The breaking through approach to low-parametric representation of multivariate functions andoperators is based on the principle of separation of variables which can be realized by usingapproximation in rank-structured tensor formats . This allows the linear complexity scalingin dimension, hence breaking the ”curse of dimensionality”. The method of quantized tensortrain (QTT) approximation is proven to provide the logarithmic data-compression on a wideclass of discretized functions and operators .
We discuss how the tensor methods based on the canonical, Tucker, TT and QTT approxi-mation apply to calculation of electrostatic potential of many-particle systems, to the (post)Hartree-Fock eigenvalue problem for large 3D lattice-structured molecules (electrostatic po-tential on a lattice in electronic structure calculations) [3,4,5], to elliptic equations with highly-oscillating coefficients (homogenization theory) , and to parametric/stochastic elliptic PDEs.The other direction is related to tensor approache for parabolic equations in space-time d + 1formulation.
The efficiency of the tensor approach is illustrated by numerical examples.
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