AC2MB Seminar: Spring 2016

## Applied and Computational Mathematics

and

Mathematical Biology seminar

**Time:**Wednesdays at 2:15pm

**Location:**CM.T01

**January, 20**

__Negative absolute temperatures: do they make sense?__

Oliver Penrose (Heriot-Watt)

__Abstract:__*Various experiments have been performed demonstrating that, under very special circumstances, it is possible to bring matter to a state that can be described in terms of negative absolute temperatures. However, these temperatures have paradoxical properties --- for example such temperatures, rather than being "colder" than absolute zero (-273 Celsius) as the name might suggest, are sometimes interpreted as being "hotter" than the interior of a star --- and two years ago a paper in "Nature Physics" claimed to cut through the confusion by showing that negative absolute temperatures were forbidden by basic principles of statistical thermodynamics. I will talk here about how we define entropy and temperature in (classical) statistical mechanics and how this definition is consistent with the way these concepts are used in thermodynamics. For some of the above-mentioned experiments, an argument based on the second law of thermodynamics leads to the conclusion that the experiments really do produce negative absolute temperatures, and a careful examination of the mathematical argument used in the "Nature Physics" paper shows that there is in fact no conflict with the basic principle of statistical thermodynamics.*

**February, 17**

__Probabilistic global well-posedness of the energy-critical defocusing nonlinear wave equation bellow the energy space__

Oana Pocovnicu (Heriot-Watt)

__Abstract:__*We consider the energy-critical defocusing nonlinear wave equation (NLW) on $R^d$, $d = 3, 4, 5$. In the deterministic setting, Christ, Colliander, and Tao showed that this equation is ill-posed below the energy space $H^1 x L^2$. In this talk, we take a probabilistic approach. More precisely, we prove almost sure global existence and uniqueness for NLW with rough random initial data below the energy space. The randomization that we use is naturally associated with the Wiener decomposition and with modulation spaces. The proof is based on a probabilistic perturbation theory and on probabilistic energy bounds. Secondly, we prove analogous results in the periodic setting, for the energy-critical NLW on $T^d$, $d = 3, 4, 5$. The main idea is to use the finite speed of propagation to reduce the problem on $T^d$ to a problem on Euclidean spaces. If time allows, we will briefly discuss how the above strategy also yields a conditional almost sure global well-posedness result below the scaling critical regularity, for the defocusing cubic nonlinear Schroedinger equation on Euclidean spaces.*

This talk is partially based on joint work with Tadahiro Oh and on joint work with Arpad Benyi and Tadahiro Oh.

**Different day and time: Friday, March, 4, 2:15pm in CM.S01**

**Ten things you should know about quadrature**

**Nick Trefethen (Oxford)**

__Abstract:__*Quadrature is the term for the numerical evaluation of integrals. It's a beautiful subject because it's so accessible, yet full of conceptual surprises and challenges. This talk will present ten of these, with plenty of history and numerical demonstrations. Some are old if not well known, some are new, and two are subjects of my current research.*

**March, 9**

__Modelling In-Situ Upgrading (ISU) of heavy oil using dimensionless numbers and Operator Splitting (OS)__

Jullien Maes (Heriot-Watt)

__Abstract:__*The In-Situ Upgrading of heavy oil and oil shale is a challenging process to model. In addition to the transport of heat by conduction and convection, and the change of properties with varying pressure and temperature, this process involves transport of mass by convection, evaporation, condensation and pyrolysis chemical reactions. The behaviours of these systems are difficult to predict as relatively small changes in the material composition can significantly change the thermo-physical properties. Accurate prediction is further complicated by the fact that many of the inputs needed to describe these processes are uncertain, e.g. the reaction constants and the temperature dependence of the material properties. Analysis using dimensionless numbers can provide a useful insight into the relative importance of different parameters and processes, especially if combined with Design of Experiments (DOE), which allows quantification of the impact of the parameters with a minimal number of numerical experiments. Ranking the different parameters enables experimental programmes to be focused on acquiring the relevant data with the appropriate accuracy. Furthermore, Numerical simulations of the ISU process are often slow since a large number of components is generally needed to describe accurately the chemical reactions. Operator splitting (OS) methods are one way of potentially improving computational performance. Each numerical operator in a process is modelled separately, allowing the best solution method to be used for the given numerical operator. A significant drawback to the approach is that decoupling the governing equations introduces an additional source of numerical error, known as splitting error. Obviously the best splitting method for modelling a given process is the one that minimises the splitting error whilst improving computational performance over that obtained from using a fully implicit approach. This works has three main objectives: (1) to quantify the main interactions between the heat conduction, the heat and mass convection and the chemical reactions, (2) to identify the primary parameters for the efficiency of the process and (3) to design a numerical method that reduces the CPU time of the simulations with limited loss in accuracy. We show that the most important parameters controlling the efficiency of the process are the activation energy of the reactions, the reaction enthalpies and the stoichiometric coefficients. We develop a new splitting algorithm, called SSO-CKA-TR-AIM that is generally more precise than FIM and can generate large speed-up.*

