Data-sparse representation of certain matrix-valued functions in high dimensional applications

Boris N. Khoromskij

Coupling 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. [2], [1], [3], [4]). Examples of such nonlocal mappings are solution operators of elliptic, parabolic and hyperbolic boundary value problems, Lyapunov and Riccati solution operators in control theory, spectral projection operators associated with the matrix sign function for solving the Hartree-Fock equation, collision integrals in the deterministic Boltzmann equation as well as the convolution integrals in the Ornstein-Zernike equation.

We focus on the functions A-1 and spectral projection sign(A), where A is the discrete elliptic operator. The asymptotic complexity of our approximations can be estimated by O(N1/d logq N), where N is the discrete problem size. Some numerical results will be addressed.

  1. W. Hackbusch, B.N. Khoromskij: Low-Rank Kronecker-Product Approximation to Multi-Dimensional Nonlocal Operators, Preprints 29/30, MPI MIS, 2005 (Computing, to appear).
  2. W. Hackbusch, B.N. Khoromskij, and E. Tyrtyshnikov: Hierarchical Kronecker tensor-product approximation, J. Numer. Math. 13 (2005), 119-156.
  3. I. P. Gavrilyuk, W. Hackbusch and B. N. Khoromskij: Tensor-Product Approximation to Elliptic and Parabolic Solution Operators in Higher Dimensions,Computing 74 (2005), 131-157.
  4. B.N. Khoromskij: An Introduction to Structured Tensor-Product Representation of Discrete Nonlocal Operators, Lecture Notes 27, MPI Leipzig, 2005.