# Tensors in modelling multi-particle interactions

## TL;DR

This paper demonstrates how low-rank tensor decompositions can efficiently model multi-particle interactions in aggregation kinetics, significantly reducing computational complexity and enabling high-accuracy solutions on supercomputers.

## Contribution

The work introduces a novel application of tensor decompositions to accelerate the evaluation of nonlinear differential equations in multi-particle kinetics.

## Key findings

- Reduced computational complexity using tensor decompositions.
- High-accuracy numerical solutions achieved with modest computational times.
- Preliminary parallel scalability results indicate efficiency on supercomputers.

## Abstract

In this work we present recent results on application of low-rank tensor decompositions to modelling of aggregation kinetics taking into account multi-particle collisions (for three and more particles). Such kinetics can be described by system of nonlinear differential equations with right-hand side requiring $N^D$ operations for its straight-forward evaluation, where $N$ is number of particles size classes and $D$ is number of particles colliding simultaneously. Such a complexity can be significantly reduced by application low rank tensor decompositions (either Tensor Train or Canonical Polyadic) to acceleration of evaluation of sums and convolutions from right-hand side. Basing on this drastic reduction of complexity for evaluation of right-hand side we further utilize standard second order Runge-Kutta time integration scheme and demonstrate that our approach allows to obtain numerical solutions of studied equations with very high accuracy in modest times. We also show preliminary results on parallel scalability of novel approach and conclude that it can be efficiently utilized with use of supercomputers.