Abstract In the recent years, multidimensional numerical simulations with tensor-structured data formats have been
recognized as the basic concept for breaking the "curse of dimensionality".
Modern applications of tensor methods include the challenging high-dimensional
problems of material sciences, bio-science, stochastic modeling, signal processing,
machine learning, and data mining, financial mathematics, etc.
The guiding principle of the tensor methods
is an approximation of multivariate functions and operators
with some separation of variables to keep the computational process in
a low parametric tensor-structured manifold.
Tensors structures had been wildly used as models of data and
discussed in the contexts of differential geometry, mechanics,
algebraic geometry, data analysis etc. before tensor methods recently
have penetrated into numerical computations. On the one hand, the existing tensor
representation formats remained to be of a limited use in many high-dimensional problems
because of lack of sufficiently reliable and fast software.
On the other hand, for moderate dimensional problems (e.g. in
"ab-initio" quantum chemistry) as well as for selected model problems
of very high dimensions, the application of traditional canonical
and Tucker formats in combination with the ideas of multilevel methods
has led to the new efficient algorithms.
The recent progress in tensor numerical methods is achieved with new
representation formats now known as "tensor-train representations"
and "hierarchical Tucker representations". Note that the formats themselves could have
been picked up earlier in the literature on the modeling of quantum systems.
Until 2009 they lived in a closed world of those
quantum theory publications and never trespassed the territory of
numerical analysis. The tremendous progress
during the very recent years shows the new tensor tools in various
applications and in the development of these tools and study of
their approximation and algebraic properties.
This special issue treats tensors as a base for efficient
numerical algorithms in various modern applications and with special
emphases on the new representation formats.
Critical two-point functions and the lace expansion for spread-out high-dimensional percolation and related models