Doing a Phd at Heriot Watt and the Maxwell Institute

Sample Phd projects I offer

• Multiscale methods and Multiscale Interacting particle systems
  Many systems of interest in the applied sciences share the common feature of possessing multiple scales, either in time or in space, or both. Some approaches to modelling focus on one scale and incorporate the effect of other scales (e.g. smaller scales) through constitutive relations, which are often obtained empirically. Multiscale modelling approaches are built on the ambition of treating both scales at the same time, with the aim of deriving (rather than empirically obtaining) efficient coarse grained models which incorporate the effects of the smaller/faster scales. Multiscale methods have been tremendously successful in applications, as they provide both underpinning for numerics/simulation algorithms and modelling paradigms in an impressive range of fields, such as engineering, material science, mathematical biology, climate modelling (notably playing a central role in Hasselmann’s programme, where climate/ whether are seen as slow/fast dynamics, respectively), to mention just a few. In this project, which is in the field of applied stochastic analysis, we will consider systems that are multiscale in time, with particular reference to multiscale interacting particle systems. We will try to understand how the multiscale approximation interacts with the mean field approximation (produced by letting the number of particles in the systems to infinity so to obtain a PDE for the evolution of the density of the particles). The motivation for this project comes especially from models in mathematical biology, but the applicabity of the framework we will investigate is broader.
  
  To apply for this project see here



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• Interacting Particle systems and Stochastic Partial Differential Equations
  Many interesting systems in physics and in the applied sciences consist of a large number of particles or agents, (e.g. individuals, animals, cells, robots) that interact with each other. When the number of agents/particles in the system is very large the dynamics of the full Particle System (PS) can be rather complex and expensive to simulate; moreover, one is quite often more interested in the collective behaviour of the system rather than in its detailed description. In this context, the established methodology in statistical mechanics and kinetic theory is to look for simplified models that retain relevant characteristics of the original PS by letting the number N of particles to infinity; the resulting limiting equation for the density of particles is a low dimensional, (in contrast with the initial high dimensional PS) non-linear partial differential equation (PDE), where the non-linearity has a specific structure, commonly referred to as a McKean-Vlasov nonlinearity. Beyond an intrinsic theoretical interest, such models were proposed with the intent to efficiently direct human traffic, to optimize evacuation times, to study rating systems, opinion formation, animal navigation strategies or, noticeably, in control engineering (e.g. collective flight of drones in artificial fireworks displays); and in all these fields they have been incredibly successful.In this project we will consider PSs modelled by Stochastic Differential Equations (SDEs) whose limiting behaviour is described by either a deterministic PDE or a stochastic PDE (SPDE) (note the greatness of this: depending on the nature of the stochasticity in the PS, the limiting equation can be either deterministic or stochastic!). Either way, the limiting equation is of McKean-Vlasov type. The overall aim of the project is the comparison of the ergodic and dynamic properties of the particle system and of the limiting PDE/SPDE. these results will help inform modelling decisions for practitioners. Keywords for this project are: Stochastic (Partial) Differential equations, McKean Vlasov evolutions, ergodic theory, mean field limits.Prerequisites: good background in either stochastic analysis/probability or analysis
  
  To apply for this project see here