Advanced graduate course on Uncertainty Quantification.
Syllabus
Theoretical Foundation
- Optimal Transport, Wasserstein Gradient Flow, Sampling [textbook]
- High Dimensional Probability [textbook]
- Functional Inequalities and Sampling, optimization in probability space
- Probabilistic Numerics [textbook]
- Well-posed inverse problems; Bayesian posterior consistency; Bernstein Von Mises Theorem.
Applications: Inverse Problems and Data Assimilation
- Sampling: Importance sampling, (continuous-time) MCMC, particle methods.
- Neural ODE and diffusion model
- Optimization: Maximum a posteriori (MAP) estimators, variational methods, approximation in Kullback-Liebler divergence, Kalman Filter.