Timeline
14:15-15:15 | Junior Richard-von-Mises-Lecture by Sven Wang Bayesian high-dimensional statistics and computational complexity in PDE models |
15:15 - 15:45 | Coffee Break |
15:45-16:45 | Richard-von-Mises-Lecture by Alexandre Ern Some recent results on the discontinuous Galerkin approximation of Maxwell's equations |
16:45-17:00 | Discussions |
Abstracts
Some recent results on the discontinuous Galerkin approximation of Maxwell's equationsAlexandre Ern (CERMICS, Ecole des Ponts and INRIA Paris, Marne la VallĂ©e and Paris, France)
We present some recent results on the discontinuous Galerkin approximation of Maxwell's equations: an asymptotically-optimal error estimate for the problem posed in the frequency domain in second-order form, and the proof of spectral correctness for the eigenvalue problem in first-order form.
Both problems hinge on a compactness property of the curl and divergence operators, and their numerical analysis crucially relies upon a duality argument originally proposed by Schatz in the context of the Helmholtz equation and conforming finite elements.
Bayesian high-dimensional statistics and computational complexity in PDE models
Sven Wang (Humboldt-UniversitĂ¤t zu Berlin)
At the heart of modern statistics lies the following question: How does
one optimally combine real-world measurement data with complex theoretical
models of some observed phenomenon? Both the amount of available data, as
well as the size, or dimension, of the employed models, have steadily
increased over the past years, leading to a growing need for theory for
high-dimensional statistics and algorithms.
In this talk, we will explore some foundational mathematical questions in
the context of Bayesian statistical methods for models with partial
differential equations (PDE), which are widely used e.g. in inverse
problems, weather modelling and geophysics. We will both discuss
statistical guarantees ("As sample size grows, how quickly does the method
converge to the ground truth?") as well as algorithmic guarantees ("How
many iterations do numerical algorithms such as optimization or sampling
algorithms require to compute relevant quantities?"), and recent progress
in both directions. Since the models at hand are typically non-linear and
ill-posed, giving such guarantees can be challenging, with many remaining
open problems.
The talk draws from the works [1], [2], [3], [4].
- [1] R. Nickl, S. van de Geer and S. Wang (2020). Convergence rates for Penalised Least Squares estimators in PDE-constrained regression problems, SIAM/ASA Journal of Uncertainty Quantification 8, 374-413
- [2] R. Nickl and S. Wang (2024). On polynomial-time computation of high-dimensional posterior measures by Langevin-type algorithms, Journal of the European Mathematical Society 26, 1031-1112.
- [3] A.S. Bandeira, A. Maillard, R. Nickl and S. Wang (2023). On free energy barriers in Gaussian priors and failure of MCMC for high-dimensional unimodal distributions. Phil. Trans. of Roy. Soc. A 381.
- [4] S. Agapiou and S. Wang (2024). Laplace priors and spatial inhomogeneity in Bayesian inverse problems. Bernoulli 30(2) 878-910