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H-Matrix approximation for the operator exponential with applications

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H-<strong>Matrix</strong> <strong>approximation</strong> <strong>for</strong> <strong>the</strong> <strong>operator</strong> <strong>exponential</strong> <strong>with</strong> <strong>applications</strong> 93<br />

Remark 2.6 The above algorithm possesses two sequential levels of parallelism:<br />

first, one can compute all resolvents at Step 2 in parallel and, second,<br />

each <strong>operator</strong> exponent at different time values (provided that we apply <strong>the</strong><br />

<strong>operator</strong> <strong>exponential</strong> <strong>for</strong> a given time vector (t 1 ,t 2 ,...,t M )).<br />

Note that <strong>for</strong> small parameters t ≪ 1, <strong>the</strong> numerical tests indicate that<br />

Step 3 in <strong>the</strong> algorithm above has slow convergence. In this case, we propose<br />

<strong>the</strong> following modification of Algorithm 2.5, which converges much faster<br />

than (2.20).<br />

√<br />

Algorithm 2.7 1 ′ . Determine h = 3 2πdk<br />

a<br />

(N +1)−2/3 ,z p (t) (p = −N,<br />

...,N) by z p (t) = a k (ph)2 + b(t) − iph, where <strong>the</strong> parameter b(t) is<br />

defined <strong>with</strong> respect to <strong>the</strong> location of sp(tL), i.e., b(t) =tb.<br />

2 ′ . Find <strong>the</strong> resolvents (z p (t)I − tL) −1 ,p= −N,...,N (it can be done<br />

in parallel).<br />

3 ′ . Find <strong>the</strong> <strong>approximation</strong> exp N (−tL) <strong>for</strong> <strong>the</strong> <strong>operator</strong> exponent exp(−tL)<br />

in <strong>the</strong> <strong>for</strong>m<br />

exp N (−tL) =<br />

h<br />

2πi<br />

N∑<br />

p=−N<br />

e −tzp(t) [ 2 a k ph − i ]<br />

(z p (t)I − tL) −1 .<br />

Though <strong>the</strong> above algorithm allows only a sequential treatment of different<br />

time values close to t =0, in many <strong>applications</strong> (e.g., <strong>for</strong> integration<br />

<strong>with</strong> respect to <strong>the</strong> time variable) we may choose <strong>the</strong> time-grid as t i = i∆t,<br />

i =1,...,n t . Then <strong>the</strong> <strong>exponential</strong>s <strong>for</strong> i =2,...,n t are easily obtained<br />

as <strong>the</strong> corresponding monomials from exp N (−∆tL).<br />

3 On <strong>the</strong> H-matrix <strong>approximation</strong> to <strong>the</strong> resolvent (zI −L) −1<br />

Below, we briefly discuss <strong>the</strong> main features of <strong>the</strong> H-matrix techniques to<br />

be used <strong>for</strong> data-sparse <strong>approximation</strong> of <strong>the</strong> <strong>operator</strong> resolvent in question.<br />

We recall <strong>the</strong> complexity bound <strong>for</strong> <strong>the</strong> H-matrix arithmetic and prove <strong>the</strong><br />

existence of <strong>the</strong> accurate H-matrix <strong>approximation</strong> to <strong>the</strong> resolvent of elliptic<br />

<strong>operator</strong> in <strong>the</strong> case of smooth data.<br />

Note that <strong>the</strong>re are different strategies to construct <strong>the</strong> H-matrix <strong>approximation</strong><br />

to <strong>the</strong> inverse A = L −1 of <strong>the</strong> elliptic <strong>operator</strong> L. The existence<br />

result is obtained <strong>for</strong> <strong>the</strong> direct Galerkin <strong>approximation</strong> A h to <strong>the</strong> <strong>operator</strong><br />

A provided that <strong>the</strong> Green function is given explicitly (we call this H-matrix<br />

<strong>approximation</strong> by A H ). In this paper, such an <strong>approximation</strong> has only <strong>the</strong><br />

<strong>the</strong>oretical significance. However, using this construction we prove <strong>the</strong> density<br />

of H-matrices <strong>for</strong> <strong>approximation</strong> to <strong>the</strong> inverse of elliptic <strong>operator</strong>s in<br />

<strong>the</strong> sense that <strong>the</strong>re exists <strong>the</strong> H-matrix A H suchthat

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