Abstract

CHAPTER 4. THEORY 104
operator.
4.2 Infinite-Dimesional Analysis of GMRES
In this section, we will consider solving the infinite-dimension fixed point problem
f(x, k) =Z(f)(x, k)+ā(x, k). This can be rewritten as (I − Z)f(x, k) − ā(x, k) =0.
Applying Newton’s method to this nonlinear equation leads to solving a series of
linear equations, where the linear operator needed to be inverted is I − Z ′ . Here, Z ′
is the Frechet derivative of Z. Using the compactness of Z, we will now show that Z ′
is a compact linear map from X to X. Letu ∈ X and {gn} be a bounded sequence
in X. Since{gn} is bounded in X, then{gn} is bounded in L 2 .SinceL 2 is a reflexive
space, then there is a subsequence of {gn} (which we will relabel as {gn}) such that
gn ⇀ ¯g ∈ L 2 .SinceZ is Frechet differentiable, then
Therefore,
So,
Z(u + γ¯g) − Z(f) − γZ ′ (u)¯g = o(γ)
Z(u + γgm) − Z(u + γgn) − γZ ′ (u)[gm − gn] =o(γ).
Z ′ (u)[gm − gn] = 1
γ [Z(u + γgm) − Z(u + γgn)+o(γ)].