Model reduction in Smoluchowski-type equations

TL;DR

It is shown in practice that there exists a low-dimensional space allowing to approximate the solutions of aggregation equations and it is demonstrated that it is possible to model the aggregation process with the complexity depending only on dimension of such a space but not on the original problem size.

Abstract

In this paper we utilize the Proper Orthogonal Decomposition (POD) method for model order reduction in application to Smoluchowski aggregation equations with source and sink terms. In particular, we show in practice that there exists a low-dimensional space allowing to approximate the solutions of aggregation equations. We also demonstrate that it is possible to model the aggregation process with the complexity depending only on dimension of such a space but not on the original problem size. In addition, we propose a method for reconstruction of the necessary space without solving of the full evolutionary problem, which can lead to significant acceleration of computations, examples of which are also presented.

Figures (4)

• Figure 1: The dependency of the new basis projection error on time. On the vertical axis are the $\left\lVert (I - V_{k-1} V_{k-1}^T) \hat{V}_k \right\rVert_2$ values; dots denote the moments when the basis is expanded. The horizontal green line shows the $\varepsilon'$ value, the red one --- $\varepsilon$.
• Figure 2: The dependency of the reduced solution relative error in Euclidean norm on time for $N = 32768$. On the vertical axis are the values of ${\left\lVert n(t) - \tilde{n}(t) \right\rVert}_2 / {\left\lVert n(t) \right\rVert}_2$, where $\tilde{n}(t) = V \tilde{x}(t)$.
• Figure 3: Solution error for $N = 131072$, $a = 0.7$ and $a = 0.6$
• Figure 4: Full particle size distribution (purple line) and reduced solutions (red, green and blue) at $t = 512$ for $N = 131072$, $a = 0.7$ and $a = 0.6$. The solution with $\varepsilon = 10^{-14}$ is close to the full solution, diverging only for the smallest concentration values