## December 17 and 18, 2018Two days of seminars about uncertainty quantification (UQ) and its use in the treatment of inverse problems.The course took place at DTU, the Technical University of Denmark, on the main campus in Lyngby located north of Copenhagen. |

The field of inverse problems is fertile ground for the development of computational uncertainty quantification methods. Inverse problems involve noisy measurements, leading naturally to statistical estimation problems, and due to their size and ill conditioning they are computationally challenging.

Regularization is a technique that provides stability for inverse problems, and in the Bayesian setting this is synonymous with the choice of the prior probability density function. Once a prior is chosen, the posterior probability density function results, and it is the solution of the inverse problem in the Bayesian setting.

The posterior maximizer - known as the MAP estimator - provides a stable estimate of the unknown parameters. However, uncertainty quantification requires that we extract more information from the posterior, which often requires sampling. The posterior density functions that arise in typical inverse problems are high-dimensional, and are often non-Gaussian, making the corresponding sampling problems challenging.

*The participants are expected to be familiar with inverse problems, basic statistics, and numerical computations.*

The presentations on the first day are based on the book:
Johnathan M. Bardsley,
Computational Uncertainty
Quantification for Inverse Problems
and the lecture notes:
Antti Salonen, Heikki Haario, and Marko Laine,
*Statistical Analysis in Modeling*.

**Organizers**

- Johnathan M. Bardsley, Univ. of Montana and visiting professor at DTU
- Per Christian Hansen, DTU
- Kim Knudsen, DTU

- 9:00-11:30 Johnathan M. Bardsley -
*Computational UQ for Inverse Problems*: Part 1, Part 2. - Lunch
- 13:00-14:00
Heikki
Haario, Lappeenranta University of Technology, Finland:
*Parameter Estimation, Monte Carlo Methods, Parametric Uncertainty Quantification (UQ)* - 14:00-15:00 Marko Laine,
Finnish Meteorological Institute, Helsinki, Finland:
*Efficient Parameter Estimation with the MCMC Toolbox* - 15:15 - 16:15
Lassi
Roininen, Lappeenranta University of Technology, Finland:
*Sparse Non-Stationary Hierarchical Priors for Bayesian Inversion* - 16:30 - 17:00 Maksim Mazuryn, DTU Compute:
*Inverse random medium scattering in 2D*

- 9:00-10:50
Klaus
Mosegaard, Copenhagen University -
*UQ in highly non-linear, multi-parameter inverse problems* - 11:00-11:45 Nicolai André Brogaard Riis, DTU Compute:
*UQ of Model Discrepancy using Gaussian Processes* - Lunch
- 13:00-14:00 Marko Laine:
*UQ for Time Series Analysis* - 14:15 - 15:15 Heikki Haario:
*MCMC for UQ: high CPU models* - Wrap up