The Experts below are selected from a list of 7689 Experts worldwide ranked by ideXlab platform
Chein-shan Liu - One of the best experts on this subject based on the ideXlab platform.
-
A double optimal Descent algorithm for iteratively solving ill-posed linear inverse problems
Inverse Problems in Science and Engineering, 2014Co-Authors: Chein-shan LiuAbstract:In the iterative solution of an ill-posed linear system , how to select a fast and easily established Descent Direction to reduce the residual is an important issue. A mathematical procedure to find a double optimal Descent Direction in , without inverting , is developed in an -dimensional Krylov subspace. The novelty is that we expand in an affine Krylov subspace with undetermined coefficients, and then two optimization techniques are used to determine these coefficients in closed form, which can greatly accelerate the convergence speed in solving the ill-posed linear problems. The double optimal Descent algorithm is proven to be absolutely convergent very fast, accurate and robust against noise, which is confirmed by numerical tests of several linear inverse problems, including the heat source identification problem, the backward heat conduction problem, the inverse Cauchy problem and the external force identification problem.
-
A vector regularization method to solve linear inverse problems
Inverse Problems in Science and Engineering, 2013Co-Authors: Chein-shan LiuAbstract:The linear inverse problem is discretized to be an n-dimensional ill-posed linear equations system . In the present paper, an invariant manifold defined in terms of the square norm of a residual vector is used to derive an iterative algorithm with a fast Descent Direction , which is close to, but not exactly equal to, the best Descent Direction . The matrix is obtained by using a vector regularization method together with a matrix conjugate gradient method to find the right inversion of : . The vector regularization iterative algorithm is proven to be Lyapunov stable, and the direct inversion method with solution expressed by converges fast. The accuracy and efficiency of them are verified through the numerical tests of linear inverse problems under a large random noise.
-
An Optimally Generalized Steepest-Descent Algorithm for Solving Ill-Posed Linear Systems
Journal of Applied Mathematics, 2013Co-Authors: Chein-shan LiuAbstract:It is known that the steepest-Descent method converges normally at the first few iterations, and then it slows down. We modify the original steplength and Descent Direction by an optimization argument with the new steplength as being a merit function to be maximized. An optimal iterative algorithm with -vector Descent Direction in a Krylov subspace is constructed, of which the optimal weighting parameters are solved in closed-form to accelerate the convergence speed in solving ill-posed linear problems. The optimally generalized steepest-Descent algorithm (OGSDA) is proven to be convergent with very fast convergence speed, accurate and robust against noisy disturbance, which is confirmed by numerical tests of some well-known ill-posed linear problems and linear inverse problems.
Dong-joo Min - One of the best experts on this subject based on the ideXlab platform.
-
Comparison of scaling methods for waveform inversion
Geophysical Prospecting, 2009Co-Authors: Ugeun Jang, Dong-joo Min, Changsoo ShinAbstract:Waveform inversion can lead to faint images for later times due to geometrical spreading. The proper scaling of the steepest-Descent Direction can enhance faint images in waveform inversion results. We compare the effects of different scaling techniques in waveform inversion algorithms using the steepest-Descent method. For the scaling method we use the diagonal of the pseudo-Hessian matrix, which can be applied in two different ways. One is to scale the steepest-Descent Direction at each frequency independently. The other is to scale the steepest-Descent Direction summed over the entire frequency band. The first method equalizes the steepest-Descent Directions at different frequencies and minimizes the effects of the band-limited source spectrum in waveform inversion. In the second method, since the steepest-Descent Direction summed over the entire frequency band is divided by the diagonal of the pseudo-Hessian matrix summed over the entire frequency band, the band-limited property of the source wavelet spectrum still remains in the scaled steepest-Descent Directions. The two scaling methods were applied to both standard and logarithmic waveform inversion. For standard waveform inversion, the method that scales the steepest-Descent Direction at every frequency step gives better results than the second method. On the other hand, logarithmic waveform inversion is not sensitive to the scaling method, because taking the logarithm of wavefields automatically means that results for the steepest-Descent Direction at each frequency are commensurate with each other. If once the steepest-Descent Directions are equalized by taking the logarithm of wavefields in logarithmic waveform inversion, the additional equalizing effects by the scaling method are not as great as in conventional waveform inversion.
