Abstract We consider the numerical solution of the multivariate aggregation population balance equation on a uniform tensor grid. This class of equations is numerically challenging to solve - the computational complexity of “straightforward” algorithms grows exponentially with respect to the number of internal coordinates describing particle properties. Here, we develop algorithms which reduce the storage and computational complexity to almost linear order, O ( d n ) and O ( d n log ⁡ ( n ) ) , respectively, where d denotes the number of internal coordinates and n the number of pivots per internal coordinate. In particular, we develop fast algorithms in tensor train format to evaluate the multidimensional aggregation integral exploiting fast Fourier transformation for the underlying convolution. A further significant result lies in the conservation of the first 2 d moments for our proposed method. Numerical tests confirm the favorable theoretical results concerning computational complexity and conservation of moments.

中文翻译：

---

人口平衡建模中多元聚合的张量训练和矩守恒

摘要 我们在均匀张量网格上考虑多元聚集种群平衡方程的数值解。这类方程在数值上很难求解——“直截了当”算法的计算复杂性相对于描述粒子特性的内部坐标的数量呈指数增长。在这里，我们开发了将存储和计算复杂度降低到几乎线性顺序的算法，分别为 O ( dn ) 和 O ( dn log ⁡ ( n ) ) ，其中 d 表示内部坐标的数量，n 表示每个内部的枢轴数量协调。特别是，我们开发了张量训练格式的快速算法，以评估利用基础卷积的快速傅立叶变换的多维聚合积分。另一个重要的结果在于我们提出的方法的前 2 d 矩的守恒。数值测试证实了有关计算复杂性和矩守恒的有利理论结果。