ИСТИНА

It is commonly accepted that high dimensionality is almost a synonym of intractability and the case where we have recourse to the Monte Carlo method. It is generally true for numerical problems with the data packed in multi-dimensional matrices, also called arrays or tensors. However, in many applications the data may and do have an intrinsic structure related with some or other form of separation of variables. In such cases the structure can be revealed by tensor decompositions, and some of recently proposed decompositions such as Tensor Train and Hierarchical Tucker decompositions allow us to exploit very efficient tensor algorithms that break the curse of dimensionality and reduce the complexity so that it depends on the number of dimensions even linearly and at least as a low degree polynom. These algorithms suggest that even some 2D and 3D problems can be converted into multi-dimensional and then be treated by those tensor algorithms. We discuss the state of the art in numerical applications of Tensor Trains and possible future developments.