Main courses
Course by Magali Ribot, Université d'Orléans
Title: Numerical studies of various models for biology
Abstract: In this course, I will present a few examples of mathematical PDE models for biology, describing the evolution in time of a population of organisms (cells, bacteria, algae…) in interaction and in interaction with its environment. All these models are based on conservation laws and need adapted numerical schemes to be efficiently solved at the numerical level. Here are two examples of the biological systems I will deal with.
In the one dimensional case, I will study different kinds of hyperbolic and parabolic PDE models to describe the phenomenon of chemotaxis, that is to say the motion of organisms due to a chemical substance. I will explain how to discretize efficiently these equations thanks to well-balanced or asymptotic-preserving numerical methods in order to obtain accurate simulations. This example will be extended to the case where chemotaxis systems are set on networks, to describe, for example, the motion of cells on a scaffold or in microfluidic devices.
In the two dimensional case, I will consider models describing the growth of biofilms, such as bacterial or phototrophic biofilms and I will explain how to deal numerically with the momentum conservation equation that contains a pressure term.
Time permitting, other numerical challenges will be presented.
Outline:
Lecture 1 : Some numerical challenges in PDE models for biology
Lecture 2 : Modeling biofilm growth : Finite volume schemes for hyperbolic PDE systems
Quick notes on numerical schemes for hyperbolic equations & monotonicity
Quick notes on relaxation schemes
Lecture 3 :Modeling chemotaxis : Well-Balanced and Asymptotic Preserving schemes
Quick notes on Asymptotic High Order schemes
Quick notes on Upwinding Sources at Interface schemes
Recordings of the lectures: Part 1, Part 2, Part 3, Part 4
Course by Marie-Therese Wolfram, University of Warwick
Title: PDE models for aggregation and segregation dynamics
Abstract: In this short course we will discuss different microscopic modelling approaches to describe the dynamics of large interacting particle systems that like to 'stick together'. We start by considering different mechanisms which lead to such aggregation and segregation dynamics and discuss how they translate in different microscopic approaches. Hereby we focus on single as well as multiple species and present how volume constraints can be included. Next we consider the corresponding mean-field models and use PDE techniques to analyse aggregation and segregation dynamics. We will focus on different applications - in the life, social and data sciences - and use Jupyter scripts to illustrate the dynamics of the respective models.
The tentative plan:
Microscopic models for active and passive interacting particle systems I
In the first lecture I will outline a framework, which allows to formulate various classes of models for self-propelled interacting particle systems. I will start with particles, that are characterised by their position in space and their direction of motion and outline the main steps to derive the respective mean-field models. The derivation of these mean-field models is only rigorous in certain scaling limits - for example in the case of point particles - while they fail for particles of finite size. I will outline the main challenges which arise in this context and present the only rigorous result by Rost; Bodnar&Velazquez as well as Gavish, Nyquist and Peletier for hard rods in 1D.
Literature:
Bruna, M., Burger, M, Pietschmann, J.F. and MTW, Active Crowds, Arxiv 2021
Rost, H. Diffusion de sphéres dures dans la droite réelle: comportement macroscopique et équilibre local, Lecture notes in Mathematics, Springer, 1984
Bodnar, M., and Velazquez, J.J.J. An integro-differential equation arising as a limit of individuell cell-based models, JDE 2006
Gavish, N, Nyquist, P., Peletier, M., Large Deviations and Gradient Flows for the One-Dimensional Hard-Rod System, Arxiv 2020
Microscopic models for active and passive interacting particle systems II
In the second part I will discuss different formally derived mean-field models for interacting agent systems with finite volume effects. I will then focus on models, in which particles can only jump to certain locations in space (also known as lattice based models), and connect them to the modeling framework presented in lecture 1. Finally I will briefly outline why the Wasserstein gradient flows arise naturally in the context of gradient flows.
Literature:
Bruna, M. and Chapman, J. Diffusion multiple species with excluded volume effects, J. Chem. Phys. 2012
Bruna, M. and Chapman, J. Excluded volume effects in the diffusion of hard spheres, Phys. Rev. E. 2012
Adams, S., Dirr, N., Peletier, M. A. and Zimmer, J.: From a large-deviations principle to the Wasserstein gradient flow: a new micro-macro passage. Commun. Math. Phys. 2011
Adams, S. Dirr, N., Peletier, M. A. and Zimmer, J. Large deviations and gradient flows, Phil Trans Roy Soc A. 2013
Macroscopic models for active and passive interacting particle systems I
In this lecture we continue with Wasserstein gradient flows and present how entropies and energies can be used to show existence of solutions or to analyse their long time behavior. I will then discuss which types of the previously derived mean-field models have a gradient flow structure as well as the lack of such. I conclude by presenting the concept of asymptotic gradient flows - these models lost their natural underlying gradient flow structure (due to the approximations made in the limiting process) - and how their closeness can be used to analyse these models.
Literature:
Bruna, M., Burger, M., Ranetbauer H. and MTW Cross diffusion systems with excluded volume effect and asymptotic gradient flow structures, J. Nonlin Sci 2017
Juengel, A. The boundedness by entropy method for cross diffusion systems, Nonlinearity 2015
Aggregation-diffusion equations
In this lecture I will focus on stationary solutions to aggregation-diffusion equations. We start with the classic Patlak-Keller-Segel model, and then consider a non-linear aggregation equation investigated by Burger et al. The authors showed that this equation has compactly supported stationary states in 1D - I will outline the main steps how to show such a result, in particular the Krein-Rutman theorem. Furthermore I will briefly discuss stability of stationary states to the non-local diffusion equation discussed in lecture 2.
Literature:
Burger, M., Haskovec, J. and MTW Individual based and mean-field modeling of direct aggregation, Physica D 2013
Burger, M., Fetecau, R. and Huang, Y. Stationary states and asymptotic behavior of aggregation models with nonlinear local repulsion SIADS 2013
Interacting particle systems in data science
In the final lecture I will give an overview on particle methods and how they can be used within the Bayesian framework for inverse problems as well as global optimisation. I will start by discussing the corresponding optimisation problems and outline the challenges related to them. Then I will focus on gradient based and gradient free ensemble/particle methods, such as the Ensemble Kalman sampler or Gaussian process samplers.
Literature:
Reich, S and Weissmann, S. Fokker-Planck Systems for Bayesian Inference, SIAM UQ 2021
Garbuno-Inigo, A. et al Interacting Langevin Diffusions: Gradient Structures and Ensemble Kalman Sampler, SIAM DS 2020
Pinnau, R., Totzeck, C and Tse, O, A consensus-based model for global optimisation and its mean-field limit, M3AS 2017
Recordings of the lectures: Part 1, Part 2, Part 3, Part 4, Part 5
