Plenanry Speakers

Andrea Barth (University of Stuttgart, Germany)

Uncertainty Quantification with Discontinuous Random Fields

In this talk I give a short introduction to random partial differential equations with discontinuous random field coefficients. Equations of this kind may be used to model for instance subsurface flow problems or microstructures. The spacial regularity of the solution requires adjustments in the approximation method in space as well as when calculating moments of the solution (as an approximation to the distribution). A full discrete method leads to stochastic convergence rates. I close the talk with divers numerical examples.

Sandra Cerrai (University of Maryland, USA)

A Smoluchowski-Kramers approximation for an infinite-dimensional system with state-dependent damping

We study the validity of a Smoluchowski-Kramers approximation for a class of wave equations in a bounded domain of $\mathbb{R}^n$ subject to a state-dependent damping and perturbed by a multiplicative noise. We prove that in the small mass limit the solution converges to the solution of a stochastic quasilinear parabolic equation where a noise-induced extra drift is created.

Serena Dipierro (University of Western Australia, AU)

Civil wars: a new Lotka-Volterra competitive system

We introduce a new model in population dynamics that describes two species sharing the same environmental resources in a situation of open hostility. The interaction among these populations is described not in terms of random encounters but via the strategic decisions of one population that can attack the other according to different levels of aggressiveness. This leads to a non-variational model for the two populations at war, taking into account structural parameters such as the relative fit of the two populations with respect to the available resources and the effectiveness of the attack strikes of the aggressive population. The analysis that we perform is rigorous and focuses on the dynamical properties of the system, by detecting and describing all the possible equilibria and their basins of attraction. Moreover, we will analyze the strategies that may lead to the victory of the aggressive population, i.e., the choices of the aggressiveness parameter, in dependence of the structural constants of the system and possibly varying in time in order to optimize the efficacy of the attacks, which take to the extinction in finite time of the defensive population.

Alison Etheridge (University of Oxford, UK)

Travelling waves in the Allen Cahn equation and how to stop them

We outline some recent work with Ian Letter (Oxford) in which we consider a special form of the Allen Cahn equation, which can be thought of as a model for `hybrid zones' in nature. Our previous work considered populations evolving on the whole of two-dimensional Euclidean space. The new departure here is that we consider the equation on different domains.

Erika Hausenblas (University of Leoben, Austria)

Stochastic Pattern Formation

Mathematical models based on systems of reaction-diffusion equations provide fundamental tools for the description and investigation of various processes in biology, biochemistry, and chemistry; in specific situations, an appealing characteristic of the arising nonlinear partial differential equations is the formation of patterns, reminiscent of those found in nature. The deterministic Gray-Scott equations, or e.g. or Gierer Meinhardt model constitute an elementary two-component system that describes autocatalytic reaction processes; depending on the choice of the specific parameters, complex patterns of spirals, waves, stripes, or spots appear. In the derivation of a macroscopic model from basic physical principles, certain aspects of microscopic dynamics, e.g. fluctuations of molecules, are disregarded; an expedient mathematical approach that accounts for significant microscopic effects relies on the incorporation of stochastic processes and the consideration of stochastic partial differential equations. The randomness leads to a variate of new phenomena and may have a highly non-trivial impact on the behaviour of the solution. E.g. it has been shown by numerical modelling that the stochastic extension leads to a broader range of parameter with Turing patterns by a genetically engineered synthetic bacterial population in which the signalling molecules form a stochastic activator-inhibitor system. The stochastic extension may lead to multistability and noise-induced transitions between different states. In the talk, we will introduce the Gray Scott system and Klausmeier system, which is a special case of an activator-inhibitor system. Then, we give shortly the proof of existence and uniqueness, and introduce its numerical modelling.

Annie Millet (University of Paris I, France)

Space-time discretization schemes for the 2D Navier Stokes equations with additive noise

We consider the incompressible 2D Navier Stokes equations on the torus subject to an additive Gaussian perturbation. We prove the strong, i.e. L^2(\Omega), convergence of a fully implicit time Euler scheme coupled by a finite element (FEM) space discretization. The speed of convergence in time is almost 1/2, which is the Holder time-regularity of the solution in L^2; the speed of convergence in space is almost1. This result is optimal and proven under some condition on the trace of the covariance operator of the noise. The results are obtained for a deterministic or a random initial condition in H^1 which has some exponential moments, and for general FEM which satisfy the LBB condition, or more specific divergence free ones. The method relies on the existence of exponential moments for the solution and the implicit time Euler scheme. This a joint work with Hakima Bessaih.

