The Randomized Kaczmarz Method with Mismatched Adjoint

TL;DR

This paper extends the randomized Kaczmarz method to cases with mismatched adjoint matrices, providing convergence analysis, rate calculations, and optimized sampling strategies, supported by numerical experiments.

Contribution

It introduces a convergence analysis for the randomized Kaczmarz method with mismatched adjoint matrices and proposes optimized probability computations.

Findings

The method converges under certain conditions even with mismatched adjoint.

Expected linear convergence rates are derived.

Numerical examples demonstrate the effectiveness of the optimized probabilities.

Abstract

This paper investigates the randomized version of the Kaczmarz method to solve linear systems in the case where the adjoint of the system matrix is not exact---a situation we refer to as "mismatched adjoint". We show that the method may still converge both in the over- and underdetermined consistent case under appropriate conditions, and we calculate the expected asymptotic rate of linear convergence. Moreover, we analyze the inconsistent case and obtain results for the method with mismatched adjoint as for the standard method. Finally, we derive a method to compute optimized probabilities for the choice of the rows and illustrate our findings with numerical example.