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Register a new userA Novel Greedy Kaczmarz Method For Solving Consistent Linear Systems
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Hanyu Li Dr.
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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With a quite different way to determine the working rows, we propose a novel greedy Kaczmarz method for solving consistent linear systems. Convergence analysis of the new method is provided. Numerical experiments show that, for the same accuracy, our method outperforms the greedy randomized Kaczmarz method and the relaxed greedy randomized Kaczmarz method introduced recently by Bai and Wu [Z.Z. BAI AND W.T. WU, On greedy randomized Kaczmarz method for solving large sparse linear systems, SIAM J. Sci. Comput., 40 (2018), pp. A592--A606; Z.Z. BAI AND W.T. WU, On relaxed greedy randomized Kaczmarz methods for solving large sparse linear systems, Appl. Math. Lett., 83 (2018), pp. 21--26] in term of the computing time.
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The famous greedy randomized Kaczmarz (GRK) method uses the greedy selection rule on maximum distance to determine a subset of the indices of working rows. In this paper, with the greedy selection rule on maximum residual, we propose the greedy randomized Motzkin-Kaczmarz (GRMK) method for linear systems. The block version of the new method is also presented. We analyze the convergence of the two methods and provide the corresponding convergence factors. Extensive numerical experiments show that the GRMK method has almost the same performance as the GRK method for dense matrices and the former performs better in computing time for some sparse matrices, and the blo
In this paper, combining count sketch and maximal weighted residual Kaczmarz method, we propose a fast randomized algorithm for large overdetermined linear systems. Convergence analysis of the new algorithm is provided. Numerical experiments show that, for the same accuracy, our method behaves better in computing time compared with the state-of-the-art algorithm.
We propose a deterministic Kaczmarz method for solving linear systems $Ax=b$ with $A$ nonsingular. Instead of using orthogonal projections, we use reflections in the original Kaczmarz iterative method. This generates a series of points on an $n$-sphere $S$ centered at the solution $x_*=A^{-1}b$. We show that these points are nicely distributed on $S$. Taking the average of several points will lead to an effective approximation to the solution. We will show how to choose these points efficiently. The numerical tests show that in practice this deterministic scheme converges much faster than we expected and can beat the (block) randomized Kaczmarz methods.
We present a novel greedy Gauss-Seidel method for solving large linear least squares problem. This method improves the greedy randomized coordinate descent (GRCD) method proposed recently by Bai and Wu [Bai ZZ, and Wu WT. On greedy randomized coordinate descent methods for solving large linear least-squares problems. Numer Linear Algebra Appl. 2019;26(4):1--15], which in turn improves the popular randomized Gauss-Seidel method. Convergence analysis of the new method is provided. Numerical experiments show that, for the same accuracy, our method outperforms the GRCD method in term of the computing time.
The sampling Kaczmarz-Motzkin (SKM) method is a generalization of the randomized Kaczmarz and Motzkin methods. It first samples some rows of coefficient matrix randomly to build a set and then makes use of the maximum violation criterion within this set to determine a constraint. Finally, it makes progress by enforcing this single constraint. In this paper, on the basis of the framework of the SKM method and considering the greedy strategies, we present two block sampling Kaczmarz-Motzkin methods for consistent linear systems. Specifically, we also first sample a subset of rows of coefficient matrix and then determine an index in this set using the maximum violation criterion. Unlike the SKM method, in the rest of the block methods, we devise different greedy strategies to build index sets. Then, the new methods make progress by enforcing the corresponding multiple constraints simultaneously. Theoretical analyses demonstrate that these block methods converge at least as quickly as the SKM method, and numerical experiments show that, for the same accuracy, our methods outperform the SKM method in terms of the number of iterations and computing time.
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