Seminar in PDE and Applications

The idea of this seminar is to bring together local and international experts in the field of partial differential equations (PDE) and related fields. In the talks we want to cover various aspects of PDE and their applications, including modeling, mathematical analysis and numerics. Our goal is to increase the visibility of work that is being done in the field of PDE across the different groups at DIAM and to serve as a meeting platform within the department.

The seminar takes place on Thursdays at 4pm.

If you are interested in talks about topics in Applied Analysis, check out the seminar page.

Organisers:

Yves van Gennip (MP)

Anna Geyer (MP)

Manuel Gnann (MP)

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28 September 2023 16:00 till 17:00

[PDE & Applications seminar] Alejandro Aragon: Enriched Finite Element Analysis and Design of Metamaterials and Structures

"Enriched finite element analysis (e-FEA) has an edge over standard FEA for problems that require complex meshing or those that involve moving boundaries. An e-FEA widely known is the eXtended/Generalized Finite Element Method (XFEM/GFEM). However, this methodology is no without issues, which include the lack of stability (arbitrarily large condition number of the stiffness matrix) when boundaries get arbitrarily close to standard mesh nodes, a non-trivial implementation of non-homogeneous essential boundary conditions, and a challenging computer implementation. These issues stem from the fact that enrichments are associated to standard mesh nodes. To circumvent these issues, while retaining the mesh-geometry decoupling property of XFEM/GFEM, a new family of interface- and discontinuity-enriched finite element methods has been proposed [1]. In these techniques, enrichments are placed directly along discontinuities, thereby solving the aforementioned issues of XFEM/GFEM.
This presentation will delve into e-FEA based on discontinuity-enriched FEMs. We will give an overview on the analysis capabilities of these methods for a wide set of problems, including the modeling of interfaces, fracture, immersed boundary problems, the coupling of non-conforming meshes and contact. We will also discuss recent developments in topology optimization, for which these e-FEA techniques are combined with a level set description of geometry. Problems investigated include compliance minimization [2], tailoring of fracture resistance in brittle solids [3], and in designing band gaps in phononic crystals [4].

References
[1] A.M. Aragón and A. Simone, The Discontinuity-Enriched Finite Element Method, International Journal for Numerical Methods in Engineering, 112(11), 1589-1613, 2017.
[2] S.J. van den Boom, J. Zhang, F. van Keulen, and A.M. Aragón, An interface-enriched generalized finite element method for level set-based topology optimization, Structural and Multidisciplinary Optimization, 63 (1), 1-20, 2021.
[3] J. Zhang, F. van Keulen, and A.M. Aragón, On tailoring fracture resistance of brittle structures: A level set interface-enriched topology optimization approach, Computer Methods in Applied Mechanics and Engineering 388, p. 114189, 2022.
[4] S.J.van den Boom, R. Abedi, F. van Keulen, A.M. Aragón, A level set-based interface-enriched topology optimization for the design of phononic crystals with smooth boundaries, Computer Methods in Applied Mechanics and Engineering 408, p. 115888, 2023."

05 October 2023 16:00 till 17:00

[PDE & Applications seminar] Kerstin Lux-Gottschalk: The effect of parameter uncertainty on climate tipping points

Several subsystems of the Earth might undergo critical transitions under sustained anthropogenic forcing, i.e. these systems are at risk of passing a so-called tipping point (TP). That induces a drastic sudden change in the system’s phase portrait that is often irreversible. We approach these tipping phenomena through the lense of nonlinear dynamical systems theory. In particular, we use tools from bifurcation theory, which provides the mathematical framework for the creation and loss of equilibria as well as changes in their stability properties under variation of a deterministic control parameter. In this talk, I will address the question how uncertainty in model parameters affects the tipping behavior of the dynamical system in terms of the location of TPs and how they are approached. I will present a workflow for climate tipping points from a Bayesian inference on the uncertain model input parameter to the forward propagation of the obtained input probability distribution through the nonlinear dynamics turning the bifurcation curve into a random object [1]. I will show numerical results on a conceptual model of the Atlantic Meridional Overturning Circulation, which is one of the identified tipping elements. This talk covers joint work with Peter Ashwin (University of Exeter, UK), Richard Wood (Met Office, UK), and Christian K ̈uhn (Technical University of Munich, Germany).

