Statistical Inverse Problems and PDEs: Progress and Challenges

Explore statistical inverse problems and partial differential equations (PDEs) in this 43-minute lecture by Richard Nickl for the International Mathematical Union. Delve into topics such as inverse regression models, PDE model examples, and challenges in inversion. Examine Bayesian inverse problems, including computation using gradient-based MCMC and Gaussian process priors. Investigate mathematical guarantees for algorithms, focusing on posterior consistency with GPS and computation in high dimensions. Learn about proof ideas and the hardness of cold-start MCMC. Gain insights into the progress made and ongoing challenges in this field of mathematical research.

Intro
Lecture Notes
Statistical inverse regression models
Inverse problems and partial differential equations (PDES)
PDE model examples
Illustration for neutron tomography
Challenges for inversion
Bayesian Inverse Problems (Stuart (2010))
Computation: gradient based MCMC
Bayesian inversion with Gaussian process priors in action
Illustration of MCMC
Mathematical guarantees for such algorithms?
Algorithmic guarantees I: Posterior consistency with GPS
Algorithmic guarantees II: Computation in high-dimensions
Proof ideas
Hardness of cold-start MCMC
Conclusions