Applied Math Colloquium

Abstract: We consider the problem of learning interaction kernels from data for stochastic interacting particle systems and their mean-field limit, described by the McKean–Vlasov PDE. We study two measurement models: observation of a single particle trajectory, and access to discrete noisy space-time measurements of the solution to the mean-field PDE. We treat both the parametric and fully nonparametric settings. We construct estimators based on the spectral analysis of the linearized McKean–Vlasov operator and present a fully Bayesian nonparametric analysis of the problem. Finally, we consider a Bayesian inverse problem for a coupled drift-diffusion–Poisson system, and show how both MCMC methods and the Ensemble Kalman filter can be used to infer the permittivity from noisy measurements of the Fokker–Planck equation.

Abstract: Shifting is a powerful technique in extremal combinatorics initiated by Erdős–Ko–Rado and further developed by Kleitman, Frankl, Füredi, and others. Algebraic shifting is a related algebraic method that I introduced in the early 1980s. It links to a wide array of topics, including convex polytopes, homological properties of simplicial complexes, high-dimensional trees, matroids, graph rigidity, bootstrap percolation, and matrix completion. My first Applied Mathematics Colloquium lecture at MIT in 1983–1984 was on algebraic shifting—a natural meeting point between algebraic and extremal combinatorics (or between Richard Stanley and Daniel Kleitman, if you wish). In this talk, I will discuss several results and open problems surrounding algebraic shifting, highlighting connections to Turán’s (4,3) problem and the Sunflower Conjecture—the focus of the Polymath10 project in 2015–2016. The presentation will be self-contained and accessible.

Abstract: The task of simultaneously reconstructing multiple physical coefficients in PDEs from observed data is ubiquitous in applications. We propose an integrated data-driven and model-based iterative reconstruction framework for such joint inversion problems where additional data on the unknown coefficients are supplemented for better reconstructions. Our method couples the supplementary data with the PDE model to make the data-driven modeling process consistent with the model-based reconstruction procedure. This coupling strategy allows us to characterize the impact of learning uncertainty on the joint inversion results for two typical inverse problems. Numerical evidence is provided to demonstrate the feasibility of using data-driven models to improve the joint inversion of multiple coefficients in PDEs.

Absratct: Representation theory tells us the ways symmetry can present itself, and building that information into discretisations of PDEs leads to trivial parallelisation with a substantial decrease in degrees of freedom. We will give a brief introduction to representation theory (for non-experts!), see how numerical linear algebra allows it to be used in a computationally efficient manner, and incorporate it into the construction of symmetry-adapted polynomial bases that enable parallelisation. Applications include Maxwell’s equation in balls and cylinders and Schrödinger equation with multiple particles, where building in symmetry enables fast sparse spectral methods for PDEs in as many as 10 dimensions.