The dynamics of particle processes can be described by population balance equations which are governed by phenomena including growth, nucleation, breakage and aggregation. Estimating the kinetics of the aggregation phenomena from measured density data constitutes an ill-conditioned inverse problem. In this work, we focus on the aggregation problem and present an approach to estimate the aggregation kernel in discrete, low rank form from given (measured or simulated) data. The low-rank assumption for the kernel allows the application of fast techniques for the evaluation of the aggregation integral (\(\mathcal {O}(n\log n)\) instead of \(\mathcal {O}(n^{2})\) where n denotes the number of unknowns in the discretization) and reduces the dimension of the optimization problem, allowing for efficient and accurate kernel reconstructions. We provide and compare two approaches which we will illustrate in numerical tests.

粒子过程的动力学可以通过种群平衡方程来描述，该方程由包括生长，成核，破裂和聚集在内的现象控制。从测得的密度数据估计聚集现象的动力学构成了病态逆问题。在这项工作中，我们关注于聚集问题，并提出了一种从给定（测量或模拟）数据中以离散，低秩形式估计聚集核的方法。内核的低秩假设允许应用快速技术来评估聚合积分（\（\ mathcal {O}（n \ log n）\）而不是\（\ mathcal {O}（n ^ { 2}）\）其中n表示离散化中的未知数），并减小了优化问题的范围，从而实现了有效而准确的内核重构。我们提供并比较了将在数值测试中说明的两种方法。