## Prof. Margaret CheneyDepartment Applied MathematicsColorado State University ## "The Radar Inverse Problem" |
||

Radar imaging is a technology that has been developed, very successfully, within the engineering community during the last 50 years. Radar systems on satellites now make beautiful images of regions of our earth and of other planets such as Venus. One of the key components of this impressive technology is mathematics, and many of the open problems are mathematical ones. This lecture will explain, from first principles, some of the basics of radar and the mathematics involved in producing high-resolution radar images. | ||

## Dr. Aku SeppänenDepartment Applied PhysicsUniversity of Eastern Finland ## "Inverse Problems in the Bayesian Framework" |
||

This workshop concentrates on ill-posed inverse problems in the framework of Bayesian inference. The emphasis is on estimation problems in which the dependence between measurable quantities and the quantities of interest is modeled computationally based on the underlying physics, and where the uncertainties are modeled statistically. In the Bayesian framework, both the measurements and model unknowns are considered as random variables. The full solution of a Bayesian inverse problem is the posterior (probability) distribution of the model unknown. The form of the posterior distribution depends on the measured data, the likelihood function describing the dependence of the measurements on the model unknown and noise, and the prior probability of the model unknown. From the posterior distribution, various point estimates (such as maximum a posteriori, MAP, or conditional mean, CM estimate) and spread estimates (such as posterior variances or credible intervals) can be computed. The Bayesian approach also offers a natural framework for handling modeling errors: In the Bayesian approximation error method, the uncertainties/inaccuracies of the models are modeled statistically, and accounted for in the solution of the inverse problem. The course starts from an introduction of basic concepts of statistical modeling (such as probability densities, multivariate models and conditional probability) and Bayesian inverse problems (e.g. the prior density, likelihood function and posterior density). We discuss various prior and likelihood models and the computation of posterior estimates. The Bayesian approximation error method for handling modeling errors is introduced. We also discuss the connection between the Bayesian MAP estimate and Tikhonov regularization, which is a widely used approach for solving inverse problems in the deterministic framework.The concepts and computational methods in Bayesian inverse problems are illustrated with various examples. Matlab codes of the low dimensional, toy examples are shared with the workshop participants, for independent studies and exercises. In addition, examples of large-scale inverse problems ‑ such as image reconstruction problems in electrical impedance tomography ‑ are shown in the lectures. |

Presentation slides (zip folder, 15 MB) | Matlab m file examples (zip folder, 9 kB) |