## Applied Algebra and Combinatorics

### Swansea, March 9th, 2018

The meeting is partially supported by the LMS New Appointments Grant. It aims at bringing together researchers from the UK with interests in Algebra, Algebraic Geometry, and Combinatorics whose work is particularly motivated by practical problems arising in areas of applications such as computational algebra, statistics or computational design. There will be three talks in the morning complemented by an afternoon session on mathematical challenges in Computer Aided Geometric Design.

**Location**

Swansea University, Singleton campus, Department of Mathematics, **Talbot 224**.

SA2 8PP, Swansea, UK.

**Program **

- 09:30 - 10:00 Registration and welcoming, tea and biscuits ☕ 🍪.
- 10:00 - 12:00
**Morning Session:****Applied algebra and combinatorics**

10: 00 - 10:40 Emilie Dufresne (University of Nottingham) ☞ Toric reparametrizations of linear compartmental models.

10:40 - 11:20 Fatemeh Mohammadi (University of Bristol ) ☞ Learning Bayesian Networks Using Generalized Permutohedra.

11:20 - 12:00 Nelly Villamizar (Swansea University) ☞ Smooth splines on surfaces with arbitrary topology.

14:00 - 16:00 **Afternoon Session:** **Applications of spline methods in CAGD**

14:00 - 14:40 Malcolm Sabin (Numerical Geometry Ltd, Cambridge) ☞ Basis Functions, IGA and Subdivision.

14:40 - 15:20 Panagiotis Kaklis (University of Strathclyde) ☞ An IGA-BEM method for the open-water flow problem around rotating marine propellers.

15:20 -16:00 Rubén Sevilla (Swansea University) ☞ Finite element mesh generation and simulations without geometric de-featuring.

- 19:00 Social dinner.

**Organizers and sponsors**

Nelly Villamizar (Swansea University).

London Mathematical Society, and Swansea University-Department of Mathematics.

**Abstracts: **

*Title of the talk:* Toric reparametrization of linear compartmental models.

*Abstract: *For ODE models with time series data, structural identifiability concerns determining if the parameters of a model are completely determined by continuous time series data given an input, or equivalently, from its input-output data. Linear compartment models are linear ODE models used in systems biology and pharmacokinetics and encoded in a labelled directed graph. Many such models are unidentifiable: parameters can take on an infinite number of values and yet yield the same input-output data. In this work, we generalise the work of Meshkat and Sullivant using techniques of Hubert and Labahn, and show how an identifiable reparametrization can be found using scaling symmetries for a certain class of unidentifiable linear compartmental models. More generally, we observe that even when an identifiable reparametrization is not possible, using scaling symmetries leads to reparametrizations that are closer to being identifiable.

Joint with Nikki Meshkat.

*Title of the talk:* Learning Bayesian Networks Using Generalized Permutohedra.

*Abstract:* Graphical models (Bayesian networks) based on directed acyclic graphs (DAGs) are used to model complex cause-and-effect systems. A graphical model is a family of joint probability distributions over the nodes of a graph which encodes conditional independence relations via the Markov properties. One of the fundamental problems in causality is to learn an unknown graph based on a set of observed conditional independence relations. In this talk, I will describe a greedy algorithm for DAG model selection that operate via edge walks on so-called DAG associahedra. For an undirected graph the set of conditional independence relations are represented by a simple polytope known as the graph associahedron, which can be constructed as a Minkowski sum of standard simplices. For any regular Gaussian model, and its associated set of conditional independence relations we construct the analogous polytope DAG associahedon which can be defined using relative entropy. For DAGs we construct this polytope as a Minkowski sum of matroid polytopes corresponding to Bayes-ball paths in graph.

Joint work with Caroline Uhler, Charles Wang, and Josephine Yu.

*Title of the talk: *Smooth splines on surfaces with arbitrary topology.

*Abstract: *We consider the space of piecewise polynomial functions which are globally differentiable on a mesh of general topology. This space is characterized by gluing data across the shared edges. Using algebraic techniques, which involve the analysis of the module of syzygies of the gluing data, we give a dimension formula for the space of geometrically smooth spline functions of degree less than or equal to a given constant, defined on surfaces of arbitrary topology. The dimension formula applies to spline spaces where the polynomial degree is sufficiently large. In the talk we will discuss this condition on the polynomial degree, and provide explicit constructions of basis functions attached respectively to vertices, edges and faces. These results can be extended to the study of the space of differentiable functions on a quad-mesh, which are composed of 4-split spline macro-patch elements on each quadrangular face. The approach is illustrated with applications to fitting and reconstruction problems.

Joint work with Ahmed Blidia, and Bernard Mourrain.

*Title of the talk: *Basis Functions, IGA and Subdivision.

*Abstract: *The theoretical basis for our commercial finite element analysis software dates from the late 1950’s and early 1960’s. Since then our computers have increased in memory size by a factor of over a million and in speed by several thousand. The computing context is different now, and we address problems of a totally different scale. The IsoGeometric analysis bandwagon has led many to look again at the fundamentals and it is now possible to ask whether there is accessible advantage in new ideas. This talk puts together the key idea from isogeometry, that splines make good basis functions (in particular because they provide conveniently nested spaces) with the range of analysis techniques now available, to suggest that maybe there are opportunities to improve the speed/accuracy tradeoff by orders of magnitude.

*Title of the talk: *An IGA-BEM method for the open-water flow problem around rotating marine propellers.

*Abstract: *Combining Iso-Geometric analysis (IGA) with Boundary Element Methods (BEM) for inviscid lifting flows imposes a number of difficulties. Firstly, an IGABEM collocation scheme has to take into account the tangent-plane discontinuity occurring along the trailing edge (TE) of such lifting geometries. More important, the scheme has to handle the non-linear Kutta condition, securing continuity of the pressure at the TE and through the a-priori unknown wake, a force-free boundary surface emanating from TE. In this presentation we shall provide some preliminary results of our work towards developing a IGABEM collocation scheme for computing steady lifting flows around 3D bodies, such as marine propellers.

*Title of the talk:* Finite element mesh generation and simulations without geometric de-featuring.

*Abstract: *The initial stage of a simulation process is the generation of an appropriate mesh that is designed to capture the required physics. When dealing with complex objects that contain multi-scale geometric details, it is often necessary to manually remove small features which, in the expert opinion of the analyst, will not have a measurable effect on the numerical solution. The major drawback of this de-featuring stage is the requirement for manual interaction with CAD systems. Furthermore, the de-featuring must be repeated for every physical system to be considered in the design, i.e. fluids, solids, electromagnetics, acoustics, heat transfer, etc. In addition, the de-featuring is often dependent on problem parameters such as the Reynolds number in a fluid mechanics problem or the frequency of the waves in electromagnetics or acoustics. In this talk, a new computational framework is presented. The proposed technique involves a new mesh generation paradigm where the element size is dictated by the required resolution of the physical phenomena, irrespective of the geometric complexity. All geometric details are encapsulated in the mesh without unnecessary mesh refinement, removing the uncertainty induced by a de-featuring process and ensuring both efficiency and robustness. The modifications of an existing finite element solver, required to utilise the meshes generated with the proposed approach, will also be presented.