Computational solution of inverse problems using large datasets of low rank

Project Description

A major challenge in computational science is how to use information from large datasets to improve computations of the large scale severly ill-posed problems that appear in inverse problems and data assimilation. Here we will explore how large datasets characterising different aspects of the solution can be used to improve approximation. The approach is to first use machine learning techniques to find a finite dimensional manifold characterising the dataset. Then we can apply recent stability results for inverse problems with solution in a finite dimensional space, and design finite element methods for which we prove rigorous error estimates up to perturbations of the data and the accuracy of the approximate manifold. Finally, we train neural networks to map from the manifold into the finite element solution space to obtain a reduced order model that can be used for the efficient iterative solution of the inverse problem. The project will give rise to three pieces of software that can be applied together or individually.