Introduction to tensor numerical methods in higher dimensions

Thu, 28/02 14:00	Dr Boris Khoromskij (MPI Leipzig)	Computational Mathematics and Applications	Gibson Grd floor SR
	Introduction to tensor numerical methods in higher dimensions
	Tensor numerical methods provide the efficient separable representation of multivariate functions and operators discretized on large $n^{\otimes d}$ -grids, providing a base for the solution of $d$ -dimensional PDEs with linear complexity scaling in the dimension, $O(d n)$ . Modern methods of separable approximation combine the canonical, Tucker, matrix product states (MPS) and tensor train (TT) low-parametric data formats. The recent quantized-TT (QTT) approximation method is proven to provide the logarithmic data-compression on a wide class of functions and operators. Furthermore, QTT-approximation makes it possible to represent multi-dimensional steady-state and dynamical equations in quantized tensor spaces with the log-volume complexity scaling in the full-grid size, $O(d \log n)$ , instead of $O(n^d)$ . We show how the grid-based tensor approximation in quantized tensor spaces applies to super-compressed representation of functions and operators (super-fast convolution and FFT, spectrally close preconditioners) as well to hard problems arising in electronic structure calculations, such as multi-dimensional convolution, and two-electron integrals factorization in the framework of Hartree-Fock calculations. The QTT method also applies to the time-dependent molecular Schr{ö}dinger, Fokker-Planck and chemical master equations. Numerical tests are presented indicating the efficiency of tensor methods in approximation of functions, operators and PDEs in many dimensions. http://personal-homepages.mis.mpg.de/bokh