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20th GAMM-Seminar Leipzig on
Numerical Methods for Non-Local Operators

Max-Planck-Institute for Mathematics in the Sciences
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  20th GAMM-Seminar
January, 22th-24th, 2004
 
     
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  Abstract Boris Khoromskij, Fri, 12.00-12.30 Previous Contents Next  
  Approximating Nonlocal Operators in Hierarchical Tensor-product Formats: Brief Survey
Boris Khoromskij (MPI Leipzig)

Coupling of the hierarchical and tensor-product formats allows an opportunity for efficient data-sparse approximation of integral and more general nonlocal operators in higher dimensions (cf. [1], [2], [3]). We discuss the H-matrix techniques combined with the Kronecker tensor-product approximation to represent a function F(A) of a discrete elliptic operator A in a hypercube (0,1)d∈Rd in the case of a high spatial dimension d. In particular, we represent the functions A-1 and sign(A) of a discrete elliptic operator with rather general location of spectrum. The asymptotic complexity of our approximations can be estimated by O(Np/dlogqN), p=1,2, where N is the discrete problem size.

Numerics presented in [2], [4], [5] will be also addressed.

Based on joint works with W. Hackbusch (MPI MIS, Leipzig), I. Gavrilyuk (Eisenach) and E. Tyrtyshnikov (Moscow)

[1] W. Hackbusch, B.N. Khoromskij, and E. Tyrtyshnikov: Hierarchical Kronecker tensor-product approximation, Preprint 35, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig, 2003.

[2] I. P. Gavrilyuk, W. Hackbusch and B. N. Khoromskij: Tensor-Product Approximation to Elliptic and Parabolic Solution Operators in Higher Dimensions. Preprint 83, Max-Planck-Institut Mathematik in den Naturwissenschaften, Leipzig 2003.

[3] W. Hackbusch, B.N. Khoromskij: Tensor-Product Based Approximation to Matrix-Valued Functions in Higher Dimensions, in progress.

[4] L. Grasedyck: Existence and computation of a low Kronecker-rank approximation to the solution of a tensor system with tensor right-hand side. Preprint 48, Max-Planck-Institut für Mathematik, Leipzig, 2003 (to appear in Computing).

[5] H-J. Flad, W. Hackbusch, B.N. Khoromskij, R. Schneider: Concept of Data-Sparse Tensor-Product Approximation in Many-Particle Models, in progress.
 

 
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