Abstract
The Kaczmarz method is presented for solving saddle point systems. The convergence is analyzed. Numerical examples, compared with classical Krylov subspace methods, SOR-like method (2001) and recent modified SOR-like method (2014), show that the Kaczmarz algorithm is efficient in convergence rate and CPU time.
Highlights
A block 2 × 2 linear system of the form BBABABATA B0BBB xxxx yyyy = bqbqbqbq (1)where A∈Rm×m and B∈Rm×n, called a saddle point system, arises in a wide variety of technical and scientific applications such as constrained optimization, the finite element method to Stokes equations, fluid dynamics and weighted linear least squares problem [2]
The Kaczmarz algorithm is a popular iterative projection method [11] as it is simple to implement and the convergence is superior to classical splitting iterative methods such as SOR-like method
The Kaczmarz algorithm is presented for symmetric saddle point system (1) with numerical comparisons to classical Krylov subspace methods [5] and splitting iterative methods SOR-like [6] (2001) and modified SOR-like [12] (MSOR-like, 2014)
Summary
Where A∈Rm×m and B∈Rm×n, called a saddle point system, arises in a wide variety of technical and scientific applications such as constrained optimization, the finite element method to Stokes equations, fluid dynamics and weighted linear least squares problem [2]. In 2001, Golub, Wu and Yuan [6] proposed the SOR-like method for solving symmetric augmented linear system (1). Golub and Liesen [2] gave a review on numerical solution of saddle point problems in 2005. The Kaczmarz algorithm is a popular iterative projection method [11] as it is simple to implement and the convergence is superior to classical splitting iterative methods such as SOR-like method. The Kaczmarz algorithm is presented for symmetric saddle point system (1) with numerical comparisons to classical Krylov subspace methods [5] and splitting iterative methods SOR-like [6] (2001) and modified SOR-like [12] (MSOR-like, 2014).
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