Introduction to tensor numerical methods for multi-dimensional PDEs

Tensor numerical approximation provides the efficient separable representation of multivariate functions and operators on large n^⊗d-grids, that allows the solution of d-dimensional PDEs with linear complexity scaling in the dimension, O(dn). Modern methods of separable approximation combine the canonical, Tucker, as well as the matrix product state (MPS) formats (also known as tensor train (TT) decomposition).
The recent quantized-TT (QTT) approximation is proven to provide the logarithmic data-compression on a wide class of functions and operators. It makes possible to solve high-dimensional steady-state and dynamical problems in quantized tensor spaces, with the log-volume complexity scaling in the full-grid size, O(dlog n), instead of O(n^d).
In this lecture we will discuss how the grid-based tensor approximation applies to hard problems arising in electronic structure calculations, such as many-electron integrals and solution of the Hartree-Fock equation.
We present the algorithms for the nonlinear Tucker decomposition of function related tensors represented in full grid size and in the canonical formats, the basic rank-structured operations with tensors, and the rank reduction algorithms. The numerical results are given for 3D tensors corresponding to functions (also with strong cusps) 1/r, e^−r, ∑c _ke^−αr_k2 , r ∈ R³, etc.
We present on-line Matlab simulations for computing the ab initio ground state energy of compact molecules including glycine and alanine amino acids.

Keywords
Tensor numerical methods, canonical tensor decomposition, multilinear algebra, quantics tensor approximation, Hartree-Fock-equation, many-electron integrals, molecular dynamics, chemical master equations, Coulomb potential, lattice sums, periodic systems

Prerequisites
PDE's, ODE's, introduction to numerics and (multi) linear algebra

Audience
MSc students, PhD students, Postdocs, Researchers

Language
English