H-Matrix approximation for the operator exponential with applications

96 I. P. Gavrilyuk et al.
of the order k ≪ N = dim V h for σ × τ ∈ P far , see [13]. The reduction
with respect to the operation count is achieved by replacing the full matrix
blocks A τ×σ (τ × σ ∈ P far ) by their low-rank approximation
k∑
H := a ν · c T ν , a ν ∈ R nτ , c ν ∈ R nσ ,
A τ×σ
ν=1
{ ∫ { ∫
where a ν =
X(τ) a ν(x)ϕ i (x)dx}
, c ν =
i∈τ
X(σ) c ν(y)ϕ j (y)dy}
.
j∈σ
Therefore, we obtain the following storage and matrix-vector multiplication
cost for the matrix blocks
( ) ( )
N st A
τ×σ
H = k(nτ + n σ ), N MV A
τ×σ
H =2k(nτ + n σ ),
where n τ =#τ, n σ =#σ. On the other hand, the approximation of the
order O(N −α ), α>0, is achieved with k = O(log d−1 N).
3.3 The error analysis
For the error analysis, we consider the uniform hierarchical cluster tree T (I)
(see [12,13] for more details) withthe depthL suchthat N =2 dL . Define
P (l)
2 := P 2 ∩ T2 l, where T 2 l is the set of clusters τ × σ ∈ T 2 suchthat blocks
τ,σ belong to level l, with l =0, 1,...,L. We consider the expansions
withthe local rank k l depending only on the level number l and defined by
k l := min{2 d(L−l) ,m d−1
l
}, where m = m l is given by
(3.4) m l = aL 1−q (L − l) q + b, 0 ≤ q ≤ 1, a,b > 0.
Note that for q =0, we arrive at the constant order m = O(L), which
leads to the exponential convergence of the H-matrix approximation, see
[16].
Introduce
{
N 0 = max max ∑
max
0≤l≤p
τ∈T (l)
τ:τ×σ∈P (l)
2
∑
}
1, max
1 .
σ∈T (l) σ:τ×σ∈P (l)
2
For the ease of exposition, we consider the only two special cases q =0
and q =1. Denote by A h : V h → V
h ′ the restriction of A onto the Galerkin
subspace V h ⊂ L 2 (Ω) defined by 〈A h u, v〉 = 〈Au, v〉 for all u, v ∈ V h .The
operator A H has the similar sense. The following statement is the particular
case of [15, Lemma 2.4].
Lemma 3.3 Let η =2 −α ,α>0, and
|s(x, y) − s τσ (x, y)| η m l
l 3−d dist(τ,σ) 2−d