  The Kaczmarz method is a popular iterative scheme for solving large-scale
linear systems. The randomized Kaczmarz method (RK) greatly improves the
convergence rate of the Kaczmarz method, by using the rows of the coefficient
matrix in random order rather than in their given order. An obvious
disadvantage of the randomized Kaczmarz method is its probability criterion for
selecting the active or working rows in the coefficient matrix. In [{\sc Z.Z.
Bai, W. Wu}, {\em On greedy randomized Kaczmarz method for solving large sparse
linear systems}, SIAM Journal on Scientific Computing, 2018, 40: A592--A606],
the authors proposed a greedy randomized Kaczmarz method (GRK). However, this
method may suffer from heavily computational cost when the size of the matrix
is large, and the overhead will be prohibitively large for big data problems.
The contribution of this work is as follows. First, from the probability
significance point of view, we present a partially randomized Kaczmarz method,
which can reduce the computational overhead needed in greedy randomized
Kaczmarz method. Second, based on Chebyshev's law of large numbers and Z-test,
we apply a simple sampling approach to the partially randomized Kaczmarz
method. The convergence of the proposed method is established. Third, we apply
the new strategy to the ridge regression problem, and propose a partially
randomized Kaczmarz method with simple random sampling for ridge regression.
Numerical experiments demonstrate the superiority of the new algorithms over
many state-of-the-art randomized Kaczmarz methods for large linear systems
problems and ridge regression problems.
