Rate of convergence of some space decomposition methods for linear and nonlinear problems. (English) Zbl 0915.65063

Tai, Xue-Cheng; Espedal, Magne

SIAM J. Numer. Anal. 35, No. 4, 1558-1570 (1998).

The authors consider the nonlinear problem \[ \text{minimize}\quad F(v)\quad\text{subject to}\quad v\in V,\tag{1} \] where \(V\) is a reflexive Banach space, and the functional \(F: V\to\mathbb{R}\) is differentiable and convex. They suppose that the space \(V\) can be decomposed into a sum of subspaces. Then two algorithms are proposed to solve the problem (1). One of the algorithms solves the minimization problems sequentially over each subspace, the other solves the minimization problems in parallel over all the subspaces. The convergence of the proposed algorithms is proved.

Reviewer: E.Duca (Cluj-Napoca)

MSC:

65K05	Numerical mathematical programming methods
90C30	Nonlinear programming
90C48	Programming in abstract spaces
65Y05	Parallel numerical computation
65J15	Numerical solutions to equations with nonlinear operators (do not use 65Hxx)

Keywords:

domain decomposition; elliptic equation; space decomposition; nonlinear programming in Banach space; parallel computation; convergence