Scalable Algorithms for Inverse Problems Governed by Dynamical Systems

In a recent HPE Data Science Institute event, professor Andreas Mang of University of Houston Department of Mathematics presented “Scalable Algorithms for Inverse Problems Governed by Dynamical Systems.”

November 12, 2023/

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Mang opened the talk by introducing the concept of inverse problems, where the source of some observational data is unknown and cannot be measured directly. His talk discussed mathematical and computational methods to recover this source from noisy observations. After introducing the conceptual idea behind inverse problems based on a simple toy example, he focused on his area of research in which the source is assumed to be parameters of a mathematical model -- in particular, parameters of a dynamical system.

Mang discussed his ongoing work in the medical field, using images to infer dynamic information about tumor progression. By fitting a model to the static data, Mang aimed to gain insights into future behavior. Similar approaches can be used for climate change prediction and weather forecasting, areas of research Mang envisioned to work on in his future research.

Emphasizing the role of the underlying dynamical system and the computational challenges involved in solving inverse problems, Mang explained why solely relying on machine learning for prediction and discovery poses significant challenges. He instead presented work on traditional optimization approaches. The proposed methods have been deployed to GPU clusters; he demonstrated that with their work they can solve problems quickly using clinical datasets on dedicated hardware.

In his presentation, Mang mostly focused on image registration, a key problem in medical imaging. He highlighted the challenges and approaches involved in solving this problem. He emphasized the need for computational resources and provided examples of solving large-scale problems on modern, heterogeneous supercomputers.

Mang also discussed mathematical formulations and techniques (including variational formulations governed by ordinary and partial differential equations for generating smooth mappings between medical images). He emphasized the importance of generating plausible mappings in the context of medical imaging, and highlighted the use of CLAIRE -- a software for constrained diffeomorphic image registration by his group and collaborators at The University of Texas at Austin (led by professor George Biros) and the University of Stuttgart (led by professor Miriam Schulte) have developed -- in several clinical applications.

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