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



Yves van Gennip (MP)

Anna Geyer (MP)

Manuel Gnann (MP)

14 december 2023 16:00 t/m 17:00

[PDE & Applications seminar] Clarice Poon: Sparsistency for inverse optimal transport

Optimal Transport is a useful metric to compare probability distributions and to compute a pairing given a ground cost. Its entropic regularization variant (eOT) is crucial to have fast algorithms and reflect fuzzy/noisy matchings. This work focuses on Inverse Optimal Transport (iOT), the problem of inferring the ground cost from samples drawn from a coupling that solves an eOT problem. It is a relevant problem that can be used to infer unobserved/missing links, and to obtain meaningful information about the structure of the ground cost yielding the pairing. On one side, iOT benefits from convexity, but on the other side, being ill-posed, it requires regularization to handle the sampling noise. This work presents an in-depth theoretical study of the $\ell_1$ regularization to model for instance Euclidean costs with sparse interactions between features. Specifically, we derive a sufficient condition for the robust recovery of the sparsity of the ground cost that can be seen as a far-reaching generalization of the Lasso’s celebrated ``Irrepresentability Condition’’. To provide additional insight into this condition, we work out in detail the Gaussian case. We show that as the entropic penalty varies, the iOT problem interpolates between a graphical Lasso and a classical Lasso, thereby establishing a connection between iOT and graph estimation, an important problem in ML. This is joint work with Francisco Andrade and Gabriel Peyré.

25 januari 2024 16:00 t/m 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.