This series of lectures will address inverse problems and parameter estimation in the Bayesian framework, with particular emphasis on the recovery of solutions that are sparse, admit a sparse representation, and when the data are much fewer than the degrees of freedom. In the Bayesian framework priors can be used to promote desirable features in the solution. The hierarchical Bayesian models that will be discussed can be very effective at promoting sparsity and the associated algorithms are computationally very efficient.

The quantification of uncertainty in the computed solutions due to, e.g., noise in the data or model reduction, is very natural in the Bayesian framework. Sampling algorithms for this purpose will be also discussed. The lectures are based on the recent monograph: D. Calvetti and E. Somersalo (2023) *Bayesian Scientific Computing*, Springer.