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Researcher: Erlend Riis, and Carola-Bibiane Schönlieb

This project  is concerned with the development and analysis of optimisation schemes based on geometric numerical integration methods. Discrete gradient methods are popular numerical schemes for solving systems of ODEs, and are known for preserving structures of the continuous system such as energy dissipation/conservation. Applying discrete gradients to dissipative ODEs/PDEs yields optimisation schemes that preserve the dissipative structure. For example, we consider a derivative-free discrete gradient method for optimising nonsmooth, nonconvex problems in a blackbox setting. This method has been shown to converge to optimal points of the objective function in a general, nonsmooth setting, while retaining favourable properties of gradient flow. This blackbox optimisation framework is useful, for instance, for bilevel optimisation of regularisation parameters in image processing.

Related Publications 

A geometric integration approach to smooth optimisation: Foundations of the discrete gradient method
MJ Ehrhardt, ES Riis, T Ringholm, C-B Schönlieb (2018)
A Geometric Integration Approach to Nonsmooth, Nonconvex Optimisation
ES Riis, MJ Ehrhardt, GRW Quispel, CB Schönlieb – Foundations of Computational Mathematics (2021) 22, 1351