Abstract. The Kaczmarz method is an iterative algorithm for solving systems of linear equations Ax = b. Theoretical convergence rates for this algorithm were largely unknown until recently when work was done on a randomized version of the algorithm. It was proved that for overdetermined systems, the randomized Kaczmarz method converges with expected exponential rate, independent of the number of equations in the system. Here we analyze the case where the system Ax = b is corrupted by noise, so we consider the system Ax ≈ b + r where r is an arbitrary error vector. We prove that in this noisy version, the randomized method reaches an error threshold dependent on the matrix A with the same rate as in the error-free case. We provide examples showing our results are sharp in the general context.
In particular, Needell demonstrates that the randomized Kaczmarz method converges, even when the linear system is inconsistent REF .
Randomized Kaczmarz solver for noisy linear systems
BIT Numerical Mathematics, 2010. The original publication is available at www.springerlink.com.
Mathematics