Analysis of H-Matrix Approximation to Operator Valued Functions in Control Theory
I. Gavrilyuk (Berufsakademie Eisenach)
In preceding papers [2,3,4,5], a class of matrices (H-matrices)
has been developed which are data-sparse and allow to approximate nonlocal
(integral) operators with almost linear complexity.
Typical examples of nonlocal mappings of interest are given by the
boundary/volume integral operators in the classical BEM/FEM as well as
by solution operators in the elliptic and parabolic problems defined by
L^-1 and exp(-t L) , respectively, where L
is a strongly elliptic partial differential operator.
The following example of operator valued functions of the elliptic
operator arises in control theory
where G is the given operator, that provides the solution to the
matrix/operator Lyapunov equation. Based on the corresponding results
for the heat semigroup exp(-t L)  we discuss the approximation theory by
H-matrices to the function F(L) mentioned above.
 I. P. Gavrilyuk, W. Hackbusch and B.N. Khoromskij:
H-Matrix Approximation for the operator exponential with applications.
Numer. Math. (to appear).
 W. Hackbusch:
A Sparse Matrix Arithmetic based on H-Matrices.
Part I: Introduction to H-Matrices.
Computing 62 (1999), 89-108.
 W. Hackbusch and B.N. Khoromskij:
A sparse H-matrix arithmetic.
Part II: Application to multi-dimensional problems.
Computing 64 (2000), 21-47.
 W. Hackbusch, B. N. Khoromskij and S. Sauter:
On H^2-matrices. In: Lectures on Applied Mathematics
(H.-J. Bungartz, R. Hoppe, C. Zenger, eds.),
Springer Verlag, 2000, 9-29.
 B.N. Khoromskij: Data-sparse Approximate Inverse in Elliptic Problems:
Green's Function Approach. Preprint MPI MIS 79, Leipzig 2001, submitted.