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Item type:Publication, Access status: without files , Tensor trains and moment conservation for multivariate aggregation in population balance modeling© 2020 IMACS 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(dn) and O(dnlog(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 2d moments for our proposed method. Numerical tests confirm the favorable theoretical results concerning computational complexity and conservation of moments.Publicationtype: Journal Article95 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , Algorithms for the Haar wavelet based fast evaluation of aggregation integrals in population balance equations(2016-10-01); In several production processes, the distribution of particles dispersed in an environmental phase may be mathematically described by the solution of population balance equations. We are concerned with the development of efficient numerical techniques for the aggregation process: It invokes an integral term that is usually numerically expensive to evaluate and often dominates the total simulation cost. We describe an approach on locally refined nested grids to evaluate both the source and the sink terms in almost linear complexity (instead of quadratic complexity resulting from a direct approach). The key is to switch from a nodal to a wavelet basis representation of the density function. We illustrate the numerical performance of this approach, both in comparison to a discretization of piecewise constant functions on a uniform grid as well as to the fixed pivot method on a geometric grid.Publicationtype: Journal Article119 - Some of the metrics are blocked by yourconsent settings
Item type:Publication, Access status: without files , The finite section method and stable subsequences(2010-04-01)The purpose of this paper is to prove a sufficient and necessary criterion on the stability of a subsequence of the finite section method for a so-called band-dominated operator on ℓp (ZN, X). We hereby generalize previous results into several directions: We generalize the subsequence theorem from dimension N = 1 (see Rabinovich, Roch and Silbermann (2008) [18]) to arbitrary dimensions N ≥ 1; and even for the case of the full sequence, our result is new in dimensions N > 2 and it corrects a mistake in the literature for N = 2. Moreover, we allow the truncations to be taken by homothetic copies of very general starlike geometries Ω ∈ RN rather than convex polytopes. © 2009 IMACS.Publicationtype: Journal Article48