Mathematics

Recent years have seen tremendous growth in the volumes of observational and experimental data. In this context, one fundamental question is: How do we extract knowledge from this data? When the data correspond to observations of physical systems (represented by mathematical models), this knowledge-from-data problem is fundamentally an inverse problem. This presentation aims to introduce the mathematical and computational aspects of inverse problems governed by partial differential equations, particularly modern developments that emphasize the quantification of uncertainty in the inverse solution within the framework of Bayesian inference. The concepts introduced in this talk will be demonstrated using the hIPPYlib - Inverse Problem Python library. hIPPYlib is an extensible software framework for the solution of large-scale deterministic and Bayesian inverse problems governed by partial differential equations with possibly infinite-dimensional parameter fields, which are high-dimensional after discretization. hIPPYlib overcomes the prohibitive nature of Bayesian inversion for this class of problems by implementing state-of-the-art scalable algorithms for PDE-based inverse problems that exploit the structure of the underlying operators, notably the Hessian of the log-posterior. The fast and scalable (with respect to both parameter and data dimensions) algorithms in hIPPYlib allow us to address critical questions in applying numerical simulations to potentially large-scale problems.