-
Efficient calculation of the steepest Descent Direction for source-independent seismic waveform inversion: An amplitude approach
Journal of Computational Physics, 2005Co-Authors: Yunseok Choi, Changsoo Shin, Dong-joo MinAbstract:In seismic waveform inversion, if we have no information on source signature, we need to invert seismic data and source signature either simultaneously or successively. In order to avoid the iterative update of the source signature in waveform inversion based on classical, local optimization techniques, we propose two source-independent objective functions using amplitude spectra of Fourier-transformed wavefields. One is constructed by normalizing the amplitude spectra of observed data and modeled data with respect to the respective reference amplitudes. The other is achieved by cross-multiplying the amplitude spectra of observed data and modeled data with the respective reference amplitudes. In the computation of the steepest Descent Direction, we circumvent explicitly computing the Jacobian by employing a matrix formalism of the wave equation in the frequency domain. Through numerical examples for the Marmousi model, we demonstrate that our inversion algorithms can reproduce the subsurface velocity structure without estimating source signature.
Changsoo Shin - One of the best experts on this subject based on the ideXlab platform.
-
Comparison of scaling methods for waveform inversion
Geophysical Prospecting, 2009Co-Authors: Ugeun Jang, Dong-joo Min, Changsoo ShinAbstract:Waveform inversion can lead to faint images for later times due to geometrical spreading. The proper scaling of the steepest-Descent Direction can enhance faint images in waveform inversion results. We compare the effects of different scaling techniques in waveform inversion algorithms using the steepest-Descent method. For the scaling method we use the diagonal of the pseudo-Hessian matrix, which can be applied in two different ways. One is to scale the steepest-Descent Direction at each frequency independently. The other is to scale the steepest-Descent Direction summed over the entire frequency band. The first method equalizes the steepest-Descent Directions at different frequencies and minimizes the effects of the band-limited source spectrum in waveform inversion. In the second method, since the steepest-Descent Direction summed over the entire frequency band is divided by the diagonal of the pseudo-Hessian matrix summed over the entire frequency band, the band-limited property of the source wavelet spectrum still remains in the scaled steepest-Descent Directions. The two scaling methods were applied to both standard and logarithmic waveform inversion. For standard waveform inversion, the method that scales the steepest-Descent Direction at every frequency step gives better results than the second method. On the other hand, logarithmic waveform inversion is not sensitive to the scaling method, because taking the logarithm of wavefields automatically means that results for the steepest-Descent Direction at each frequency are commensurate with each other. If once the steepest-Descent Directions are equalized by taking the logarithm of wavefields in logarithmic waveform inversion, the additional equalizing effects by the scaling method are not as great as in conventional waveform inversion.
-
Efficient calculation of the steepest Descent Direction for source-independent seismic waveform inversion: An amplitude approach
Journal of Computational Physics, 2005Co-Authors: Yunseok Choi, Changsoo Shin, Dong-joo MinAbstract:In seismic waveform inversion, if we have no information on source signature, we need to invert seismic data and source signature either simultaneously or successively. In order to avoid the iterative update of the source signature in waveform inversion based on classical, local optimization techniques, we propose two source-independent objective functions using amplitude spectra of Fourier-transformed wavefields. One is constructed by normalizing the amplitude spectra of observed data and modeled data with respect to the respective reference amplitudes. The other is achieved by cross-multiplying the amplitude spectra of observed data and modeled data with the respective reference amplitudes. In the computation of the steepest Descent Direction, we circumvent explicitly computing the Jacobian by employing a matrix formalism of the wave equation in the frequency domain. Through numerical examples for the Marmousi model, we demonstrate that our inversion algorithms can reproduce the subsurface velocity structure without estimating source signature.
Yixun Shi - One of the best experts on this subject based on the ideXlab platform.