Barbara Niethammer (University of Bonn, Germany)

An obstacle problem for cell polarization

We consider a simple bulk-surface reaction diffusion model to describe the response of biological cells to an outer signal. In this talk we will discuss steady states of this model and show that in a certain scaling limit solutions converge to solutions of an obstacle type problem. For this limiting problem we can characterize the the parameter regime in which polarization occurs. The related time-dependent problem will be discussed in the talk by Anna Logioti. Joint work with Anna Logioti (Bonn), Matthias Roeger (Dortmund) and Juan Velazquez (Bonn).

Postdoctoral Researchers

Petra Bacova (Cyprus Institute & IACM/FORTH)

Connecting mathematics, chemistry and physics: on modelling of complex polymer materials

Computational design of novel materials is a rapidly developing field which couples various aspects of mathematics, chemistry and physics. In an attempt to get a detailed description of the system under study and thus to be able to provide a quantitative bottom-up approach to obtain materials with predefined properties, atomistic molecular dynamics simulations are becoming the leading simulation technique. This technique preserves all chemical characteristics by assigning a series of chemistry-specific equations to each type of interaction. Such detailed description of macromolecular systems comes at high cost and the complexity of the problem requires the usage of supercomputer facilities and advanced analysis methods such as machine learning. Inspired by the versatility of the lego bricks, bottom-up approach uses the same principles: smaller building blocks such as polymers with complex architectures or composition are modelled first and then the acquired knowledge is applied in design of macrostructures.

Amrita Gosh (University of Bonn, Germany)

A brief overview of Contact Line Problem

We will discuss the general moving contact line problem which arises in many uid mechanics phenomena. Modelling this free boundary problem mathematically correctly is itself a challenging problem. The classical formulation of the model with the no-slip (Dirichlet) boundary condition for velocities at the interface of the solid and the liquid gives rise to a non-integrable singularity of the shear stress, which is known as the "moving contact line paradox". We will give an overview of different models available in the literature, giving solution to this paradoxical issue.

Afroditi Talidou ( University of Ottawa, Canada)

Propagation of pulses along cylindrical surfaces

The generation of an action potential that propagates along a nerve axon has been a problem of significant interest since the early '50s. In this talk, I will discuss the FitzHugh-Nagumo model on a surface of a long, thin cylinder that represents the axonal membrane of a single neuron. This model is a system of a partial differential equation coupled with an ordinary differential equation in two dimensions (plus time). Key questions are the existence of a pulse -- a special solution that travels along the length of the axon -- and its stability under small perturbations of the initial conditions and the geometry.

Graduate Students

Elisa Affili (University of Milan, Italy)

Population diffusion in periodic media with a fast diffusion line

In 1951, Skellam was the first to apply reaction-diffusion equations to predict animal dispersion, reporting great success. From that moment, mathematicians have studied sophisticated models to analyse the effects of environmental elements such as physical barriers or unfavourable zones. In this talk, I will present some results on ecological diffusion in periodic media exhibiting a fast diffusion line.

Maria Arnitalli (University of Crete, Greece)

Molecular Dynamics Simulations of Biomolecules

Mathematical and computational methods have been applied in biology, for instance in examination of behaviour of proteins in aqueous solution. Nowadays, Molecular Dynamics (MD) simulation constitutes a widely used computational technique in biology, like studying proteins in aqueous solution. MD generates a trajectory by numerical integration of classical equations of motion. The big asset of performing MD simulations is the capacity to obtain insight at atomic-level detail beyond what could be provided only through experiments. The produced trajectory contains all dynamical information, that it is necessary for our analysis. Here, we present the modeling of two proteins at their native state conformation through all-atom MD simulations and we show how the increase of temperature affects.

