Greedy randomized and maximal weighted residual Kaczmarz methods with oblique projection

For solving large-scale consistent linear system, a greedy randomized Kaczmarz method with oblique projection and a maximal weighted residual Kaczmarz method with oblique projection are proposed. By using oblique projection, these two methods greatly reduce the number of iteration steps and running time to find the minimum norm solution, especially when the rows of matrix are highly linearly correlated. Theoretical proof and numerical results show that the greedy randomized Kaczmarz method with oblique projection and the maximal weighted residual Kaczmarz method with oblique projection are more effective than the greedy randomized Kaczmarz method and the maximal weighted residual Kaczmarz method respectively.