Course: Math 277A: Topics in Computational and Applied Math : Gradient Systems and Algorithms
Class meet time and location: TTh 2pm-3:20pm, APM B402A Office hours: Tuesday 3:30- 4pm, Thursday 3:30- 5pm, or by appointment, location: APM 5755
Credit Hours: 4 Prerequisites: Graduate standing Catalog Description: This course develops a unified framework for understanding continuous and discrete gradient flows. The guiding principle is that many models in PDEs and optimization share a common structure: state space + energy + geometry + kinetics (dissipation) + Lyapunov/EVI + discretization. A key goal is to see how continuous viewpoints help explain and design algorithms. Depending on time and student interests, topics may include:
- gradient flows in Euclidean space and Lyapunov functions
- gradient descent (explicit Euler), proximal methods (implicit Euler), and splitting schemes (IMEX)
- accelerated and nonsmooth (l1) optimization
- gradient flows in Hilbert, Banach, and metric spaces
- long-time behavior and convergence theory
- mean-field models in neural networks
Recommended Texts/Papers :
- S. Chewi: Lectures on Optimization
- Lyapunov analysis of optimization algorithms: [Chen–Luo–Wei, 2025], [Moucer–Taylor–Bach, 2023]
- A. Mielke: An introduction to the analysis of gradient systems
- Y. Nesterov: Introductory Lectures on Convex Optimization
- F. Santambrogio: {Euclidean, Metric, and Wasserstein} Gradient Flows: an overview
- M. A. Peletier: Variational modeling: energies, gradient flows, and large deviations
- Chizat–Bach, 2018
- Rotskoff–Vanden-Eijnden, 2018
- Mei–Misiakiewicz–Montanari, 2019
- Liu–Zhu–Belkin, 2022
- Wang–Yu–Perdikaris, 2022
- Bonfanti–Bruno–Cipriani, 2024
- Bolte–Daniilidis–Ley–Mazet, 2009
- Hauer–Mazón, 2019
Grading: Your grade in the course is based on your project and a class presentation. The schedule is flexible, and students are welcome to present at any time during the quarter, including early weeks. The topic of your project should be decided in consultation with the instructor no later than the end of the 7th week.
Academic Integrity: Academic integrity is highly valued at UCSD and academic dishonesty is considered a serious offense. Students involved in an academic integrity violation will face an administrative sanction which may include suspension or, in very serious cases, expulsion from the university. Your integrity has great value: Cultivate and protect your academic integrity. For more about academic integrity and its value, visit the UCSD Academic Integrity Website.