem from a quite different a... |

67 | On the Barzilai and Borwein choice of steplength for the gradient method
- Raydan
- 1993
(Show Context)
Citation Context ... [1] were able to produce a substantial improvement of the convergence speed for a certain test. The results of Barzilai and Borwein [1] have initiated many researches on the SDM, for example, Raydan =-=[18, 19]-=-, Friedlander et al. [6], Raydan and Svaiter [20], Dai et al. [3], Dai and Liao [2], Dai and Yuan [4], Fletcher [5], and Yuan [26]. In this paper we will approach this problem from a quite different a... |

55 |
Iterative Methods for Nonlinear Optimization Problems
- Jacoby, Kowalik, et al.
- 1972
(Show Context)
Citation Context ... large perturbation to the solution on the left-hand side. 2 The ODEs on an invariant manifold 2.1 The steepest descent and conjugate gradient methods Solving Eq. (1.1) by the steepest descent method =-=[7]-=- is equivalent to solve the following minimum problem: min x2Rn φ(x) = min x2Rn [ 1 2 xTAx− bTx ] . (2.7) Then, by using the Ritz variational principle we can derive the following algorithm: (i) Give ... |

42 |
Regularization of ill-posed problems in Banach spaces: convergence rates. Inverse Problems
- Resmerita
(Show Context)
Citation Context ...finite difference and finite element methods for partial differential equations, etc. To account of the sensitivity to noise it is usually using a regularization method to solve the ill-posed problem =-=[8, 24, 25, 21]-=-, where a suitable regularization parameter is used to depress the bias in the computed solution by a better balance of approximation error and propagated data error. There are several methods develop... |

38 | Gradient method with retards and generalizations
- Friedlander, Martínez, et al.
- 1998
(Show Context)
Citation Context ...ubstantial improvement of the convergence speed for a certain test. The results of Barzilai and Borwein [1] have initiated many researches on the SDM, for example, Raydan [18, 19], Friedlander et al. =-=[6]-=-, Raydan and Svaiter [20], Dai et al. [3], Dai and Liao [2], Dai and Yuan [4], Fletcher [5], and Yuan [26]. In this paper we will approach this problem from a quite different aspect of invariant manif... |

37 |
R-linear convergence of the Barzilai and Borwein gradient method
- Dai, Liao
- 2002
(Show Context)
Citation Context ...dual error for SDM. Communications in Numerical Analysis 15 Figure 7: For Example 4.3 comparing (a) and (c) for a0, and (b) and (d) of the residual errors for MSDM and random SDM with the interval of =-=[0, 2]-=-. 16 Communications in Numerical Analysis Figure 8: For Example 4.3 comparing accuracies of random SDM, SDM and MSDM. Example 4.4. The Runge phenomenon illustrates that the error can occur when one em... |

22 | Iterative choices of regularization parameters in linear inverse problems
- Kunisch, Zhou
- 1998
(Show Context)
Citation Context ...finite difference and finite element methods for partial differential equations, etc. To account of the sensitivity to noise it is usually using a regularization method to solve the ill-posed problem =-=[8, 24, 25, 21]-=-, where a suitable regularization parameter is used to depress the bias in the computed solution by a better balance of approximation error and propagated data error. There are several methods develop... |

22 | Relaxed steepest descent and Cauchy-Barzilai-Borwein method
- Raydan, Svaiter
- 2002
(Show Context)
Citation Context ...f the convergence speed for a certain test. The results of Barzilai and Borwein [1] have initiated many researches on the SDM, for example, Raydan [18, 19], Friedlander et al. [6], Raydan and Svaiter =-=[20]-=-, Dai et al. [3], Dai and Liao [2], Dai and Yuan [4], Fletcher [5], and Yuan [26]. In this paper we will approach this problem from a quite different aspect of invariant manifold and bifurcation, and ... |

15 |
Alternate minimization gradient method
- Dai, Yuan
- 2003
(Show Context)
Citation Context ...ts of Barzilai and Borwein [1] have initiated many researches on the SDM, for example, Raydan [18, 19], Friedlander et al. [6], Raydan and Svaiter [20], Dai et al. [3], Dai and Liao [2], Dai and Yuan =-=[4]-=-, Fletcher [5], and Yuan [26]. In this paper we will approach this problem from a quite different aspect of invariant manifold and bifurcation, and propose a new strategy to modify the steplength. We ... |

12 | A Fictitious time integration method for the numerical solution of the Fredholm integral equation and for numerical differentiation of noisy data, and its relation to the filter theory, CMES
- Liu, Atluri
(Show Context)
Citation Context ...sly, the author and his coworkers have developed several methods to solve the ill-posed linear problems, like that using the fictitious time integration method as a filter for ill-posed linear system =-=[12]-=-, a modified polynomial expansion method [13], the nonstandard group preserving scheme [15], a vector regularization method [16], the relaxed steepest descent method [11], as well as the optimal itera... |

12 |
An improved model function method for choosing regularization parameters in linear inverse problems, Inverse Problems
- Xie, Zou
(Show Context)
Citation Context ...finite difference and finite element methods for partial differential equations, etc. To account of the sensitivity to noise it is usually using a regularization method to solve the ill-posed problem =-=[8, 24, 25, 21]-=-, where a suitable regularization parameter is used to depress the bias in the computed solution by a better balance of approximation error and propagated data error. There are several methods develop... |

