Numerical treatment of the high-dimensional problems by using the traditional
numerical methods suffers from the so-called “curse of dimensionality”.
This exponential growth $O({n}^{d})$ of the complexity of numerical algorithms
in the dimension parameter *d* can be only slightly relaxed by parallel computations and
high performance computing.
The tensor numerical methods, bridging the multilinear algebra and nonlinear approximation theory,
allow to reduce solution of multidimensional problems to one-dimensional calculations.
They rely on an efficient separable representation of multivariate functions and operators on
large ${n}^{\otimes d}$ grids, leading to algorithms of low computational cost
that scale polynomially or linearly in the dimension parameter *d*.
Thus, tensor methods may be understood as a discrete analogue of the separation of variables,
which may be efficiently maintained at all steps of calculations.
The target function can be the solution of some operator equation $Au=f$, in particular PDE,
and it can be represented through the so-called solution operator $u=S(A)f$.
In tensor numerical methods the entities *u*, *f*, *A* and also $S(A)$
are gainfully approximated in the low rank tensor formats.

Traditional methods of separable approximation combine the canonical, Tucker,
as well as the matrix product states (MPS) formats, the latter known as the tensor train (TT)
decomposition [22, 21].
The recent tensor methods in combination with exponentially accurate sinc-based approximations
are proven to provide the $\mathcal{\mathcal{O}}(d)$ data-compression on a wide class of
functions and operators [5, 6, 9, 7, 4].
The quantized-TT (QTT) tensor approximation of functions [13]
makes it possible to solve high-dimensional PDEs in quantized tensor spaces,
with the $\mathrm{log}$-volume complexity scaling in the full-grid size,
i.e. $\mathcal{\mathcal{O}}(d\mathrm{log}n)$, instead of $\mathcal{\mathcal{O}}({n}^{d})$.

At present, tensor numerical methods and multilinear algebra continue to expand
rapidly to a wide range of theoretical and applied fields, see for example
[11, 14, 10].
We also refer to the recent research monographs [15, 12],
where the tensor numerical methods in scientific computing with the particular
focus on multi-dimensional PDEs and electronic structure calculations have been presented.
These trends are reflected also in the papers from the present issue of CMAM.

This special issue is a collection of papers which demonstrate that the tensor techniques allow
to solve various hard theoretical and computational problems
including approximation of multi-dimensional elliptic/parabolic PDEs.
This issue includes ten invited contributions on
theoretical analysis and applications of tensor-based numerical methods.
These papers cover a broad range of topics including construction of computational schemes
for steady-state and dynamical problems as well as for stochastic and parametric equations, separation rank estimates for classes of
functions and operators, numerical simulations etc.
Below we briefly describe the content of the special issue.

The goal of the paper [1] is the efficient numerical solution of stochastic
eigenvalue problems.
Such problems often lead to prohibitively high-dimensional systems with tensor product structure
when discretized with the stochastic Galerkin method. The authors exploit this inherent tensor product
structure to develop a globalized low-rank inexact Newton method with which they tackle the stochastic
eigenvalue problem. The effectiveness of the solver is illustrated by numerical experiments.

The paper [2] deals with an algorithm for solution of high-dimensional evolutionary
equations (ODEs and discretized time-dependent PDEs) in the TT
decomposition, assuming that the solution and the right-hand side of the ODE
admit such a decomposition with a low rank parameter. A linear ODE, discretized via
one-step or Chebyshev differentiation schemes, turns into a large linear system.
The tensor decomposition allows to solve this system for several time points
simultaneously.
In numerical experiments with the transport and the chemical
master equations, the author demonstrates that the new method is faster than traditional
time stepping and stochastic simulation algorithms.

The paper [3] examines a completely non-intrusive, sample-based
method for the computation of functional low-rank solutions
of high-dimensional parametric random PDEs which have become
an area of intensive research in Uncertainty Quantification. In
order to obtain a generalized polynomial chaos representation of the
approximate stochastic solution, a novel black-box rank-adapted
tensor reconstruction procedure is proposed. The performance
of the described approach is illustrated with several numerical
examples and compared to Monte Carlo sampling.

