Hierarchical tensor-product approximation to the inverse and related operators for high-dimensional elliptic problems

The class of H-matrices allows an approximate matrix arithmetic with almost linear complexity. In the present paper, we apply the H-matrix technique combined with the Kronecker tensor-product approximation ( cf. [ 2, 20]) to represent the inverse of a discrete elliptic operator in a hypercube (0,1)(d) is an element of R-d in the case of a high spatial dimension d. In this data-sparse format, we also represent the operator exponential, the fractional power of an elliptic operator as well as the solution operator of the matrix Lyapunov-Sylvester equation. The complexity of our approximations can be estimated by O(dn log(q) n), where N = n(d) is the discrete problem size.