Spline approximation, Kronecker products and multilinear forms

Abstract

SUMMARYSums of Kronecker products occur naturally in high-dimensional spline approximation problems, which arise, for example, in the numerical treatment of chemical reactions. In full matrix form, the resulting non-sparse linear problems usually exceed the memory capacity of workstations. We present methods for the manipulation and numerical handling of Kronecker products in factorized form. Moreover, we analyze the problem of approximating a given matrix by sums of Kronecker products by making use of the equivalence to the problem of decomposing multilinear forms into sums of one-forms. Greedy algorithms based on the maximization of multilinear forms over a torus are used to obtain such (finite and infinite) decompositions that can be used to solve the approximation problem. Moreover, we present numerical considerations for these algorithms.