We consider a stochastic slow-fast nonlinear dynamical system derived from computational neuroscience. Independently, we uncover the mechanisms that underlie two different forms of weak-noise-induced resonance phenomena, namely, self-induced stochastic resonance (SISR) and inverse stochastic resonance (ISR) in the system. We then show that SISR and ISR are actually mathematically related through the relative geometric positioning (and stability) of the fixed point and the generic folded singularity of the system's critical manifold. This result could explain the experimental observation in which real biological neurons with identical physiological features and synaptic inputs sometimes encode different information.
Signalling pathways can be modelled as a biochemical reaction network. When the kinetics are to follow mass-action kinetics, the resulting mathematical model is a polynomial dynamical system. I will overview approaches to analyse these models using computational algebraic geometry and statistics. Then I will present how to analyse such models with time-course data using differential algebra and geometry for model identifiability, specifically focusing on parameter identifiability for parameter inference. I will briefly mention how topological data analysis can help distinguish models and data.
In this talk we briefly review some basic PDE models that are used to model phase separation in materials science. They have since become important tools in image processing and over the last years semi-supervised learning strategies could be implemented with these PDEs at the core. The main ingredient is the graph Laplacian that stems from a graph representation of the data. This matrix is large and typically dense. We illustrate some of its crucial features and show how to efficiently work with the graph Laplacian. In particular, we need some of its eigenvectors and for this the Lanczos process needs to be implemented efficiently. Here, we suggest the use of the NFFT method for evaluating the matrix vector products without even fully constructing the matrix. We illustrate the performance on several examples.
In this presentation we give an overview of several aspects related to the analysis and numerical simulation of elliptic and parabolic PDEs on (evolving) surfaces. We consider both scalar valued and vector valued PDEs. Surface partial differential equations are used in models from, for example, computational fluid dynamics and computational biology. Two examples of such applications are briefy addressed. Concerning the analysis of surface PDEs, a few results related to well-posedness of certain weak formulations are explained. We discuss a class of recently developed finite element discretization methods for an accurate numerical simulation of surface PDEs. The key idea of these methods is explained and numerical simulation results are presented.
Smooth isometric immersions of flat domains into ٰEuclidean spaces are known to enjoy a rigidity property referred to as developability, provided that the dimension of the target space is not too high. In particular, the images of such isometries from 2d domains into the 3d space are locally ruled surfaces. It has been known since a few years that this rigidity property survives if the second fundamental form of the immersion is merely L^2 integrable. On the other hand, convex integration methods à la Nash and Kuiper show that this would no more be the case for C^1α immersions if α < 1/5. We will show that a geometrically meaningful second fundamental form can be defined for such immersions if α is large enough in order to prove their developability for α > 2/3.
Reaction-diffusion equations are ubiquitous in a variety of fields including combustion and population dynamics. There is an extensive mathematical literature addressing the analysis of steady state solutions, traveling waves, and their stability, among other properties. Control problems involving these models arise in many applications . Often times, control and/or state constraints emerge as intrinsic requirements of the processes under consideration. There is also a broad literature on the control of those systems, addressing, in particular, issues such as the possibility of driving the system to a given final configuration in finite time. But, the necessity of preserving the natural constraints of the process are rarely taken into account. In this lecture we shall present the recent work of our team on the Fisher-KPP and Allen-Canh or bistable model, showing results of two different types depending of the initial and final states under consideration. First, the fact that, in some cases, the target can be reached for large enough time horizons, but that there is a minimal waiting time for this property to hold. And, second, negative results showing the existence of threshold effects, making some targets unreachable. We shall also present some numerical experiments showing that optimal trajectories are often quite complex, and hard to guess from purely analytical arguments.
Potent immunosuppressant drugs suppress the body’s ability to reject a transplanted organ up to the point that a transplant across blood- or tissue-type incompatibility becomes possible. In contrast to the standard kidney exchange problem, our setting also involves the decision about which patients receive from the limited supply of immunosuppressants that make them compatible with originally incompatible kidneys. We firstly present a general computational framework to model this problem. Our main contribution is a range of efficient algorithms that provide flexibility in terms of meeting meaningful objectives. We also show that these algorithms satisfy desirable axiomatic and strategic properties.
Dynamical systems have been immensely successful in describing and predicting behaviours of a wide range of applications especially in cases where they are well-modelled by a closed (unforced or autonomous) dynamical system. However, not all systems are closed; for example, elements in the climate system respond to changes in greenhouse gas forcing in quite a nontrivial manner over a range of timescales. For such open (forced or nonautonomous) systems with time-varying inputs, it is hard to get much insight out of dynamical systems theory and usually one resorts to numerical simulations. This is especially the case when inputs vary on a similar timescale to the system itself. I will discuss some recent progress in understanding critical transitions or tipping points in classes of nonautonomous dynamical systems, and an application to modelling a specific climate tipping point.