Mosaic-skeleton approximation is all you need for Smoluchowski equations

TL;DR

This work tackles the computational bottleneck of solving Smoluchowski coagulation equations for large numbers of particle sizes by introducing mosaic-skeleton approximations. The authors combine low-rank kernel representations with a blockwise mosaic partitioning to compress and efficiently compute the two Smoluchowski operators, achieving a per-step cost of $\mathcal{O}(RM\log^2 M)$ for kernels with off-diagonal rank $R$. They provide theoretical memory and rank bounds, report extensive numerical experiments across diverse kernels, and demonstrate substantial speedups compared with Monte Carlo methods while maintaining accuracy, enabling simulations with up to $M=2^{20}$. The approach is implemented as an open-source solver and shows promise for broad applicability in coagulation dynamics and related kinetic models, with future work aimed at extending the technique to accelerate stochastic methods as well.

Abstract

In this work we demonstrate a surprising way of exploitation of the mosaic--skeleton approximations for efficient numerical solving of aggregation equations with many applied kinetic kernels. The complexity of the evaluation of the right-hand side with $M$ nonlinear differential equations basing on the use of the mosaic-skeleton approximations is $\mathcal{O}(M \log^2 M)$ operations instead of $\mathcal{O}(M^2)$ for the straightforward computation. The class of kernels allowing to make fast and accurate computations via our approach is wider than analogous set of kinetic coefficients for effective calculations with previously developed algorithms. This class covers the aggregation problems arising in modelling of sedimentation, supersonic effects, turbulent flows, etc. We show that our approach makes it possible to study the systems with $M=2^{20}$ nonlinear equations within a modest computing time.

Figures (4)

• Figure 1: Smoluchowski kernels low--rank approximation error in the Frobenius norm depending on the rank $R$ for different matrix sizes $N$
• Figure 2: Mosaic partitioning depending on the low--rank criteria
• Figure 3: The mutual error $\varepsilon_1$ of the FDMSk and LRMC with increasing accuracy $\varepsilon_2$ of each of them for kernels $K_{ij}^{2}, K_{ij}^3$. Linear decreasing of mutual error is observed.
• Figure 4: Evaluation time of the operators $f_1$ and $f_2$ in mosaic-skeleton format compared to the theoretical estimates.