ITERATIVE ALGORITHMS FOR THE SYMMETRIC AND LEAST-SQUARES SYMMETRIC SOLUTION OF A TENSOR EQUATION

2021 ◽  
Vol 50 (2) ◽  
pp. 179-212
Author(s):  
Qun Meng ◽  
Yu-Zhu Xie
Keyword(s):  
Least Squares ◽  
Symmetric Solution ◽  
Iterative Algorithms ◽  
Tensor Equation

scholarly journals On weighted Strand’s iteration

2018 ◽  
Vol 34 (2) ◽  
pp. 183-190
Author(s):  
D. CARP ◽  
◽  
C. POPA ◽  
T. PRECLIK ◽  
U. RUDE ◽  
...  
Keyword(s):  
Iterative Method ◽  
Least Squares ◽  
Numerical Approximation ◽  
Iterative Algorithms ◽  
Linear Least Squares ◽  
Minimal Norm ◽  
Least Squares Problem ◽  
Minimal Norm Solution ◽  
Linear Least Squares Problem

In this paper we present a generalization of Strand’s iterative method for numerical approximation of the weighted minimal norm solution of a linear least squares problem. We prove convergence of the extended algorithm, and show that previous iterative algorithms proposed by L. Landweber, J. D. Riley and G. H. Golub are particular cases of it.

Migration deconvolution using domain decomposition

Geophysics ◽  
2021 ◽  
pp. 1-61
Author(s):  
Luana Nobre Osorio ◽  
Bruno Pereira-Dias ◽  
André Bulcão ◽  
Luiz Landau
Keyword(s):  
Domain Decomposition ◽  
Least Squares ◽  
Iterative Algorithms ◽  
Image Resolution ◽  
Image Domain ◽  
Hessian Operator ◽  
Incomplete Coverage ◽  
And Migration ◽  
Least Squares Migration ◽  
Data Frequency

Least-squares migration (LSM) is an effective technique for mitigating blurring effects and migration artifacts generated by the limited data frequency bandwidth, incomplete coverage of geometry, source signature, and unbalanced amplitudes caused by complex wavefield propagation in the subsurface. Migration deconvolution (MD) is an image-domain approach for least-squares migration, which approximates the Hessian operator using a set of precomputed point spread functions (PSFs). We introduce a new workflow by integrating the MD and the domain decomposition (DD) methods. The DD techniques aim to solve large and complex linear systems by splitting problems into smaller parts, facilitating parallel computing, and providing a higher convergence in iterative algorithms. The following proposal suggests that instead of solving the problem in a unique domain, as conventionally performed, we split the problem into subdomains that overlap and solve each of them independently. We accelerate the convergence rate of the conjugate gradient solver by applying the DD methods to retrieve a better reflectivity, which is mainly visible in regions with low amplitudes. Moreover, using the pseudo-Hessian operator, the convergence of the algorithm is accelerated, suggesting that the inverse problem becomes better conditioned. Experiments using the synthetic Pluto model demonstrate that the proposed algorithm dramatically reduces the required number of iterations while providing a considerable enhancement in the image resolution and better continuity of poorly illuminated events.

An Iterative algorithm for $\eta$-(anti)-Hermitian least-squares solutions of quaternion matrix equations

2015 ◽  
Vol 30 ◽  
Author(s):  
Fatemeh Beik ◽  
Salman Ahmadi-Asl
Keyword(s):  
Least Squares ◽  
Iterative Algorithm ◽  
Numerical Experiments ◽  
Iterative Algorithms ◽  
Iterative Approach ◽  
Matrix Equations ◽  
Quaternion Matrix ◽  
Hermitian Matrices ◽  
Convergence Properties ◽  
Least Squares Problems

Recently, some research has been devoted to finding the explicit forms of the η-Hermitian and η-anti-Hermitian solutions of several kinds of quaternion matrix equations and their associated least-squares problems in the literature. Although exploiting iterative algorithms is superior than utilizing the explicit forms in application, hitherto, an iterative approach has not been offered for finding η-(anti)-Hermitian solutions of quaternion matrix equations. The current paper deals with applying an efficient iterative manner for determining η-Hermitian and η-anti-Hermitian least-squares solutions corresponding to the quaternion matrix equation AXB + CY D = E. More precisely, first, this paper establishes some properties of the η-Hermitian and η-anti-Hermitian matrices. These properties allow for the demonstration of how the well-known conjugate gradient least- squares (CGLS) method can be developed for solving the mentioned problem over the η-Hermitian and η-anti-Hermitian matrices. In addition, the convergence properties of the proposed algorithm are discussed with details. In the circumstance that the coefficient matrices are ill-conditioned, it is suggested to use a preconditioner for accelerating the convergence behavior of the algorithm. Numerical experiments are reported to reveal the validity of the elaborated results and feasibility of the proposed iterative algorithm and its preconditioned version.

Zur Gitterkonstantenbestimmung mit Ausgleichsmethoden

1979 ◽  
Vol 149 (3-4) ◽  
Author(s):  
Heike von Benda
Keyword(s):  
Least Squares ◽  
Iterative Algorithms ◽  
Lattice Parameters ◽  
The Other ◽  
Normal Equation ◽  
Linear Approach ◽  
Other Hand ◽  
Necessary And Sufficient

AbstractFor determination of lattice parameters by least squares methods iterative algorithms are often used for monoclinic and triclinic symmetries because of apparent nonlinearity of the normal equation. On the other hand, several authors have used straightforward linear solutions, but without proof of linearity. This paper gives conditions necessary and sufficient for the linearity of the problem. In practice, the direct linear approach is possible in general for the determination of lattice parameters.

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