336 Designing Scientific Applications on GPUs
Note that A
i
.U =
α
X
j=1
A
i,j
.U
j
, where A
i,j
denote block matrices of A.
The parallel asynchronous iterations of the projected Richardson method
for solving the obstacle problem (14.8) are defined as follows: let U
0
∈ E, U
0
=
(U
0
1
, . . . , U
0
α
) be the initial solution, then for all p ∈ N, the iterate U
p+1
=
(U
p+1
1
, . . . , U
p+1
α
) is recursively defined by
U
p+1
i
=
(
F
i,γ
(U
ρ
1
(p)
1
, . . . , U
ρ
α
(p)
α
) if i ∈ s(p),
U
p
i
otherwise,
(14.13)
where
∀p ∈ N, s(p) ⊂ {1, . . . , α} and s(p) 6= ∅,
∀i ∈ {1, . . . , α}, {p | i ∈ s(p)} is enumerable,
(14.14)
and ∀j ∈ {1, . . . , α},
(
∀p ∈ N, ρ
j
(p) ∈ N, 0 ≤ ρ
j
(p) ≤ p and ρ
j
(p) = p if j ∈ s(p),
lim
p→∞
ρ
j
(p) = +∞.
(14.15)
The previous asynchronous scheme of the projected Richardson method
models computations ...