Elliptic Variational Inequalities - Computational Methods for Quantitative Finance - page 272

Let ρ> 0 be a parameter. Then ( A.22 ) is equivalent to
Find u ∈ K such that

(A.23)

ρf − ρBu + u − u, v − u V ≤ 0 ∀ v ∈ K ,

or to the fixed point problem

u = P K (ρf − ρBu + u) in K . (A.24)

Define, for v ∈ K , S v = P (v − ρ(Bv − f)) . Then, by ( A.6 ), for every

K

v 1 ,v 2 ∈ K ,

S v 1 − S v 2 V ≤ (v 1 − v 2 ) − ρB(v 1 − v 2 ) V ,

and hence

2 2

ρ 2

2
V

S v 1 − S v 2

V ≤

v 1 − v 2

V −

2 ρ B(v 1 − v 2 ), v 1 − v 2 V + B(v 1 − v 2 )

2
V

2 ρC 2 + ρ 2 C 1 ). ≤ v 1 − v 2 ( 1 −

2 ρC 2 + ρ 2 C 1 ) 2 < 1. Then

We fix ρ> 0 such that κ := ( 1 − S is a contraction on

K and ( A.24 ) has a unique solution.

Remark A.3.4 The proof is constructive—the contraction S converges with rate κ<

1. The rate is maximal, if

ρ 2 C 1 }= C 2 /C 1 , ρ = ρ

∗ =

arg min { 1 − 2 ρC 2 + (A.25)

then

C 2 /C 1 ) 2 < 1 .

κ = κ

∗ =

( 1 − (A.26)

In particular, if v ( 0 )

∈ K is arbitrary, the sequence

v (i + 1 )

v (i)

f) v (i) ρ(Bv (i) = S = P − − (A.27)

K

converges to u ∈ K , the solution of ( A.17 ), and

v (i) Cκ i v ( 0 ) u − V ≤

∗

u − V ,i ≥ 0 . (A.28)

Corollary A.3.5 If M ⊆ V is a closed , linear subspace , the problem

∈ M ≥ ∀ ∈ M

Find u such that b(u, v) (v) v

admits a unique solution .