H-Matrix approximation for the operator exponential with applications

102 I. P. Gavrilyuk et al.
withrespect to j = −N,...,N, to compute the approximation u H (t).
The second level of parallelisation appears if we are interested to calculate
the right-hand side of (4.2) for different time values.
4.2 Dynamical systems and control theory
In the second example, we consider the linear dynamical system of equations
dX(t)
= AX(t)+X(t)B + C(t), X(0) = X 0 ,
dt
where X, A, B, C ∈ R n×n . The solution is given by
∫ t
X(t) =e tA X 0 e tB +
0
e (t−s)A C(s)e (t−s)B ds.
Suppose that we can construct the H-matrix approximations of the corresponding
matrix exponents
exp H (tA) =
2N∑
0 −1
l=1
γ al e −a lt A l , exp H (tB) =
A l , B j ∈M H,k (I × I,P 2 ).
2N∑
0 −1
j=1
γ bj e −b jt B j ,
Then the approximate solution X H (t) may be computed in parallel as in the
first example,
[
N∑
X H (t) = γ alJ0 γ bjJ0 e −(a lJ 0
+b jJ0 )t A lJ0 X 0 B jJ0
l,j=−N
∑J 0
∫
+ γ alα γ bjα e −(a lα+b jα )t A lα
α=1
δ α
e (a lα+b jα )s C(s)dsB jα
]
Let C be constant and the eigenvalues of A, B have negative real parts, then
X(t) → X ∞ as t →∞, where
X ∞ =
∫ ∞
satisfies the Lyapunov-Sylvester equation
0
e tA Ce tB dt
AX ∞ + X ∞ B + C =0.
.