The Dual Kaczmarz Algorithm

Autor: Mary K. Vaughan, Anna Aboud, Steven N. Harding, Eric Weber, Emelie Curl
Rok vydání: 2019
Předmět:
Zdroj: Acta Applicandae Mathematicae. 165:133-148
ISSN: 1572-9036
0167-8019
DOI: 10.1007/s10440-019-00244-6
Popis: The Kaczmarz algorithm is an iterative method for solving a system of linear equations. It can be extended so as to reconstruct a vector $x$ in a (separable) Hilbert space from the inner-products $\{\langle x, \phi_{n} \rangle\}$. The Kaczmarz algorithms defines a sequence of approximations from the sequence $\{\langle x, \phi_{n} \rangle\}$; these approximations only converge to $x$ when $\{\phi_{n}\}$ is ${effective}$. We dualize the Kaczmarz algorithm so that $x$ can be obtained from $\{\langle x, \phi_{n} \rangle\}$ by using a second sequence $\{\psi_{n}\}$ in the reconstruction. This allows for the recovery of $x$ even when the sequence $\{\phi_{n}\}$ is not effective; in particular, our dualization yields a reconstruction when the sequence $\{\phi_{n}\}$ is $almost$ $effective$. We also obtain some partial results characterizing when the sequence of approximations from $\{\langle x, \phi_{n} \rangle\}$ using $\{\psi_{n}\}$ converges to $x$, in which case $\{(\phi_n, \psi_n)\}$ is called an $effective$ $pair$.
Comment: 15 pages
Databáze: OpenAIRE
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