Nonnegative tensor train for the multicomponent Smoluchowski equation

TL;DR

The paper addresses nonnegativity in tensor-train (TT) based numerical solutions of the multidimensional Smoluchowski coagulation equations for multicomponent systems. It introduces a rank-one nonnegative correction in the TT-format, determined via global TT-optimization, and integrates this correction into a TT-based predictor-corrector time-stepping scheme, enabling nonnegative solutions without leaving the compressed representation. The authors implement a fast TT-convolution-based discretization for the coagulation operators and validate the approach on constant and ballistic kernels, with and without sources, demonstrating accurate results at modest computational cost and enabling higher-dimensional simulations. The work advances scalable, deterministic modeling of complex coagulation processes while preserving physical nonnegativity, with potential extensions to QTT representations and fragmentation phenomena.

Abstract

We propose an efficient implementation of the numerical tensor-train (TT) based algorithm solving the multicomponent coagulation equation preserving the nonnegativeness of solution. Unnatural negative elements in the constructed approximation arise due to the errors of the low-rank decomposition and discretization scheme. In this work, we propose to apply the rank-one corrections in the TT-format proportional to the minimal negative element. Such an element can be found via application of the global optimization methods that can be fully implemented within efficient operations in the tensor train format. We incorporate this trick into the time-integration scheme for the multicomponent coagulation equation and also use it for post-processing of the stationary solution for the problem with the source of particles.

Figures (4)

• Figure 1: The two-component problem \ref{['eq:SmoluchowskiMultidim']} with the constant kernel and exponential initial conditions with $h=0.1, \tau=0.05$. Left panel: Dynamics of the relative number of negative elements; Right panel: negative elements in the numerical solution at $t=10$.
• Figure 2: Plots of the total density for the two-component problem with the ballistic kernel (left) and constant kernel (right) and exponential initial conditions for $t\in[5,10]$ simulations time with $V_{max} = 20$. The major difference between the numerical total density $N(t)$ corresponds to the convergence of the finite-difference scheme but not to the introduction of the nonnegative corrections.
• Figure 3: Total density (left) and mass (right) for experiments with source for constant and ballistic kernels for $V_{max}=20$ and $t\in[0,10]$.
• Figure 4: Nonnegative solution for the two-component coagulation problem with the ballistic kernel without (left) and with source (right), $V_{max}=20$ and $t=5$.