Hierarchical Kronecker tensor-product approximations

The goal of this work is the presentation of some new formats which are useful for the approximation of (large and dense) matrices related to certain classes of functions and nonlocal (integral, integrodifferential) operators, especially for high-dimensional problems. These new formats elaborate on a sum of few terms of Kronecker products of smaller-sized matrices (cf. 34, 35). In addition to this we need that the Kronecker factors possess a certain data-sparse structure. Depending on the construction of the Kronecker factors we are led to so-called “profile-low-rank matrices” or hierarchical matrices (cf. 17, 18). We give a proof for the existence of such formats and expound a gainful combination of the Kronecker-tensor-product structure and the arithmetic for hierarchical matrices.