A weighted randomized Kaczmarz method for solving linear systems
نویسندگان
چکیده
The Kaczmarz method for solving a linear system $Ax = b$ interprets such as collection of equations $\left \langle a_i, x\right \rangle b_i$, where $a_i$ is the $i$-th row $A$. It then picks an equation and corrects $x_{k+1} x_k + \lambda a_i$ $\lambda$ chosen so that satisfied. Convergence rates are difficult to establish. Strohmer & Vershynin established if order randomly (with likelihood proportional size $\|a_i\|^2_{\ell ^2}$), $\mathbb {E}~ \|x_k - x\|_{\ell ^2}$ converges exponentially. We prove selected with |\left \right b_i\right |^{p}$, $0<p<\infty$, {E}~\|x_k faster than purely random method. As $p \rightarrow \infty$, de-randomizes explains, among other things, why maximal correction works well.
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 2021
ISSN: ['1088-6842', '0025-5718']
DOI: https://doi.org/10.1090/mcom/3644