Deriving the X-Z Identity from Auxiliary Space Method*

Abstract

posed on a finite dimensional Hilbert space V equipped with an inner product (·, ·). Here A : V 7→ V is a symmetric positive definite (SPD) operator, f ∈ V is given, and we are looking for u ∈ V such that (1) holds. The X-Z identity for the multiplicative subspace correction method for solving (1) is introduced and proved in Xu and Zikatanov [2002]. Alternative proves can be found in Cho et al. [2008] and Vassilevski [2008]. In this paper we derive the X-Z identity from the auxiliary space method Nepomnyaschikh [1992], Xu [1996]. A basic linear iterative method for solving (1) can be written in the following form: starting from an initial guess u, for k = 0, 1, 2, · · ·