In this thesis, we investigate the results of gradient flows in Euclidean, metric, and Wasserstein space. Our primary objective is to provide a comprehensive and self-contained analysis of minimizing non-convex functions using the 2-Wasserstein gradient flow and some modified gradient flows. Firstly, we establish the equivalence between minimizing a continuous function in $\mathbb{R}^{d}$ and minimizing a relaxed functional $F[m]$ in the set of all probability measures in $\mathbb{R}^{d}$. Subsequently, we thoroughly examine minimizing the relaxed functional F[m] using the approach of gradient flows. To further enhance our understanding, we apply Swarm calculus and particle methods to numerically solve the gradient flow of $F[m]$.
| Identifer | oai:union.ndltd.org:kaust.edu.sa/oai:repository.kaust.edu.sa:10754/692270 |
| Date | 29 May 2023 |
| Creators | Abdulaziz, Hussain H. Al |
| Contributors | Gomes, Diogo A., Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division, Urbano, Miguel, Al-Naffouri, Tareq Y. |
| Source Sets | King Abdullah University of Science and Technology |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Relation | N/A |
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