In the upcoming weeks we shall be having regular meetings to expose our work in our team. The format is, approximately, 20 min talk plus 10 min questions.

Where? At Pierre's office.

When? The dates are the following:

• Wednesday 5th of December 2018, Angelika Manhart and Yuhua Zhu from 10 am to 11 am.

Angelika Manhart: Traveling Waves in Cell Populations. Abstract:Transport-reaction equations are abundant in the description of movement of motile organisms. In this talk I will focus on a system of coupled transport-reaction equations that arises from an age-structuring of a species of turning individuals. The highlight consists of the explicit construction and characterization of counter-propagating traveling waves, patterns which have been observed in bacterial colonies, e.g. in earth-dwelling myxobacteria. Fascinatingly, while the wave profiles do not change, the wave composition does and the fractions of reversible and non-reversible bacteria form waves traveling in the opposite direction. Stability analysis reveals conditions for wave formation as well as for pulsating-in-time spatially constant solutions. Slides

Yuhua Zhu: Toward the Theoretical Understanding of Large-batch Training in Stochastic Gradient Descent. Abstract: Stochastic gradient descent (SGD) is almost ubiquitously used for training nonconvex optimization tasks including deep neural networks. Recently, a hypothesis that "large batch SGD tends to converge to sharp minimizers of training function" has received increasing attention. We develop some new theory to give a justification of this hypothesis. In particular, we provide new properties of SGD in both finite-time and asymptotic regimes, with the tools from empirical processes and Partial Differential Equations. A connection between the stochasticity in SGD and the idea of smoothing splines in nonparametric statistics is also built. We include numerical experiments to corroborate these theoretical findings. Slides

Past Events

• Monday 12th of February 2018, Guanglian and Sophie from 11 am to 12 am.

Guanglian Li: Multiscale Model Reduction to Flow Problems with High-Contrast Heterogeneous Random Coefficient. Abstract: This talk is concerned with deriving the efficient numerical methods for flow problems with high contrast heterogeneous random coefficients, which arise from many applications, for example, reservoir simulation and material science. In specific, we are interested in two types of model order reduction: the truncation estimate with respect to the random variables and the multiscale model reduction. There are many approaches to arrive at the truncation estimate in the literature, including the polynomial chaos expansion, the localized wavelets expansion and the Karhunen-Loeve (KL) expansion. Here, we will stick to the Karhunen-Loeve (KL) expansion which yields the optimal truncation estimate in the mean square error. The numerical estimate for KL expansion will be examined in details. Next, we will present the efficient numerical methods for solving the flow problem with each realization of the random coefficient. This is realized by multiscale analysis. We will present the construction of local multiscale basis functions, the randomized snapshot algorithm and the adaptive GMsFEM algorithm. Finally, I will close my talk with a brief discussion on the optimality of local multiscale basis functions. Slides

Sophie Hecht: Mathematical model for tissue growth. Abstract: The study of size and shape control of tissue is a major issue because it could lead to improvement in the understanding of tumor proliferation or tissue regeneration. Some possible regulators of the growth are mechanical feedback and morphogens diffusion. We aims to understand the influence of mechanical forces such as stretching and compression during the development, by creating models for cell division at different scales of precision. At the continuous macroscopic level, we have first modeled the growth of a tissue with two categories of cell unable to mix, to then be able to observe if stretching could appear in the tissue. For the microscopic model, we are considering each cell individually confined in a box with a local attraction between each other and aimed to see if the nucleus are compressed during the development. Slides

• Monday 19th of February 2018, Lifan and Rafael from 11 am to 12 am.

Rafael Bailo: Optimal Consensus Control of the Cucker-Smale Model. Abstract: This talk is concerned with the study of the numerical realisation of optimal consensus control laws for agent-based models. For a nonlinear multi-agent system of Cucker-Smale type, consensus control is cast as a dynamic optimisation problem for which we derive first-order necessary optimality conditions. In the case of a smooth penalization fo the control energy, the optimality system is numerically approximated via a gradient-descent method. For an increasing number of agents, we discuss the approximation of the consensus control problem by following a mean-field modelling approach. Slides Preprint

Lifan Xuan: A model for the dynamics of limit order book. Abstract: In modern financial markets, more than half of the trades are executed through limit order book(LOB) mechanism, which leads to the massive studies into the dynamics of limit order book. In this presentation, a measure-based model will be introduced, in which we describe the evolution of limit order book as a two-time step process. Compared to the past model, a market clearing operator is introduced which makes the model a general framework for other models. In the first section the background is discussed followed by the introduction about the model. Then a simple example to used to show how our model can be applied followed by another simple example of time scaling limits. Finally we conclude. Slides

• Monday 12th of March 2018, Pedro Aceves-Sanchez and Matt Barker from 11 am to 12 am.

Pedro Aceves-Sanchez: Emergence of vascular networks. Abstract: This talk is concerned with the formation of vascular networks. The generation of vascular networks is a long standing problem which has been the subject of intense research in the past decades. One of the main reason being the wide spread applications that it has in tissue regeneration, wound healing, cancer treatments, among many others. The mechanisms involved in the formations of vascular networks are complex and despite the vast amount of research devoted to it there are still many mechanisms involved which are poorly understood. Our aim is to bring insight into the study of vascular networks by defining heuristic rules, as simple as possible, and to simulate them numerically to test their relevance in the vascularization process. We introduce a hybrid agent-based/continuum model coupling blood flow, oxygen flow, capillary network dynamics and tissues dynamics.

Matt Barker: Linking mean field games to the best reply strategy. Abstract: Mean field games (MFGs) are used to describe the evolution of large numbers of weakly interacting agents strategically optimising their behaviour. Under the framework of MFGs, agents are assumed to be able to change their control on a continuous basis with a long-term view to their optimisation. I will discuss a more realistic approach to describing the behaviour of human agents, known as the best reply strategy (BRS), explain how it can be derived from MFGs and discuss some toy models. Finally, I will provide insight into future research prospects, these include extending the BRS to a wider variety of cost functions, potential applications to promoting a low-carbon economy, and comparing the difference in the behaviour of agents between MFG and BRS approaches.

• Tuesday 20th of November 2018, Pedro Aceves-Sanchez from 11 am to 12 am.

Pedro Aceves-Sanchez: Emergence of vascular networks. Abstract: This talk is concerned with the formation of vascular networks. The generation of vascular networks is a long standing problem which has been the subject of intense research in the past decades. One of the main reason being the wide spread applications that it has in tissue regeneration, wound healing, cancer treatments, among many others. The mechanisms involved in the formations of vascular networks are complex and despite the vast amount of research devoted to it there are still many mechanisms involved which are poorly understood. Our aim is to bring insight into the study of vascular networks by defining heuristic rules, as simple as possible, and to simulate them numerically to test their relevance in the vascularization process. We introduce a hybrid agent-based/continuum model coupling blood flow, oxygen flow, capillary network dynamics and tissues dynamics. Out latest results shall be presented.

If you have any question please contact P. Aceves-Sanchez (pacevess@ic.ac.uc) or Prof. P. Degond (p.degond@ic.ac.uk).