Addendum: On stochastic Kaczmarz type methods for solving large scale systems of ill-posed equations (2022 Inverse Problems 38 025003)

Abstract

Abstract We address the convergence analysis derived in (Rabelo et al 2022 Inverse Problems 38 025003) for the sPLWK method; a SGD type method for solving large scale systems of ill-posed equations. The assumption constraining the growth rate of the stopping index function k * : δ ↦ k * ( δ ) ∈ N is removed; this assumption was needed in the proof of the semi-convergence result. The most important consequence of our findings is the fact that, what semi-convergence concerns, in the sPLWK method the growth rate of k*(δ), as δ goes to zero, is independent of the decay rate of the noise level. This is in strong contrast to the deterministic theory, where one needs additional assumptions of the type lim δ→0‖δ‖2 k*(δ) = 0 for many iterative schemes, i.e., the stopping index should not grow to fast.