H-Matrix approximation for the operator exponential with applications

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104 I. P. Gavrilyuk et al. 5 On the choice of computational parameters and numerics In this section we discuss how the parameters of the parabola influence our method. Let Γ 0 = {z = ξ ± iη = aη 2 + γ 0 ± iη} be the parabola containing the spectrum of L whose parameters a, γ 0 are determined by the coefficients of L. Given a, we choose the integration path Γ b(k) = {z = ξ 1 ± iη 1 = a k η2 1 + b ± iη 1 : η 1 ∈ (−∞, ∞)} with b(k) =γ 0 − k−1 4a . In this case ( the integrand ) can be extended analytically into the strip D d with d = 1 − √ 1 k k 2a and the estimate (2.18) holds with constants given by (2.19). First, let us estimate the constant M 1 in (2.19). Since the absolute value of an analytic function attains its maximum on the boundary, we have { } (5.1) M 1 = max where (5.2) f ± (η) = It is easy to see that sup η∈(−∞,∞) f − (η), sup η∈(−∞,∞) f + (η) |2 a k (η ± id) − i| ∣∣ a 1+√ k (η ± id)2 + b − i(η ± id) ∣ . (5.3) f± 2 (2 a k ≤ η)2 +(2 a k d ∓ 1)2 1+ ∣ a k (η2 ± 2ηdi − d 2 )+b + d ∓ iη ∣ . Further we have for the function f + f+(η) 2 ≤ 4 a2 η 2 +(2 a k 2 k d − 1)2 1+ √ ( a k (η2 − d 2 )+b + d) 2 +(2 a k d − 1)2 η 2 (5.4) ≤ 4 a2 η 2 +(2 a k 2 k d − 1)2 1+ a k (η2 − d 2 )+b + d = 4 a2 η 2 + 1 k 2 k a k η2 +1+γ 0 = 4a +4γ 0 − 1 k(ξ +1+γ 0 ) 2 , where ξ = aη 2 /k, ξ ∈ (0, ∞). The latter function increases monotonically and max f + = f + (∞) = 4a/k provided that 4a +4γ 0 > 1, while it decreases monotonically and max f + = f + (0) = 1 k(1+γ 0 ) provided that 0 < 4a +4γ 0 < 1. Similarly one can see that max f − = f − (∞) =4a/k provided that 4a(1 − γ 0 )/k > (2 − 1/k) 2 , while max f − = f − (0) = (2 − 1/ √ k) 2 /(1 + γ 0 ) if 4a(1 − γ 0 )/k < (2 − 1/k) 2 . Thus, we have { } 4a (5.5) M 1 ≤ max k , 1 k(1 + γ 0 ) , (2 − 1/√ k) 2 . 1+γ 0 ,
H-Matrix approximation for the operator exponential with applications 105 Other constants can be rewrite as (5.6) c = M 1 e t[ad2 /k+d−b] = M 1 e tk(1−1/√ k)/a , s = 3 √π 2 k(1 − 1/ √ k) 2 /a ≍ 3√ k/a. One can see that the multiplicative constant c in the estimate (2.18) increases linearly as a →∞, whereas the multiplicative coefficient s in the front of (N +1) 2/3 in the exponent tends to zero, i.e., the convergence rate becomes worse. On the other hand, s tends to infinity as a tends to zero, but the constant c tends exponentially to infinity. Given a, we can influence the efficacy of our method by changing the parameter k>1. Denoting t ∗ = min {1,t}, we see that for k large enough, the leading term of the error is (5.7) e −[ 3√ k/at ∗(N+1) 2/3 −tk/a] . In order to arrive at a given tolerance ε>0, we have to choose ( 1 N ≍ ln 1 ) 3/2 mt ∗ ε + tm2 /t ∗ , where m = 3√ k/a. It is easy to find that N (i.e., the number of resolvent inversions for various z i ) becomes minimal if we choose k ≍ a 2t ln 1 ε . In this case the number of resolvent inversions is estimated by N min ≍ 3√ ( t/t ∗ ln 1 ) 2/3 . ε To complete this section, we present numerical examples on the H- matrix approximation of the exponential for the finite difference Laplacian ∆ h on Ω =(0, 1) d , d =1, 2 (withzero boundary conditions) defined on the uniform grid with the mesh-size h =1/(n +1), where n d is the problem size. Table 4 presents the relative error of the H-matrix approximation for 1D Laplacian by Algorithm 2.3 versus the number N of resolvents involved. The relative error is measured by ‖ exp(−t∆ h ) − exp N (−t∆ h )‖ 2 /‖ exp(−t∆ h )‖ 2 . The local rank is chosen as k 0 =8, while b =0.9 λ min (∆ h ),a=4.0, k = 5.0. Our calculations indicate the robust exponential convergence of (2.18) withrespect to N for the range of parameters b ∈ (0, 0.95λ min (∆ h )) and a ∈ (0,a 0 ) with a 0 = O(1) and confirm our analysis. The computational time (in sec.) corresponding to the H-matrix evaluation of eachresolvent in (2.18) at a 450MHz SUN-UltraSPARC2 station is presented in the last
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H-Matrix approximation for the operator exponential with applications

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