10 |
Novel methods for solving severely ill-posed linear equations system
- Liu, Chang
- 2009
(Show Context)
Citation Context ...ear problems, like that using the fictitious time integration method as a filter for ill-posed linear system [12], a modified polynomial expansion method [13], the nonstandard group preserving scheme =-=[15]-=-, a vector regularization method [16], the relaxed steepest descent method [11], as well as the optimal iterative algorithm [14]. A measure of the ill-posedness of Eq. (1.1) can be performed by using ... |

9 |
Fast realization algorithms for determining regularization parameters in linear inverse problems,”
- Wang, Xiao
- 2005
(Show Context)
Citation Context |

9 |
A new stepsize for the steepest descent method
- Yuan
- 2006
(Show Context)
Citation Context ...1] have initiated many researches on the SDM, for example, Raydan [18, 19], Friedlander et al. [6], Raydan and Svaiter [20], Dai et al. [3], Dai and Liao [2], Dai and Yuan [4], Fletcher [5], and Yuan =-=[26]-=-. In this paper we will approach this problem from a quite different aspect of invariant manifold and bifurcation, and propose a new strategy to modify the steplength. We also compare our results with... |

7 | Modified two-point stepsize gradient methods for unconstrained optimization
- Dai, Yuan, et al.
- 2002
(Show Context)
Citation Context ... speed for a certain test. The results of Barzilai and Borwein [1] have initiated many researches on the SDM, for example, Raydan [18, 19], Friedlander et al. [6], Raydan and Svaiter [20], Dai et al. =-=[3]-=-, Dai and Liao [2], Dai and Yuan [4], Fletcher [5], and Yuan [26]. In this paper we will approach this problem from a quite different aspect of invariant manifold and bifurcation, and propose a new st... |

6 | A highly accurate technique for interpolations using very highorder polynomials and its applications to some ill-posed linear problems, CMES
- Liu, Atluri
- 2009
(Show Context)
Citation Context ...ped several methods to solve the ill-posed linear problems, like that using the fictitious time integration method as a filter for ill-posed linear system [12], a modified polynomial expansion method =-=[13]-=-, the nonstandard group preserving scheme [15], a vector regularization method [16], the relaxed steepest descent method [11], as well as the optimal iterative algorithm [14]. A measure of the ill-pos... |

4 |
Intermittent transition to quasiperiodicity demonstrated via a circular differential equation
- Liu
(Show Context)
Citation Context ...nding to 107. Thus we can observe that when γ varies from zero to a positive value, the iterative dynamics given by Eq. (3.51) undergoes a Hopf bifurcation, like as the ODEs behavior observed by Liu =-=[9, 10]-=-. The original stable slow manifold existent for γ = 0 now becomes a ghost manifold for γ = 0.05, and thus the iterative orbit generated from the algorithm with the case γ = 0.05 does not be attracted... |

4 |
A revision of relaxed steepest descent method from the dynamics on an invariant manifold
- Liu
(Show Context)
Citation Context ...r for ill-posed linear system [12], a modified polynomial expansion method [13], the nonstandard group preserving scheme [15], a vector regularization method [16], the relaxed steepest descent method =-=[11]-=-, as well as the optimal iterative algorithm [14]. A measure of the ill-posedness of Eq. (1.1) can be performed by using the condition number [22]: cond(A) = ∥A∥∥A1∥, (1.2) where ∥A∥ is the Frobenius... |

3 |
A study of type I intermittency of a circular differential equation under a discontinuous right-hand side
- Liu
(Show Context)
Citation Context ...nding to 107. Thus we can observe that when γ varies from zero to a positive value, the iterative dynamics given by Eq. (3.51) undergoes a Hopf bifurcation, like as the ODEs behavior observed by Liu =-=[9, 10]-=-. The original stable slow manifold existent for γ = 0 now becomes a ghost manifold for γ = 0.05, and thus the iterative orbit generated from the algorithm with the case γ = 0.05 does not be attracted... |

2 | An iterative method using an optimal descent vector, for solving an ill-conditioned system Bx = b, better and faster than the conjugate gradient method, CMES
- Liu, Atluri
(Show Context)
Citation Context ...lynomial expansion method [13], the nonstandard group preserving scheme [15], a vector regularization method [16], the relaxed steepest descent method [11], as well as the optimal iterative algorithm =-=[14]-=-. A measure of the ill-posedness of Eq. (1.1) can be performed by using the condition number [22]: cond(A) = ∥A∥∥A1∥, (1.2) where ∥A∥ is the Frobenius norm of A. For arbitrary ϵ > 0, there exists a m... |

2 | Novel algorithms based on the conjugate gradient method for inverting ill-conditioned matrices, and a new regularization method to solve ill-posed linear systems, CMES
- Liu, Hong, et al.
- 2010
(Show Context)
Citation Context ...titious time integration method as a filter for ill-posed linear system [12], a modified polynomial expansion method [13], the nonstandard group preserving scheme [15], a vector regularization method =-=[16]-=-, the relaxed steepest descent method [11], as well as the optimal iterative algorithm [14]. A measure of the ill-posedness of Eq. (1.1) can be performed by using the condition number [22]: cond(A) = ... |

1 |
On the Barzilai-Borwein gradient method
- Fletcher
- 2005
(Show Context)
Citation Context ... and Borwein [1] have initiated many researches on the SDM, for example, Raydan [18, 19], Friedlander et al. [6], Raydan and Svaiter [20], Dai et al. [3], Dai and Liao [2], Dai and Yuan [4], Fletcher =-=[5]-=-, and Yuan [26]. In this paper we will approach this problem from a quite different aspect of invariant manifold and bifurcation, and propose a new strategy to modify the steplength. We also compare o... |