Abstract

Joint work with Jakub Skrzeczkowski, Błażej Miasojedow and Piotr Gwiazda. We present a new proliferation model of cells living within a colony, that is a non-local equation with a discontinuous interaction kernel. We discuss the range of applicability of the model, select suitable data and apply the Bayesian method to perform parameter estimation. We discuss proof of the problem’s well-posedness and investigate the convergence of the EBT algorithm applied to solve the equation. The main difficulty lies in the kernel’s low regularity, 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 prove the problem’s well-posedness and investigate the convergence of the EBT algorithm applied to solve the equation. Finally, we prove the stability of posterior distributions in the total variation norm which exploits the theory of spaces of measures equipped with the weighted flat norm. References¹ Szymańska, Z., Skrzeczkowski, J., Miasojedow, B., Gwiazda, P. “Bayesian inference of a non-local proliferation model”, R. Soc. Open Sci. 8(11), 211279, 2021;[2] Gwiazda, P., Miasojedow, B., Skrzeczkowski, J., Szymańska, Z., “Convergence of the EBT method for a non-local model of cell proliferation with discontinuous interaction kernel”, IMA J . Numer. Anal. 43(1), 590-626, 2023;