An overview of the universality of interacting particle systems

An overview of the universality of interacting particle systems

Abstract: In the 1970s, Frank Spitzer introduced interacting particle systems (IPS) to the mathematical community. These models describe particles evolving randomly according to memoryless (Markovian) dynamics that conserve certain quantities. IPS were already familiar in the physics and biophysics communities, and they serve as simplified models for a wide range of complex phenomena. One of the most classical and extensively studied examples is the exclusion process, in which particles move in a discrete space according to prescribed transition probabilities, subject to the constraint that each site can be occupied by at most one particle.
A central objective in the study of such systems is to derive their hydrodynamic limit-that is, to rigorously obtain the macroscopic partial differential equations that describe the space-time evolution of conserved quantities from the underlying stochastic microscopic dynamics. In this talk, I will review how these limits can be derived in the context of the exclusion process. I will also discuss equilibrium fluctuations, namely the stochastic fluctuations around the typical macroscopic profile when the system is initialised under the invariant measure.
We will see that, depending on the specific transition probabilities of the particles, a wide variety of limiting equations can emerge. Remarkably, these equations exhibit universal behaviou r, appearing across a broad class of distinct microscopic dynamics.

A Keller-Segel-flow model for cancer cell migration: numerical analysis, simulation and validation

A Keller-Segel-flow model for cancer cell migration: numerical analysis, simulation and validation

Abstract: The seminal 1970s Keller-Segel model was proposed to describe the evolution of cell density under the influence of a chemical signal concentration. This phenomenon is known as autologous chemotaxis. Over the years, researchers have adapted the Keller-Segel model to describe a variety of biological processes. For a comprehensive overview of these developments, see reference [AT2021].
Recent studies have investigated the impact of interstitial flow on cancer cell dynamics. The interstitial flow is defined as the fluid motion through the porous extracellular matrix (ECM) driven by the pressure gradient between the surrounding blood and lymphatic capillaries. In [WGRE2021], the authors use a multiphase cell migration model to study how ECM heterogeneities influence on a tumour's metastatic propensity. In [RCLG2020], the authors propose a mechanobiological model to analyse the mechanical factors that drive cell migration. In [LK2023], the focus is on how a growing cancer responds to combination chemotherapy, which is a treatment that combines two or more chemotherapy drugs.
Our focus is on biological processes in which convective flow plays a significant role. The central objective of this investigation is the migration of cancer cells in the presence of interstitial flow. In this particular context, we consider that Darcy's law governs the pressure field, which is dependent on the ECM's permeability and the fluid exchanges that occur between capillaries and the ECM. The convective motion of the cells is driven by their sensitivity to interstitial flow and concentration of the chemical. In this talk we study, from a numerical point of view, a system of nonlinear partial differential equations composed of two parabolic equations for the density of the cancer cells and for the concentration of the chemical and an elliptical equation for the pressure of the interstitial fluid. The system is completed with convenient initial and boundary conditions. The behaviour of the model, as well the main results, are numerically illustrated and the model is validated taking into account the results of cancer cell migration in vitro presented in [SKTRS2010].
Joint work with A. Fernandes, L. Pinto (Univ. Coimbra)
References:
[AT2021] G. Arumugam and J. Tyagi, Jagmohan, Keller-Segel chemotaxis models: a review, Acta Applicandae Mathematicae, 171,6, 2021.
[LK2023] I. Lampropoulos and M. Kavousanakis, Application of combination chemotherapy in two dimensional tumor growth model with heterogeneous vasculature. Chemical Engineering Science, 280:118965, 2023.
[RCLG2020] G. S. Rosalem, E. B. Las Casas, T. P. Lima, and L. A. Gonzalez-Torres, A mechanobiological model to study upstream cell migration guided by tensotaxis. Biomechanics and Modeling in Mechanobiology, 19:1537-1549, 2020.
[SKTRS2010] J. D. Shields, I. C. Kourtis, A. A. Tomei, J. M. Roberts, and M. A. Swartz, Induction of lymphoidlike stroma and immune escape by tumors that express the chemokine ccl21. Science, 328:749-752, 2010.
[WGRE2021] J. O. Waldeland, J.-V. Gaustad, E. K. Rofstad, and S. Evje, In silico investigations of intra tumoral heterogeneous interstitial fluid pressure. Journal of Theoretical Biology, 526:110787, 2021.

