## PDE-constrained optimization in scientific processes

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** Abstract:** A vast number of important and challenging applications in mathematics and engineering are governed by inverse problems. One crucial class of these problems, which has significant applicability to real-world processes, including those of fluid flow, chemical and biological mechanisms, medical imaging, and others, is that of PDE-constrained optimization. However, whereas such problems can typically be written in a precise form, generating accurate numerical solutions on the discrete level is a highly non-trivial task, due to the dimension and complexity of the matrix systems involved. In order to tackle practical problems, it is essential to devise strategies for storing and working with systems of huge dimensions, which result from fine discretizations of the PDEs in space and time variables. In this project, "all-at-once" solvers coupled with appropriate preconditioning techniques will be derived for these systems, in such a way that one may achieve fast and robust convergence in theory and in practice. This project is related to the EPSRC Fellowship.

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