the Landweber Method is imho nothing more than gradient descent on a specific quadratic form.
https://en.wikipedia.org/wiki/Landweber_iteration
so why does it warrant an own name?
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Just speculating, but maybe when it was introduced there was less awareness of gradient descent so the equivalence wasn't widely recognized. Or maybe when discussing methods for solving $Ax=b$ people needed a shorter name than "using gradient descent to minimize the norm of the residual". – littleO Apr 21 '18 at 10:40