Viscoelastic effects in epithelial tissue dynamics: a particle-based modelling approach

Viscoelastic effects in epithelial tissue dynamics: a particle-based modelling approach

Abstract: Pseudostratified epithelial tissues are composed of tightly packed cells with minimal intercellular space. During the cell cycle, individual cells undergo stretching or compression depending on spatial constraints. To maintain tissue integrity, cells actively release mechanical stress by reorganizing their cytoskeleton and altering their shape. These active and passive responses produce viscoelastic behaviour locally at the cell level, which then propagates to larger scales, influencing the global shape and dynamics of the tissue. However, the precise contributions of the mechanisms at the cell level remain poorly understood and challenging to study in vivo.
To address this, we developed and validated a particle-based modelling framework in which cells are represented as hard spheres connected by springs with adaptive rest lengths. The dynamics is described by a non-autonomous generalized gradient flow under nonconvex constraints and implemented using a position-based dynamics algorithm recently developed in computer graphics.
Through a series of in silico simulations complemented by in vivo experiments, we investigated how cell distribution and tissue morphology evolve over time. Moreover, by introducing cell heterogeneity into the model, we explored how defects in individual cells can trigger cell extrusion - an essential process in both embryonic development and cancer metastasis.
Joint work with Steffen Plunder (Kyoto), Sara Merino-Aceituno (Vienna), Pierre Degond (Toulouse) and Eric Theveneau's lab (CBI, Toulouse)

Adaptive isogeometric method with hierarchical splines for multi-patch Kirchhoff-Love shells

Adaptive isogeometric method with hierarchical splines for multi-patch Kirchhoff-Love shells

Abstract: Isogeometric analysis is a method for the solution of partial differential equations in which spline functions are used both for the representation of the geometry and for the approximation of the discrete solution. The high continuity of splines has proven advantageous to the solution of fourth order partial differential equations, and in particular one of the most successful applications is the analysis of Kirchhoff-Love shell structures. While $C^1$ continuity is easily obtained in single-patch domains, where the basis functions are defined by tensor-product, the extension of the method to multi-patch surfaces has been done for many years based on weak continuity coupling (with penalty methods or Lagrange multipliers, for instance), while recently new spline spaces with strong $C^1$ continuity across patches have been developed and applied.
In this work we introduce an adaptive isogeometric method for Kirchhoff-Love shell structures with hierarchical splines, and with $C^1$ continuity across patches. The method relies on the definition of smooth multi-patch splines over analysis suitable $G^1$ surfaces from [1], along with the construction of hierarchical splines for multi-patch planar geometries from [2]. Although the splines from [1] are not locally linearly independent, as it is required in the definition of standard hierarchical splines, we introduce a refinement algorithm with linear complexity that guarantees that the constructed hierarchical splines are linearly independent. This refinement algorithm is coupled with an a posteriori error estimator to obtain a high order adaptive method.
This is a joint work with Cesare Bracco, Andrea Farahat, Carlotta Giannelli and Mario Kapl
[1] A. Farahat, B. Jüttler, M. Kapl, T. Takacs. Isogeometric analysis with C1-smooth functions over multi-patch surfaces. Comput. Methods Appl. Mech. Engrg. 403 (2023), 115706.
[2] C. Bracco, C. Giannelli, M. Kapl. R. Vázquez. Adaptive isogeometric methods with C1 (truncated) hierarchical splines on planar multi-patch domains, Math. Models Methods Appl. Sci. 33 (2023), 9, 1829-1874.