Eirini Gkolfi (University of Crete, Greece)

Computational study of complex polymeric systems through molecular dynamics simulations

Computer simulations serve as a powerful tool to imitate real-life problems without any laboratory cost. In this talk I will introduce Molecular Dynamics (MD) algorithm, used to simulate molecular systems, such as polymers, in different length and time scales. I will explain the way we combine mathematics, programming, physics and chemistry in order to represent a computational model of a complex molecular system, and to equilibrate it despite computational difficulties due to the huge number of particles (big data analysis problem). Finally, I will present few structural properties related to the cooperative motion of star-like polymers which we studied by using detailed, atomistic simulations and analyzed by applying computational algorithms tailored for this type of geometrical problems.

Rania Kousovista (University of Crete, Greece)

Non-linear mixed effect modeling and simulations of hydrochlorothiazide

The aim of this study was to investigate the pharmacokinetics of hydrochlorothiazide (HCTZ) using non-linear mixed effect modeling approaches. The model building was based on the concentration-time data of 32 subjects. In addition, external validation was carried out using an alternate dataset containing data of an independent group of patients. For comparative purposes, previously published population pharmacokinetic modelof HCTZ was also applied to the same data. Finally, a robust population pharmacokinetic model for HCTZ in adult patients was built and externally validated. Subsequently, this model was used to perform dosing simulationsto optimize personalized HCTZ dosing regimens in clinical practice.

Angeliki Koutsimpela (National Technical University of Athens, Greece)

The shallow lake problem

We discuss the welfare function of the deterministic and stochastic shallow lake problem, which concludes to be the viscosity solution of a Hamilton-Jacobi-Bellman equation. We present several properties including its asymptotic behaviour at infinity and we numerically investigate the equilibrium distribution and the path properties of the optimally controlled lake.

Anna Logioti (University of Bonn, Germany)

A free boundary problem modeling cell polarization

We are interested in a model for cell polarization as a response to an external signal which results in a system of PDEs for different variants of a protein on the cell surface and interior respectively. We study the evolution of this model for several reaction rates on the membrane as well as the diffusion coefficient inside the cell being large. It turns out that, in suitable scaling limits, solutions converge to solutions of obstacle type problems, for which we show uniqueness. Further, we prove the global stability of steady states. For the steady states case the occurrence of polarization has been shown by Niethammer, Röger and Velázquez (2020) provided that the total mass of protein is sufficiently small.

Georgia Mpaxevani (University of Crete, Greece)

Modeling dynamics of coarse-grained molecular systems, described by Generalized Langevin equation, using all-atom molecular dynamics simulations. .

In recent years, the development of coarse-grained models for studying large-scale physicochemical processes that cannot be practically studied with atomically detailed molecular dynamics simulations is an active research field. Defining the new effective coarse-grained system, which reduces dimensionality, means finding the model that best represents the reference system in structure and dynamic properties. In the present work, we estimate the coarse-grained system dynamics, described by the Generalized Langevin equation. Based on the effective approximation of the conservative force through a path-space variational inference method, for the Markovian approximation, at equilibrium, the friction coefficient is determined by utilizing information from the all-atom representation of the system, calculating correlation functions for the coarse-grained particles. The model's effectiveness is examined by comparing its structural and dynamical properties with the corresponding reference system. The methodology is illustrated in the example of a water molecular system.

Sofia Ntousi (University of Athens, Greece)

In Silico Clinical Trials for Anticancer Drugs

The aim of this study was to apply mathematical modeling & simulation techniques in order to show how pharmacokinetics (Concentration vs. time profile) and pharmacodynamics (Effect vs. time) can be used in clinical practice for dose personalized medicine. The case of a new anticancer drug was used in the simulations of new dosage regimens.

Nastia Tsyplakova (University of Athens, Greece)

Mathematical modelling and simulations towards donepezil individualized dosage regimens

The purpose of this study was the development of a population pharmacokinetic model of donepezil (used in the Alzheimer's disease) including the evaluation of the impact of crucial demographic characteristics of the population in the levels of donepezil concentrations. Secondly, based on the final model of this study, the purpose is to carry out simulations of the dosing regimens of donepezil in case of multiple administration.