This year’s SIAM Student Chapter Symposium takes place on Friday, 31 May starting at 13:00 at the Bayes Centre seminar room 5.02.

The afternoon will consists of four presentations (see schedule below) by distinguished lecturers from St Andrews and Edinburgh who will provide an insight into their research and might even delve into more recreational topics. The talks will be accessible to a wide mathematical audience.

Call for Posters: After the talks, we will have a poster session where you will be able to showcase and discuss your work with fellow PhD students over some drinks and nibbles. If you would like to contribute a poster, please get in touch.

Attendance of the event is free, but registration by 26/05 is required for catering purposes.

We’re hoping to see many of you at the event!

Schedule:
12:45 - 13:00: Registration
13:00 – 14:00: Heiko Gimperlein (Heriot-Watt, MACS): Nonlocal diffusion: From swarming robots to the analysis of fractional PDEs
14:00 – 15:00: Ahmed Elsheikh (Heriot-Watt, IPE): Machine learning approaches for quantifying uncertainty in subsurface flows
15:00 – 15:30: Coffee break
15:30 – 16:30: Ben Goddard (University of Edinburgh): Recreational maths for pleasure, pedagogy and (possibly) profit
16:30 – 17:30: Tommaso Lorenzi (University of St Andrews): Dissecting the evolutionary and spatial dynamics of cancer through nonlinear partial differential equations.
17:30: Wine reception & poster session

Abstracts:

Heiko Gimperlein (Heriot-Watt, MACS): Nonlocal diffusion: From swarming robots to the analysis of fractional PDEs
Abstract: This talk discusses diffusion processes beyond Brownian motion and their description by nonlocal differential operators, such as the fractional Laplacian. We consider their microscopic derivation for interacting particle systems, as well as the pure and numerical analysis of the resulting nonlocal heat equations. The results have applications to the design of swarm robotic systems and the chemotactic movement of bacteria.

Ahmed Elsheikh (Heriot-Watt, IPE): Machine learning approaches for quantifying uncertainty in subsurface flows
Abstract: Computational models for multi-phase flow in porous media relies on a number of subsurface parameters that are poorly known (e.g. porosity and permeability fields). In practical setting, these parameters are observed at sparse set of points (e.g wells) and/or indirectly using low resolution imaging techniques (seismic surveys). Quantifying the impact of these parameters on the model outputs is an important task for robust decision support and risk assessment. In this talk, I will present two different techniques to handle this challenging uncertainty propagation task given the high dimensionality of the input parameter space and the non-polynomial nonlinearities in the subsurface flow model.
In the first part of my talk, I will introduced a machine learning based multiscale method [1] for solving elliptic equations (e.g. pressure equation in subsurface flow problems). Several multiscale methods account for sub-grid scale heterogeneities using coarse scale basis functions for upscalling. For example, in the Multiscale Finite Volume method (FVM), coarse scale basis functions are obtained by solving a set of local problems over a dual-grid. In this work, we introduce a data-driven approach for estimating the coarse scale basis functions using a neural network (NN) predictor fitted using a set of training data. For uncertainty propagation tasks, the trained NN learns to generate basis functions at a lower computational cost when compared to solving the local problems. The computational advantage of this approach is realized when a large number of realizations has to be evaluated. We attribute the ability to learn these basis functions to the modularity of the local problems and the redundancy of the permeability patches between samples. The proposed method is evaluated on single phase flow problems yielding very promising results.
In the second part of my talk, I will introduce a deep residual recurrent neural network (DR-RNN) as an efficient model reduction technique for subsurface multi-phase flow problems [2]. DR-RNN is a physics-aware recurrent neural network for modeling the evolution of dynamical systems. The architecture of DR-RNN is inspired by iterative update techniques of line search methods where a fixed number of layers are stacked together to minimize the residual (or reduced residual) of the physical model under consideration. For subsurface flow models, we combine DR-RNN with proper orthogonal decomposition (POD) and discrete empirical interpolation method (DEIM) to reduce the computational complexity associated with high-fidelity numerical simulations and thus reduce the total cost of uncertainty quantification tasks. Our numerical evaluations show that DR-RNN combined with POD–DEIM provides an accurate and stable reduced models with a fixed computational budget that is much less than the computational cost of standard POD–Galerkin reduced model combined with DEIM.

References:

[1] S Chan, AH Elsheikh, "A machine learning approach for efficient uncertainty quantification using multiscale methods", Journal of Computational Physics, 354, 493-511, 2018.

[2] JN Kani, AH Elsheikh, "Reduced-Order Modeling of Subsurface Multi-phase Flow Models Using Deep Residual Recurrent Neural Networks", Transport in Porous Media, 126 (3), 713-741, 2019.

Ben Goddard (University of Edinburgh): Recreational maths for pleasure, pedagogy and (possibly) profit
Abstract: Martin Gardner, one of the leading figures of recreational maths, described it as the part of maths that 'includes anything that has a spirit of play about it'. As well as providing enjoyment [pleasure], many recreational maths problems provide excellent opportunities for explaining mathematical concepts in clear and engaging ways [pedagogy]. After attempting to give a definition of recreational maths, we'll briefly discuss some of its long and wide-ranging history. The remainder of the talk will be devoted to some examples of recreational maths problems, including seeing how they can be (elegantly?) solved with university-level maths. There will be time to try the problems yourself - please bring a pen and paper - with the chance of winning a prize [possibly profit].

Tommaso Lorenzi (University of St Andrews): Dissecting the evolutionary and spatial dynamics of cancer through nonlinear partial differential equations.
Abstract: Mathematical modelling can complement experimental cancer research by offering alternative means of interpreting extant data and enabling extrapolation beyond empirical observation. In this talk, I will present a number of results illustrating how analysis and numerical simulation of mathematical models formulated in terms of nonlinear partial differential equations can uncover fresh insights into the underpinnings of the evolutionary and spatial dynamics of cancer.

List of Posters:

Ilnaz Ashayeri: Structural behaviour of aged timber elements under loading
Emine Atici Endes: E and N Cadherin-Dependent Non-Local Model For Wound Healing
Andres Barajaz Paz: Age heaping in population data of emerging countries
Lukas Eigentler: Metastability as a coexistence mechanism in a model for dryland vegetation patterns
Gissell Estrada Rodriguez: Space-time fractional equations in chemotaxis and immunology
Masoud Ghaderi Zefreh: Mesh-free simulation for precipitation and dissolution in a 2D discrete fracture network
Panagiotis Kaklamanos: Regularization and geometry of piecewise smooth systems with intersecting discontinuity sets
Panagiotis Kaklamanos: Local & global implications of multiple time-scale interactions
Stefania Lisai: Smooth solutions of the surface semi-geostrophic equations
Jakub Stocek: Fractional PKS in Immunology
Cat Wedderburn: Burn, Baby, Burn: Mathematical Optimisation of Contact Networks to Limit Disease Spread

SIAM Student Chapter Symposium 2019