H-Matrix approximation for the operator exponential with applications

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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
H-Matrix approximation for the operator exponential with applications 97 for each τ × σ ∈ P (l) 2 , where the order of expansion m l is defined by (3.4) with q =0, 1 and with a given a>0 such that −αa +2< 0. Then, for all u, v ∈ V h there holds (3.5) 〈(A h − A H )u, v〉 h 2 N 0 δ(L, q)||u|| 0 ||v|| 0 , where δ(L, 0) = η L and δ(L, 1)=1and d =2, 3. Note that in the case of constant order expansions, i.e., for q =0,we obtain the exponential convergence 〈(A h − A H )u, v〉 N 0 L 4−d η L ||u|| 0 ||v|| 0 (u, v ∈ V h ) for any a in (3.4). The first important consequence of Lemma 3.3 is that for the variable order expansions with q =1the asymptotically optimal convergence is verified only for trial functions from L 2 (Ω). On the other hand, the exponential convergence in the operator norm ||·|| H −1 →H1 may be proven for any 0 ≤ q
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H-Matrix approximation for the operator exponential with applications

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