# AC2MB Seminar: Fall 2015

## Applied and Computational Mathematics and Mathematical Biology seminar

Time:          Wednesdays at 2:15pm
Location:   CM.T01
• 30.09.15, DIFFERENT TIME: 4:15pm in CM.T01
Numerical Analysis of backward SPDE's and related problems
Andreas Prohl (U Tuebingen)

Abstract: Backward SPDE's e.g. arise as part of the optimality system for stochastic control problems. In this talk, I discuss the finite element discretization of the backward stochastic heat equation, and related simulation aspects. The results are then used to discretize the forward-backward SPDE system describing the stochastic control problem, and to provide corresponding error estimates. Algorithmic aspects will be detailed, and simulations are reported.
This is joint work with Thomas Dunst (U Tuebingen).

• 07.10.15
Tissue Mechanics: from Microstructure to Function - tissue diagnostics using mathematical and engineering approaches
Yuhang Chen (HWU)

Abstract: Unlike the classical biomechanics and one of its major sub-fields cell mechanics, instead of focusing on either system or cell/sub-cellular behaviours, tissue mechanics considers interaction between them at meso-scale which bridges our understanding on human physiology at multiple length scales. One of the critical questions that remain unresolved in tissue mechanics is how cell activities express at system level and how changes taking place at organ level regulate the cellular behaviours. In our group we focus on tissue microstructure as an intermediate player and attempt to answer this question using analytical and engineering approaches. In this talk, some examples that are currently being studied in our group will be presented, including enhanced clinical diagnostics for bone by nonlinear mechanics, diagnostics of cancer in soft tissue and coronary artery disease using dynamic mechanical sensing as well as in vivo design optimisation of tissue scaffold microarchitectures.

• 14.10.15
Instabilities in Evaporating Pools and Sessile Droplets: Ultrahigh Resolution DNS
P. Valluri (1), P. Saenz (1,2), O. K. Matar (2), K. Sefiane (1) and J. Kim (3)

(1) School of Engineering, The University of Edinburgh, United Kingdom
(2) Department of Chemical Engineering, Imperial College London United Kingdom
(3) Department of Mechanical Engineering, University of Maryland, USA

Abstract: This talk will discuss detailed evolution of thermocapillary instabilities (leading to hydrothermal waves) during evaporation of liquids under three geometrical configurations of pure fluids: i) planar pools ii) hemispherical sessile droplets and iii) non-hemispherical sessile droplets. The results presented are obtained via 3D, two-phase direct numerical simulations (DNS) that are validated rigorously against experiments and theory. We have used our home-grown DNS Solver TPLS-DIM (http://sourceforge.net/projects/tpls/). We will present our latest findings, which show the break of symmetry and the consequent development of a preferential direction for thermocapillary convection. As a result, counter-rotating whirling currents emerge in the drop playing a critical role in regulating the interface thermal motion. Our DNS show good agreement with experiments and reveal the intricate drop dynamics due to this geometry-induced phenomenon. The triggering mechanism is analysed along with the resulting bulk flow.

• 21.10.15
Photo-dissociation of molecules: mathematics meets quantum chemistry
Ben Goddard (UoE)

Abstract: Photo-dissociation is a chemical reaction in which molecules are broken down into atoms or smaller molecules by interaction with light (photons). Important examples include the formation and removal of the ozone layer, the break down of CFCs in the atmosphere, and part of photosynthesis in plants. More fundamentally, the photo-dissociation of diatomic molecules is one of the paradigmatic chemical reactions of quantum chemistry, typically used to benchmark new methods. The associated mathematical problem is the study of transitions in a two-level partial differential equation, with one effective spatial degree of freedom - the internuclear separation. Given a wavepacket that travels on the upper level, the challenge is to determine the size and shape of the part of the wavepacket transmitted to the lower level at large times. Such problems are highly multi-scale; the transmitted wavepacket is typically very small with rapid oscillations. This leads to great difficulty in performing accurate numerical calculations, and an alternative method is required. Fortunately, there exists a small parameter $\epsilon$ which is the square root of the the ratio of the electron and nuclear masses. In the standard adiabatic representation, widely used in chemistry, the transmitted wavepacket is of order $\epsilon$ globally in time but exponentially small (order $\exp(-1/\epsilon)$) for large times. This strongly suggests that the adiabatic representation is not the optimal one in which to study the problem. Using the more general superadiabatic representations, and an approximation of the dynamics in the region where the two energy levels become close but do not cross, we obtain an explicit formula for the transmitted wavepacket. Our results agree extremely well with high precision ab-initio calculations, in particular for the real-world sodium iodide molecule.
Joint work with Volker Betz (TU Darmstadt) and Stefan Teufel (U Tuebingen)

• 11.11.15
The essential numerical range for unbounded linear operators
Sabine Boegli (Cardiff)

Abstract: We study the approximation of a linear operator T by compression to an n-dimensional subspace of the domain of T (called Galerkin/finite section/projection method). It is well known that, in the limit n to infinity, the eigenvalues of the n-by-n matrix T_n may accumulate at a point that does not belong to the spectrum of T. The occurrence of such a spurious eigenvalue is commonly known as spectral pollution. In this talk we present methods to identify regions in the complex plane that enclose the set of spectral pollution (as tightly as possible). A useful tool is the notion of essential numerical range W_e(T) which was introduced in the late 1960s for bounded T. It was proved to contain the set of spectral pollution that arises for the Galerkin method. Moreover, it is a sharp enclosure in the sense that every point in W_e(T) can be arranged to be a spurious eigenvalue (by a clever choice of subdomains to which we truncate). We discuss the generalization of the notion of essential numerical range to unbounded operators, including equivalent characterizations and the above (sharp) property of enclosing the set of spectral pollution. Moreover, we prove that for selfadjoint operators the essential numerical range coincides with the convex hull of the (extended) essential spectrum, thus confirming the well-known fact that in the selfadjoint case spurious eigenvalues typically occur in gaps of the (extended) essential spectrum.

• 25.11.15, DIFFERENT TIME: 1:15pm in CM.T01
The Vagaries of Granular Flow
Raffaella Ocone (Chemical Engineering, HWU)

Abstract: Existing theories for particulate flow lack the robustness, predictability and flexibility required to handle the totality of phenomena that such flow may exhibit. Some unwanted industrial issues (such as particle agglomeration) and their management still remain an “art”. Current practice is based mainly on ad-hoc models for each specific flow condition and on operator experience. The talk will describe the journey from the first rigorous effort to model granular flow to recent theories and models to explain the vagaries observed in practical applications. Open problems, in need of a solution, will be presented and the search for a “unifying” theory will be discussed.

• 02.12.15
Asymptotic statistics of cycles in Surrogate-Spatial Random Permutations
Dirk Zeindler (Lancaster)

Abstract: We propose a natural approximation of the probability measure induced on the symmetric group by the so-called spatial random permutations, recently studied by V.Betz and Ueltschi. We show that this approximation shares some important properties with the original spatial model; in particular, under the thermodynamic limit both measures have the same critical density as well as the same fraction of points in infinite cycles. Using the greater analytic tractability of our model, we obtain a few new results about the asymptotic distribution of the cycle lengths.

• Tuesday, 15.12.15, 2pm, CM.T01
Musings on the properties of a mobile CO2 layer flowing in a porous sand: integrating monitoring and modelling
Gareth Williams (Geolocial Survey)

Abstract: