Solving non-stationary problems is hard, even if they are linear. There exist numerous techniques for the efficient solution of such kind of problems. The task becomes much harder, when the problem in question is high-dimensional. Chemical master equation for stochastic modelling of chemical kinetics is of the most vivid examples of such kind. Using Finite State Projection (FSP) the problem is reduced to a linear non-stationary system of equations, that has very high dimensionality, i.e. the solution can be represented as a multidimensional array (tensor) with large mode sizes. Application of tensor decompositions to such kind of problems is very promising, and that has been shown by recent work of Dolgov and Khoromskij, and also Kazeev, Schwab, Khammash and Nip. Tensor techniques allow to break the curse of dimensionality and reduce the complexity. Still, there is a lot of to on the algorithmic side. In this talk I will present several new approaches for solving high-dimensional non-stationary problem by means of tensor techniques. The new approach is related to the so-called global time-stepping scheme and we will show that it boils down to a simple Krylov-type approximation scheme. Finally we will show how the Tensor Train Toolbox (TT-Toolbox, http://github.com/oseledets/TT-Toolbox and Python version http://github.com/oseledets/ttpy) can be applied to routinely solve high-dimensional chemical master equations.

tensor decomposition; tensor trains; chemical master equation; high-dimensional problems