Numerical treatment of high-dimensional PDEs requires nonstandard approaches since the computational complexity of traditional numerical methods scales exponentially in the dimension refered as the "curse of dimensionality", O(n^{d }).
The separable tensor product low parametric representations of multidimensional arrays in particular, the canonical and Tucker formats, since long have been used in the computer science community (F.L. Hitchcock (1927), L.R. Tucker (1966), ... L. De Lathauwer, et al (2000)).
In 2006 it was shown and proved by B. Khoromskij that the Tucker tensor format has exceptional properties for approximating some multivariate functions discretized on d-dimensional Cartesian grid (Newton kernel, Slater function, etc). Next, the canonical-to-Tucker algorithm (C2T) and the reduced higher order singular value decomposition (RHOSVD) have been introduced in 2006, and 2008 by B. Khoromskij and V. Khoromskaia. In these papers it was shown that the canonical and Tucker formats can be efficiently used for the low-rank tensor representation of functions of several variables and for their numerical treatment. RHOSVD can be used for building the tensor formats "avoiding the curse of dimensionality".
The tensor-structured methods have been initiated by the problem of the grid-based numerical solution of the 3D Hartree-Fock equation in a general basis (V. Khoromskaia, B. Khoromskij, H.-J. Flad). These methods, in particular, include fast and accurate tensor calculation of 3D and 6D convolution operators with the Newton kernel in 1D complexity, (2008), (2009) and (2010). Recent approach for factorized calculation of the two-electron integrals tensor is efficient in electronic structure calculations for large molecular systems. (2012). Our algorithms resulted in a package (3D grid-based "black-box" solver) for the numerical solution of the Hartree-Fock equation in a general basis (2013) .
The tensor-train (TT) format of O( n d) complexity for the treatment of multidimensional tensors was developed by I. Oseledets and E. Tyrtyshnikov in 2009.
The quantics N-d tensor approximation in high-dimensional numerical modeling (QTT) was invented by B. Khoromskij in 2009 as a tool to compress function related vectors/tensors with O(d log n ) complexity scaling. In this paper the QTT method was also justified by the proof of uniform rank estimates for a wide class of discretized functions.
We focus on O(d log n ) -complexity numerical methods for the efficient solution of high-dimensional steady-state and dynamical PDEs. They are based on the QTT approximation of multivariate functions and integral-differential operators. In particular, tensor methods gainfully apply to the large scale problems in molecular dynamics, for the Focker-Planck and master equations, as well as for parametric/stochastic PDEs (B. Khoromskij, Ch. Schwab, I. Oseledets, S. Dolgov, V. Kazeev).
Superfast QTT-Fourier and QTT-convolution transforms of O(d log n ) complexity open the way to the fast treatment of huge data arrays (B. Khoromskij, E. Tyrtyshnikov, D. Savostyanov, S. Dolgov, V. Kazeev).
Numerical efficiency and accuracy of our tensor numerical algorithms is confirmed on the examples of real-life problems and in comparison with the existing alternative approaches (analytically based multidimensional integration in quantum chemistry, Monte Carlo and model reduction methods).
Since 2006: development of the MATLAB library implementing multi-linear algebra and the rank-structured tensor approximations to a class of multivariate functions, integral transforms and Hamiltonians (electronic structure calculations), many-particle modeling, as well as the fast tensor-truncated iterative solution methods for high-dimensional stationary and dynamical equations in scientific computing.