Richard von Mises Lecture 2024stoA

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 equations
Alexandre Ern (CERMICS, Ecole des Ponts and INRIA Paris, Marne la Vallée 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-Universität 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].


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