Mathematical Modelling: Perspectives on large scale systems, machine learning, and applications

The workshop will take place on the 26th November online.

Speakers and titles:

Organiser: Andrés Miniguano-Trujillo

Abstracts

Claudia Sagastizábal:
What is the cost of water? How can we price wind?

Sustainable sources of electricity are, without doubt, the future of energy generation worldwide.
When dealing with a resource like electricity, that cannot be stored in large quantities, generation needs to be done taking into account that energy that is generated and not used will be wasted.
This feature leads to challenging optimization problems whose solution has been considered in different formats by researchers since the 1970s. In order to make the mathematical models suitable for the transition to energy systems with low carbon emissions, those proposals require to be revisited and adapted.
Specifically, when choosing which technology to employ at each given time, an important element is the generation cost. Setting this cost appears as a difficult task for sustainable sources, as either hydroelectricity nor energy produced by wind farms use combustible. By means of examples, and without getting into technicalities, we shall explain how elegant tools of Variational Analysis can help in this sense.

Carola-Bibiane Schönlieb:
Mathematical imaging: From geometric PDEs and variational modelling to deep learning for images

Images are a rich source of beautiful mathematical formalism and analysis. Associated mathematical problems arise in functional and non-smooth analysis, the theory and numerical analysis of nonlinear partial differential equations, inverse problems, harmonic, stochastic and statistical analysis, and optimisation.

In this talk we will learn about some of these mathematical problems, about variational models and PDEs for image analysis and inverse imaging problems as well as recent advances where such mathematical models are complemented and replaced by deep neural networks.

The talk is furnished with applications to art restoration, forest conservation and cancer research.

Martin Stoll:
Learning with the graph Laplacian

In this talk we briefly review some basic PDE models that are used to model phase separation in materials science. They have since become important tools in image processing and over the last years semi-supervised learning strategies could be implemented with these PDEs at the core. The main ingredient is the graph Laplacian that stems from a graph representation of the data. This matrix is large and typically dense. We illustrate some of its crucial features and show how to efficiently work with the graph Laplacian. We then show how it can be applied in studying signed networks as well as focussing on learning with time-series where various distance measures play an important role.

Annegret Wagler:
About routing and spectrum assignment in optical networks, driven by combinatorial properties

The emergence of a new generation of optical networks, called elastic optical networks, allows a more flexible spectrum allocation for routing traffic demands within telecommunication networks. From this context arises the routing and spectrum assignment problem, which consists of routing a given set of origin-destination traffic demands and assigning them to contiguous frequency slots within an optical spectrum such that no frequency slot is assigned to more than one demand within a network link. In this talk, we discuss several approaches to solve this problem, including integer linear programming models and underlying combinatorial properties.