Mosaic-skeleton approximation is all you need for Smoluchowski equations

TL;DR

This paper introduces a novel mosaic-skeleton approximation method that significantly accelerates the numerical solution of aggregation equations with many kernels, reducing computational complexity from quadratic to near-linear.

Contribution

The paper presents a new mosaic-skeleton approximation technique that enables efficient and accurate numerical solutions for a broader class of aggregation kernels in kinetic equations.

Findings

Reduces computational complexity from O(M^2) to O(M log^2 M)

Allows solving systems with up to 2^20 equations efficiently

Applicable to aggregation problems in sedimentation, turbulence, and supersonic flows

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.