Concepts of Data-Sparse Tensor-Product Approximation in Many-Particle Modelling

AbstractWe present concepts of data-sparse tensor approximations to the functions and operatorsarising in many-particle models of quantum chemistry. Our approach is based on thesystematic use of structured tensor-product representations where the low-dimensionalcomponentsare representedin hierarchicalor waveletbased matrix formats. The modernmethods of tensor-product approximation in higher dimensions are discussed with thefocus on analytically based approaches. We give numerical illustrations which confirmthe efficiency of tensor decomposition techniques in electronic structure calculations. AMS Subject Classification: 65F30, 65F50, 65N35, 65F10Key words: Schr¨odinger equation, Hartree-Fock method, density functional theory, tensor-product approximation 1 Introduction Among the most challenging problems of scientific computing nowadays are those of high di-mensions, for instance, multi-particle interactions, integral or differential equations on [0,1] d and the related numerical operator calculus for d≥ 3. Many standard approaches have acomputational complexity that grows exponentially in the dimension dand thus fail becauseof the well known “curse of dimensionality”. To get rid of this exponential growth in thecomplexity one can use the idea of tensor-product constructions (cf. [85]) on all stages ofthe solution process. Hereby we approximate the quantity of interest in tensor-product for-mats and use other approximation methods for the remaining low-dimensional components.Depending on the specific properties of the problem, these low-dimensional components are1