19 October 2023 16:00 till 17:00

[PDE & Applications seminar] Zachary Adams: A quasi-stationary approach to metastability for weakly interacting particle systems

We consider systems of N particles moving as Brownian motions interacting via an attractive potential. For instance, the Glauber dynamics associated with the classical mean-field O(2) spin system of statistical mechanics. In the large particle limit, the empirical measure of such systems is known to converge to a nonlocal parabolic PDE of McKean-Vlasov type. While the McKean-Vlasov system is known to possess no non-trivial stationary solutions, numerical experiments demonstrate the existence of an almost-synchronized state that persists over a long time scale. In this talk, we characterize this almost-synchronized state and the time scale on which it persists using methods involving sub-Markov semigroups, quasi-stationary distributions, and the spectral theory of Schrödinger operators. Control on the time scale in terms of the noise amplitude and particle number are obtained. This is joint work with Maximilian Engel (FU Berlin) and Rishabh Gvalani (Max Planck Institute for Mathematics in the Sciences).

02 November 2023 16:00 till 17:00

[PDE & Applications seminar] Julie Rowlett

16 November 2023 16:00 till 17:00

[PDE & Applications seminar] Laura Scarabosio

30 November 2023 16:00 till 17:00

[PDE & Applications seminar] Marianne Bauer

14 December 2023 16:00 till 17:00

[PDE & Applications seminar] Clarice Poon

11 January 2024 16:00 till 17:00

[PDE & Applications seminar] Oliver Tse: Generalized gradient structures for interacting population dynamics

In this talk, we discuss a class of non-local evolution equations arising from reversible interacting birth-and-death processes that can be given a generalized gradient structure, and further motivate the structure via the large-population limit of measure-valued processes in population dynamics.

25 January 2024 16:00 till 17:00

[PDE & Applications seminar] Daniele Avitabile: Uncertainty Quantification for Neurobiological Networks

This talk presents a framework for forward uncertainty quantification problems in spatially-extended neurobiological networks. We will consider networks in which the cortex is represented as a continuum domain, and local neuronal activity evolves according to an integro-differential equation, collecting inputs nonlocally, from the whole cortex. These models are sometimes referred to as neural field equations.

Large-scale brain simulations of such models are currently performed heuristically, and the numerical analysis of these problems is largely unexplored. In the first part of the talk I will summarise recent developments for the rigorous numerical analysis of projection schemes [1] for deterministic neural fields, which sets the foundation for developing Finite-Element and Spectral schemes for large-scale problems.

The second part of the talk will discuss the case of networks in the presence of uncertainties modelled with random data, in particular: random synaptic connections, external stimuli, neuronal firing rates, and initial conditions. Such problems give rise to random solutions, whose mean, variance, or other quantities of interest have to be estimated using numerical simulations. This so-called forward uncertainty quantification problem is challenging because it couples spatially nonlocal, nonlinear problems to large-dimensional random data.

I will present a family of schemes that couple a spatial projector for the spatial discretisation, to stochastic collocation for the random data. We will analyse the time- dependent problem with random data and the schemes from a functional analytic viewpoint, and show that the proposed methods can achieve spectral accuracy, provided the random data is sufficiently regular. We will showcase the schemes using several examples.

Acknowledgements This talk presents joint work with Francesca Cavallini (VU Amsterdam), Svetlana Dubinkina (VU Amsterdam), and Gabriel Lord (Radboud University).

[1] Avitabile D. (2023). Projection Methods for Neural Field Equations. https://www.danieleavitabile.com/projection-methods-for-neural-field-equations/