
Analysis of HMatrix Approximation to Operator Valued Functions in Control Theory
I. Gavrilyuk (Berufsakademie Eisenach)
In preceding papers [2,3,4,5], a class of matrices (Hmatrices)
has been developed which are datasparse 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) [1], 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) [1] we discuss the approximation theory by
Hmatrices to the function F(L) mentioned above.
[1] I. P. Gavrilyuk, W. Hackbusch and B.N. Khoromskij:
HMatrix Approximation for the operator exponential with applications.
Numer. Math. (to appear).
[2] W. Hackbusch:
A Sparse Matrix Arithmetic based on HMatrices.
Part I: Introduction to HMatrices.
Computing 62 (1999), 89108.
[3] W. Hackbusch and B.N. Khoromskij:
A sparse Hmatrix arithmetic.
Part II: Application to multidimensional problems.
Computing 64 (2000), 2147.
[4] W. Hackbusch, B. N. Khoromskij and S. Sauter:
On H^2matrices. In: Lectures on Applied Mathematics
(H.J. Bungartz, R. Hoppe, C. Zenger, eds.),
Springer Verlag, 2000, 929.
[5] B.N. Khoromskij: Datasparse Approximate Inverse in Elliptic Problems:
Green's Function Approach. Preprint MPI MIS 79, Leipzig 2001, submitted.

