14 december 2023 14:00 t/m 15:00

[DMO] Esther Julien: Neur2RO: Neural two-stage robust optimization

Robust optimization provides a mathematical framework for modeling and computing solutions to decision-making problems under worst-case uncertainty. In this talk I will present recent work in two-stage robust optimization (2RO) problems, wherein first-stage and second-stage decisions are made before and after uncertainty is realized. This results in a nested min-max-min optimization problem, which generally means that we are dealing with computationally challenging problems, especially in case of integer decisions. Together with my co-authors, we propose Neur2RO, an efficient machine learning-based algorithm. We learn to estimate the value function of the second-stage problem via a neural network architecture designed to construct an easy-to-solve surrogate optimization problem. Our computational experiments on two 2RO benchmarks demonstrate that we can find near-optimal solutions among different sizes of instances, often within orders of magnitude less computing time.
In this talk we mostly focus on multi-colouring. An (n,k)-colouring of a graph is an assignment of a k-subset of {1, 2, . . . , n} to each vertex such that adjacent vertices receive disjoint subsets. And the question we are looking at is: given a graph G that is (n,k)-colourable, how large can we guarantee an (n',k')-colourable induced subgraph to be (for some given (n',k'))? Answering that question leads to having to look at combinatorial objects such as set systems and Kneser graphs, and is connected to several open problems in those areas.
This is joint work with Xinyi Xu.

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.

15 december 2023 12:30 t/m 13:15

[NA] Stefan Kurz: Observers in relativistic electrodynamics

"We introduce a relativistic splitting structure to map fields and equations of electromagnetism from four-dimensional spacetime to three-dimensional observer's space. We focus on a minimal set of mathematical structures that are directly motivated by the language of the physical theory. Space-time, world-lines, time translation, space platforms, and time synchronization all find their mathematical counterparts. The splitting structure is defined without recourse to coordinates or frames. This is noteworthy since, in much of the prevalent literature, observers are identified with adapted coordinates and frames. Among the benefits of the approach is a concise and insightful classification of observers. The application of the framework to Schiff's ""Question in General Relativity"" [1] further illustrates the advantages of the framework, enabling a compact, yet profound analysis of the problem at hand. 

[1] Schiff, L. I. ""A question in general relativity."" Proceedings of the National Academy of Sciences 25.7 (1939): 391-395.
Consider two concentric spheres with equal and opposite total charges uniformly distributed over their surfaces. When the spheres are at rest, the electric and magnetic fields outside the spheres vanish. [...] Then an observer traveling in a circular orbit around the spheres should find no field, for since all of the components of the electromagnetic field tensor vanish in one coordinate system, they must vanish in all coordinate systems. On the other hand, the spheres are rotating with respect to this observer, and so he should experience a magnetic field. [...] It is clear in the above arrangement that an observer A at rest with respect to the spheres does not obtain the same results from physical experiments as an observer B who is rotating about the spheres."

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.