The authors of [8] consider the abstract differential equations of the
heat and Schrödinger
type and discuss various *N*-parametric approximations on the base of the Cayley
transform and of the Laguerre expansion providing a sub-exponential accuracy,
i.e. the accuracy of the order $\mathcal{\mathcal{O}}({e}^{-N}\mathrm{log}N)$.
They propose a new approximation using the combination of the Gauss–Lobatto–Chebyshev
interpolation and the
Cayley transform and obtain a purely exponential accuracy of the order $\mathcal{\mathcal{O}}({e}^{-N})$.
The rank-structured tensor form of this approximation for a *d*-dimensional
spatial operator coefficient results in an algorithm having a linear complexity in *d*.

The paper [16] study a dynamical low-rank approximation on the manifold of
fixed-rank tensor trains, and analyze projection
methods for the time integration of such problems. The authors prove error estimates
for the explicit Euler method, amended with quasi-optimal projections to the
manifold, under suitable approximability assumptions. Then they discuss the
possibilities and difficulties with higher order explicit and implicit projected
Runge–Kutta methods, in particular, the ways for limiting rank growth
in the increments, and robustness with respect to small singular values.

The paper [17] deals with a new algorithm for spectral learning of Hidden Markov
Models (HMM). In contrast to standard approach, the parameters
of the HMM are not approximated directly, but through an estimate for the joint probability
distribution.
Using TT-format, the authors get an approximation by minimizing the Frobenius
distance between the empirical joint probability distribution and tensors with low
TT-ranks with core tensors normalization constraints. An algorithm
for the solution of optimization problem that is based on the alternating least
squares (ALS) approach is proposed and its fast version for sparse tensors is developed.
The authors compare the
performance of the proposed algorithm with the existing schemes
and found that it is much more robust if the number of hidden states is overestimated.

The paper [18] describes advanced numerical tools for working with multivariate
functions and for the analysis of large data sets.
In particular, covariance
matrices are crucial in spatio-temporal statistical tasks, but are often very
expensive to compute and store, especially in 3D.
Therefore, one can alternatively use a low-rank tensor formats, which reduce the
computing and storage costs essentially.
The authors apply the Tucker and canonical tensor decompositions to a family of
Matérn-type radial functions with varying parameters and demonstrate
theoretically and numerically that their tensor approximations exhibit exponentially
fast convergence in the rank parameter, thus providing low computational complexity.

The paper [19] deals with a space-time isogeometric analysis scheme
for the discretization of parabolic evolution equations with diffusion coefficients
depending on both time and space variables. The problem is considered in a
space-time cylinder in ${\mathbb{R}}^{d+1}$, with $d=2,3$ and is discretized using higher-order
and highly-smooth spline spaces. This makes the matrix formation task very
challenging from a computational point of view. The authors overcome this problem by
introducing a low-rank decoupling of the operator into space and time components.
Numerical experiments demonstrate the efficiency of this approach.

In [20] the authors propose an efficient algorithm to compute a low-rank
approximation to the solution of so-called “Laplace-like” linear systems. The
idea is to transform the problem into the frequency domain, and then to use cross
approximation. In this case, we do not need to form explicit approximation to
the inverse operator and can approximate the solution directly, which leads to
reduced complexity. It is demonstrated that the proposed method is fast and robust by using
it as a solver inside Uzawa iterative method for solving the Stokes problem.

The problem of approximately solving a system of univariate polynomials with
one or more common roots and its coefficients corrupted by noise is studied in [23].
New Rayleigh quotient methods are proposed
and evaluated for estimating the common
roots. Using tensor algebra, reasonable starting values for the Rayleigh quotient methods
can be computed. The new methods are compared to Gauss–Newton, solving an eigenvalue
problem obtained from the generalized Sylvester matrix, and building a cluster among the roots
of all polynomials. It is shown in a simulation study that Gauss–Newton and a new Rayleigh
quotient method perform best, where the latter is more accurate when other roots than the true
common roots are close together.

## Acknowledgements

We would like to thank Professor Carsten Carstensen,
the editor-in-chief of the Journal of Computational Methods in Applied Mathematics, for his
kind support of this special issue and for the useful comments on this overview paper.
We thank the managing editor Professor Piotr Matus, and
Dr. Almas Sherbaf for the effective assistance of the review and production process.
We appreciate the authors of all articles published in this special
issue for the excellent contributions as well as the reviewers for their work on refereeing
the manuscripts.

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## About the article

**Received**: 2018-03-05

**Revised**: 2018-06-01

**Accepted**: 2018-06-06

**Published Online**: 2018-06-26

**Published in Print**: 2019-01-01

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