Tensor train versus Monte Carlo for the multicomponent Smoluchowski coagulation equation

In this paper we present a novel numerical algorithm for the space-homogeneous multicomponent (multidimensional) Smoluchowski coagulation equation, the number of components is considered as dimensionality. The new methodology is based on the classical finite-difference predictor-corrector scheme. In a straightforward implementation of this scheme, however, one would have to compute and store prohibitively many values of the grid function at the nodes of a multidimensional grid. We propose to use special low-parametric representations for the grid functions and as well for the coagulation kernel. The corresponding multidimensional arrays are approximated by low-rank tensor-train decompositions reducing them to combinations of small low-dimensional arrays, eventually to matrices for which we can use fast algorithms of linear algebra. Instead of O(N2d) operations in the classical scheme, we propose a new method that requires only O(d2Nlog⁡N) operations, where N is the number of nodes per axis in the space grid and d is the number of components. In this work we accelerate the predictor-corrector time-scheme and use the trapezoidal rule for the computation of multidimensional integral operators. Thus, the accuracy of the new method is O(h2+τ2), where h is the space grid step and τ is the time step.