18th GAMM-Seminar Leipzig on
Multigrid and related methods for optimization problems

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  18th GAMM-Seminar
January, 24th-26th, 2002
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  Abstract I. Gavrilyuk, Thu, 12.00-12.25 Previous Contents Next  
  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) [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 H-matrices to the function F(L) mentioned above.

[1] I. P. Gavrilyuk, W. Hackbusch and B.N. Khoromskij: H-Matrix Approximation for the operator exponential with applications. Numer. Math. (to appear). [2] W. Hackbusch: A Sparse Matrix Arithmetic based on H-Matrices. Part I: Introduction to H-Matrices. Computing 62 (1999), 89-108.

[3] W. Hackbusch and B.N. Khoromskij: A sparse H-matrix arithmetic. Part II: Application to multi-dimensional problems. Computing 64 (2000), 21-47.

[4] 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.

[5] B.N. Khoromskij: Data-sparse Approximate Inverse in Elliptic Problems: Green's Function Approach. Preprint MPI MIS 79, Leipzig 2001, submitted.

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