Vortex-Induced Transport: Optimal Strategies for Passive Particles

Vortex-Induced Transport: Optimal Strategies for Passive Particles

Abstract: Following a brief introduction to point vortices and passive particles - both in planar and spherical settings - and their role in modeling real-world fluid flows, we will explore optimization strategies for passive particle transport driven by point vortices. Additionally, we will examine methods for estimating the circulation and position of point vortices under Gaussian noise, based on passive particle trajectory data.

Nonlinear diffusion with the fractional porous medium equation with a discontinuous coefficient

Nonlinear diffusion with the fractional porous medium equation with a discontinuous coefficient

Abstract: We consider a nonlinear PDE model in a porous medium with local and non-local effects, which incorporates a spatially varying and discontinuous coefficient $\beta(x) \geq \beta* > 0$, appearing in the fractional gradient to capture the multiscale phenomenon and the long-range effects of anomalous diffusion in many complex hydrodynamic systems. The fractional operator conveniently describes the short-range non-local effects, typically observed in the laboratory settings or at the Darcy scale. Still, its properties, such as symmetry and invariance, break down when we consider field scale and multiscale porous medium heterogeneity of the real-life porous media flow systems, such as groundwater flow in hydrogeology and porous carbon for CO2 capture technology in energy transition resources.
We consider a family of approximate problems by introducing a linear diffusion term with coefficient $\delta>0$ and mollifying $\beta$. We establish the existence and positivity of weak solutions for these problems. We then derive adequate a priori estimates for the solutions and, passing to the limit as $\delta\to0$, obtain a positive solution to the original problem.
Work in collaboration with Eduardo Abreu, Assis Azevedo, Julio Guevara, and Lisa Santos.

On the well-posedness for a Quasi Geostrophic equation on a large Besov-weak-Morrey space and numerical insights

On the well-posedness for a Quasi Geostrophic equation on a large Besov-weak-Morrey space and numerical insights

Abstract: We present a local in-time well-posedness result for a two-dimensional Quasi Geostrophic equation on a large Besov-weak-Morrey space, recently introduced in [1]. We develop some key estimates which allow us to get well-known results in classical Besov spaces. We also recover Bernstein-type inequalities for modified weak-Morrey spaces and Sobolev embedding for modified Besov-Weak-Morrey spaces. Some very interesting numerical insights are presented on the initial data belonging to the Besov framework in the sense of presenting many oscillations. These numerics have been realized via the novel method of type Lagrangian-Eulerian based on the concept of no-flow curves [2] and by means of a new approach for numerical approximation of the nonlocal operator Riesz transform.
References
[1] Ferreira, Lucas CF, and Jhean E. P´erez-L´opez. "On the well-posedness of the incompressible Euler equations in a larger space of Besov-Morrey type." Dynamics of Partial Differential Equations 19.1 (2022): 23-49.
[2] Abreu, Eduardo, et al. On the conservation properties in multiple scale coupling and simulation for Darcy flow with hyperbolic-transport in complex flows. Multiscale Modeling & Simulation 18.4 (2020): 1375-1408.

Function space theory and numerics of a problem involving irregular objects: the case of acoustic scattering

Function space theory and numerics of a problem involving irregular objects: the case of acoustic scattering

Abstract: This talk focuses on recent work on acoustic scattering problems which involve solving the Helmholtz equation for irregular - often fractal - domains or boundaries. After introducing the general scattering framework and the variational formulation we employ, we present the types of irregular geometries considered and compare theoretical predictions with numerical computations in selected examples. As an illustrative case, let us mention here the Koch snowflake domain, which exemplifies the type of geometric complexity that can be tackled within this framework.
The core of the approach follows a standard Galerkin-type discretization, the novelty being in addressing the new mathematical and numerical challenges posed by the fractal nature of the domains. These challenges led us to develop formulations based on Hausdorff measures associated with the underlying fractals, giving rise to new theoretical questions in function space theory, and to practical issues such as how to compute integrals with respect to such measures numerically.
We will highlight some of the new results obtained - on Sobolev spaces on fractals, wavelet decompositions, and interpolation theory - which may be of interest beyond the specific scattering context.