  In this paper, we focus on efficient methods to solve discretized linear
systems obtained from eddy current optimal control problems in an all-at-once
approach. We construct a new low-rank matrix equation method based on a special
splitting of the coefficient matrix and the Krylov-plus-inverted-Krylov (KPIK)
algorithm. Firstly, we rewrite the resulting discretized linear system in a
matrix-equation form. Then using the KPIK algorithm, we can obtain the low-rank
approximation solution. The new method is named the splitting-based
Krylov-plus-inverted-Krylov (SKPIK) method. The SKPIK method can not only solve
the large and sparse discretized systems fast but also overcomes the storage
problem. Theoretical results about the existence of the low-rank solutions are
given. Numerical experiments are used to illustrate the performance of the new
low-rank matrix equation method by compared with some existing classical
efficient methods.