-
A projected‐steepest‐Descent potential‐reduction algorithm for convex programming problems
Numerical Linear Algebra with Applications, 2004Co-Authors: Yixun ShiAbstract:A recent work of Shi (Numer. Linear Algebra Appl. 2002; 9: 195–203) proposed a hybrid algorithm which combines a primal-dual potential reduction algorithm with the use of the steepest Descent Direction of the potential function. The complexity of the potential reduction algorithm remains valid but the overall computational cost can be reduced. In this paper, we make efforts to further reduce the computational costs. We notice that in order to obtain the steepest Descent Direction of the potential function, the Hessian matrix of second order partial derivatives of the objective function needs to be computed. To avoid this, we in this paper propose another hybrid algorithm which uses a projected steepest Descent Direction of the objective function instead of the steepest Descent Direction of the potential function. The complexity of the original potential reduction algorithm still remains valid but the overall computational cost is further reduced. Our numerical experiments are also reported. Copyright © 2004 John Wiley & Sons, Ltd.
-
On the projected Descent Direction methods for solving convex programming problems
Neural Parallel & Scientific Computations archive, 2003Co-Authors: Yixun ShiAbstract:A recent paper [14] has considered the possibility of combining interior point strategy with steepest Descent method when solving convex programming problems, in such a way that the convergence property of the interior point method remains valid but many iterations do not request the solution of a system of equations. Motivated by this general idea, the paper [14] proposed a hybrid algorithm which combines a primal-dual potential reduction algorithm with the use of the steepest Descent Direction of the potential function. The O(√n|ln e|) complexity of the potential reduction algorithm remains valid but the overall computational cost can be reduced. In this paper, we discuss the relation between this method and general projected Descent Direction methods, and compare it with a projected steepest Descent Direction method for solving complex programming problems.
-
Globally Convergent Algorithms for Unconstrained Optimization
Computational Optimization and Applications, 2000Co-Authors: Yixun ShiAbstract:A new globalization strategy for solving an unconstrained minimization problem is proposed based on the idea of combining Newton's Direction and the steepest Descent Direction WITHIN each iteration. Global convergence is guaranteed with an arbitrary initial point. The search Direction in each iteration is chosen to be as close to the Newton's Direction as possible and could be the Newton's Direction itself. Asymptotically the Newton step will be taken in each iteration and thus the local convergence is quadratic. Numerical experiments are also reported. Possible combination of a Quasi-Newton Direction with the steepest Descent Direction is also considered in our numerical experiments. The differences between the proposed strategy and a few other strategies are also discussed.
Xiao-ming Yuan - One of the best experts on this subject based on the ideXlab platform.
-
An improved proximal alternating Direction method for monotone variational inequalities with separable structure
Computational Optimization and Applications, 2009Co-Authors: Xiao-ming YuanAbstract:To solve a class of variational inequalities with separable structure, this paper presents a new method to improve the proximal alternating Direction method (PADM) in the following senses: an iterate generated by the PADM is utilized to generate a Descent Direction; and an appropriate step size along this Descent Direction is identified. Hence, a Descent-like method is developed. Convergence of the new method is proved under mild assumptions. Some numerical results demonstrate that the new method is efficient.
-
An improved Goldstein's type method for a class of variant variational inequalities
Journal of Computational and Applied Mathematics, 2008Co-Authors: Xiao-ming YuanAbstract:This paper aims at presenting an improved Goldstein's type method for a class of variant variational inequalities. In particular, the iterate computed by an existing Goldstein's type method [He, A Goldstein's type projection method for a class of variant variational inequalities J. Comput. Math. 17(4) (1999) 425-434]. is used to construct a Descent Direction, and thus the new method generates the new iterate by searching the optimal step size along the Descent Direction. Some restrictions on the involving functions of the existing Goldstein's type methods are relaxed, while the global convergence of the new method is proved without additional assumptions. The computational superiority of the new method is verified by the comparison to some existing methods.
-
A Descent method for structured monotone variational inequalities
Optimization Methods and Software, 2007Co-Authors: Xiao-ming YuanAbstract:This article presents a Descent method for solving monotone variational inequalities with separate structures. The Descent Direction is derived from the well-known alternating Directions method. The optimal step size along the Descent Direction also improves the efficiency of the new method. Some numerical results demonstrate that the new method is effective in practice.