**March, 30**

__New horizons in multiscale modelling and analysis: the novel concept of three-scale convergence__

Dumitru Trucu (Dundee)

__Abstract:__*In this work we propose a new notion of multiscale convergence, called "three-scale", which aims to give a topological framework in which to assess complex processes occurring at three different scales or levels within a heterogeneous medium. While these kind of three-scale processes occur in many physical, chemical and biological situations, this work was motivated by our specific research interest that is focused on the development of a novel multiscale modelling framework for cancer cells invasion of human body tissue. The genuinely multiscale nature of the cancer invasion process is explored in this new framework via a novel three-scale moving boundary modelling approach. Naturally, this modelling leads to a question concerning the establishment of a fundamental framework that would enable a rigorous analysis of involved operators. The new multiscale analysis concept that we introduce here generalises and extends the notion of two-scale convergence, a well-established concept that is commonly used for obtaining an averaged, asymptotic value (homogenization) of processes that exist on two different spatial scales. The well-posedness of this new concept is justified via a compactness theorem which ensures that all bounded sequences of square integrable functions over a bounded domain are relative compact with respect to the three-scale convergence. This is taken further by giving a boundedness characterisation of three-scale convergent sequences and is then continued with the introduction of the notion of "strong three-scale convergence" whose well-posedness is also discussed. Finally, the three-scale convergence of the gradients is established.*

**April, 13**

__Universal asymptotic of clone size distributions for arbitrary population growth__

Tibor Antal (UoE)

__Abstract:__*We are interested in modeling the evolution of the size distribution of metastasis. For this we consider a deterministically growing primary tumor which randomly seeds metastatic lesions, which lesions then evolve stochastically. The time evolution of the size of the tumor before detection, however, is hardly ever accessible from clinical data. Fortunately, the details of this growth function are irrelevant. To show this, we first provide exact expressions for exponential (usually termed as Luria-Delbruck or Lea-Coulson model), power-law and logistic population growth. We then prove that the large time limit of the clone size distribution has a general two-parameter form for any type of population growth. Moreover, the clone size distribution always has a power law (fat) tail, with infinite moments. We show that for sub-exponential primary growth the probability of a give clone size is inversely proportional to the clone size. We support our findings by analyzing a dataset on tumor metastasis sizes, and we find that a power-law tail is more likely than an exponential one. Joint work with Michael Nicholson.*

**May, 11**

__Dispersion and front propagation in the large-deviation regime__

Jacques Vanneste (UoE)

__Abstract:__*A passive scalar, e.g. a pollutant, released in a fluid flow disperses under the action of advection by the flow and molecular diffusion. In the long-time limit, the combined effect of advection and diffusion can often be approximated as an enhanced diffusion, with an effective diffusivity that may be obtained using homogenisation techniques. This approximation applies only to the (Gaussian) core of the scalar distribution. I will discuss how the form of the non-Gaussian tails can be computed by applying the probabilistic theory of large deviations. The tail behaviour is particularly important for a class of reacting scalars (FKPP reactions) which leads to the formation of propagating fronts whose speed is determined by the large-devation rate function. I will consider two types of periodic flows — cellular (Taylor-Green) flows, and flows in Manhattan-style network geometries — for which explicit results can be derived.*

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JOINT WORK?