Journal ArticleDOI
An iterative procedure for the solution of constrained nonlinear equations with application to optimization problems
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In this paper, the convergence of a general iterative technique for finding the zeros of G:D?H 2 subject to the general constraintP(x)=x, whereP:D?D is established.Abstract:
LetH 1 andH 2 denote Hilbert spaces and suppose thatD is a subset ofH 1. This paper establishes the local and linear convergence of a general iterative technique for finding the zeros ofG:D?H 2 subject to the general constraintP(x)=x, whereP:D?D. The results are then applied to several classes of problems, including those of least squares, generalized eigenvalues, and constrained optimization. Numerical results are obtained as the procedure is applied to finding the zeros of polynomials in several variables.read more
Citations
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Journal ArticleDOI
Randomized Extended Kaczmarz for Solving Least Squares
TL;DR: A randomized iterative algorithm that exponentially converges in the mean square to the minimum $\ell_2$-norm least squares solution of a given linear system of equations.
Journal ArticleDOI
Randomized Extended Kaczmarz for Solving Least-Squares
TL;DR: In this paper, a randomized iterative algorithm that exponentially converges in expectation to the minimum Euclidean norm least squares solution of a given linear system of equations is presented, where the expected number of arithmetic operations required to obtain an estimate of given accuracy is proportional to the square condition number of the system multiplied by the number of non-zeros entries of the input matrix.
Journal ArticleDOI
On Greedy Randomized Kaczmarz Method for Solving Large Sparse Linear Systems
Zhong-Zhi Bai,Wen-Ting Wu +1 more
TL;DR: For solving large-scale systems of linear equations by iteration methods, an effective probability criterion for selecting the working rows from the coefficient matrix is introduced.
Book ChapterDOI
Convergence Rates Results for Iterative Methods for Solving Nonlinear Ill-Posed Problems
Heinz W. Engl,Otmar Scherzer +1 more
TL;DR: An overview over Tikhonov regularization and iterative regularization techniques is given and the analysis of iterative methods for the solution of well-posed problems is put into perspective.
Journal ArticleDOI
A greedy block Kaczmarz algorithm for solving large-scale linear systems
Yu-qi Niu,Bing Zheng +1 more
TL;DR: A greedy block Kaczmarz algorithm is proposed by using a greedy strategy to construct control index set and choosing row submatrix in each iteration, which has a linear rate of convergence that can be expressed in terms of the geometric properties of the matrix and itsRow submatrices.
References
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Journal ArticleDOI
A unified convergence theory for a class of iterative processes.
TL;DR: Unified theory for convergence results based on nonlinear estimates for iteration function and on majorizing sequences concept is presented in this article, where the convergence results are based on the nonlinear estimate of the iteration function.
Journal ArticleDOI
A gradient method for the matrix eigenvalue problemAx=λBx
TL;DR: In this paper, a gradient method for the minimization of the functional F(x)=?Ax?(?Ax, Bx?/?Bx, bx?)Bx?2 is developed.