Convergence of the EBT method for a non-local model of cell proliferation with discontinuous interaction kernel

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Authors
Gwiazda, Piotr
Miasojedow, Błażej
Skrzeczkowski, Jakub
Szymańska, Zuzanna
Date
2022-02-02
Journal Title
Journal ISSN
Volume Title
Publisher
Oxford University Press
License
Creative Commons License
Creative Commons Attribution 4.0 International License
Abstract
We consider the EBT algorithm (a particle method) for the nonlocal equation with a discontinuous interaction kernel. The main difficulty lies in the low regularity of the kernel, which is not Lipschitz continuous, thus preventing the application of standard arguments. Therefore, we use the radial symmetry of the problem instead and transform it using spherical coordinates. The resulting equation has a Lipschitz kernel with only one singularity at zero. We introduce a new weighted flat norm and prove that the particle method converges in this norm. We also comment on the two-dimensional case that requires the application of the theory of measure spaces on general metric spaces and present numerical simulations confirming the theoretical results. In a companion paper we apply the Bayesian method to fit parameters to this model and study its theoretical properties.
Description
Keywords
particle method    EBT algorithm    measure solutions    flat metric    nonlocal equation    convergence analysis    cancer modelling
Citation
Gwiazda, P., Miasojedow, B., Skrzeczkowski, J., Szymańska, Z. (2023). Convergence of the EBT method for a non-local model of cell proliferation with discontinuous interaction kernel, IMA Journal of Numerical Analysis, 43(1), 590–626. https://doi.org/10.1093/imanum/drab102
URI
https://depot.ceon.pl/handle/123456789/22689
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Artykuły ICM / ICM Articles