Applied Mathematics and Analysis Seminar
As part of the Duke University
Department of Mathematics, the Program in Applied Mathematics
hosts this ongoing series of seminars. The presentations cover a broad range of topics including numerical analysis, ordinary and partial differential equations, nonlinear systems, scientific computing, dynamical systems theory, mathematical biology, pattern formation, and complex physical systems.
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As a convenience, some selected seminars and presentations can be viewed live via the web. Further, we have video archives of past talks, which are also publicly available for you to view at any time.
- Monday, August 20, 2018, 9:30am, Physics 047, Applied Math And Analysis Seminar
Primal dual methods for Wasserstein gradient flows
Li Wang (University of Minnesota)
- In this talk, I will introduce a variational method for nonlinear
equations with a gradient flow structure, which arise widely in
applications such as porous median flows, material science, animal
swarms, and chemotaxis. Our method builds on the JKO framework and a
reformulation of the Wasserstein distance into a convex optimization
with a linear PDE constraint. As a result, we end up with one nested
structure of optimization problem with two time scales, and we adopt a
recent primal dual three operator splitting scheme. Thanks to the
variational structure, our method has a built-in positivity
preserving, entropy decreasing properties, and overcomes stability
issue due to the strong nonlinearity and degeneracy. Upon
discretization of the PDE constraint, we also show the Γ−convergence
of the fully discrete optimization towards the semi-discrete JKO
scheme. This is a joint work with Jose Carrillo, Katy Craig, and
- Monday, August 20, 2018, 11:00am, Physics 047, Applied Math And Analysis Seminar
Scaling limit analysis of Stein variational gradient descent
Yulong Lu (Duke University)
- The Stein variational gradient descent (SVGD) was proposed by Liu and Wang as a deterministic algorithm for sampling from a given probability density with unknown normalization. The key idea is to involve a system of interacting particles in an optimized way so that the empirical measure approximates a target distribution. In this talk, I will first introduce the algorithm and compare it with some stochastic-dynamics-based sampling methods. I will also present some recent rigorous analysis results on the mean field limit and long time behavior of the resulting mean field partial differential equation. This is a joint work with Jianfeng Lu and James Nolen.
- Monday, August 20, 2018, 2:00pm, Physics 047, Applied Math And Analysis Seminar
Aggregation diffusion to constrained interaction: minimizers and gradient flows in the slow diffusion limit
Kathy Craig (University of California, Santa Barbara)
- Nonlocal interactions arise throughout the natural world, from collective dynamics in biological swarms to vortex motion in superconductors. Over the past fifteen years, there has been significant interest in aggregation diffusion equations, which consider the competing effects of nonlocal interactions and local diffusion. More recently, interest has also emerged in constrained aggregation equations, which consider the competition between nonlocal interactions and a hard height constraint on the density.
In joint work with Ihsan Topaloglu, we prove that aggregation diffusion equations converge to the constrained aggregation equation in the slow diffusion limit. As an application of this theoretical result, we adapt Carrillo, Craig, and Patacchini’s blob method for diffusion to develop a numerical method for constrained aggregation equations, which we use to explore open conjectures in geometric shape optimization.
- Monday, August 20, 2018, 3:30pm, Physics 047, Applied Math And Analysis Seminar
Fourth order models for crystal surface fluctations
Jeremy Marzuola (University of North Carolina Chapel Hill)
- We’ll discuss derivations, dynamics, numerical approximations, recent analytical advances and open questions for a family of 4th order nonlinear PDEs that arise when modeling the fluctuations of a crystal surface. The microscopic problem follows from a continuous time jump Markov process where the jumps occur randomly with rates set from a generalized broken-bond Kinetic Monte Carlo model. The PDEs have a similar look to those of the thin film equations that have been studied by a large number of authors. We will discuss work with on this problem with Jonathan Weare, as well as Jian-Guo Liu, Jianfeng Lu and Dio Margetis; and Anya Katsevich.
- Wednesday, September 5, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
Primal-Dual Pi Learning Using State and Action Features
Mengdi Wang (Princeton University)
- We survey recent advances on the complexity and methods for solving Markov decision problems (MDP) and Reinforcement Learning (RL) with finitely many states and actions - a basic mathematical model for reinforcement learning.
For model reduction of large scale MDP in reinforcement learning, we propose a bilinear primal-dual pi learning method that utilizes given state and action features. The method is motivated from a saddle point formulation of the Bellman equation. The sample complexity of bilinear pi learning depends only on the number of parameters and is variant with respect to the dimension of the problem.
In the second part we study the statistical state compression of general Markov processes. We propose a spectral state compression method for learning the state features from data. The state compression method is able to “ sketch” a black-box Markov process from its empirical data and output state features, for which we provide both minimax statistical guarantees and scalable computational tools.
- Tuesday, September 11, 2018, 3:15pm, Physics 119, Applied Math And Analysis Seminar
Kui Ren (Columbia University)
- Wednesday, September 12, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
Andrej Zlatos (University of California-San Diego)
- Wednesday, September 19, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
Alexander Cloninger (UC San Diego)
- Wednesday, September 26, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
Roman Shvydkoy (University of Illinois-Chicago)
- Wednesday, October 3, 2018, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
Jun Kitagawa (Michigan State University)
- Wednesday, October 10, 2018, 12:00pm, 119 Physics, Applied Math And Analysis Seminar
Ilse Ipsen (NC State University)
- Wednesday, October 17, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
Jiequn Han (Princeton University)
- Wednesday, October 24, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
Almut Burchard (U of Toronto)
- Wednesday, October 31, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
Guillaume Bal (University of Chicago)
- Wednesday, November 7, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
Weijie Su (University of Pennsylvania)
All seminars take place on Mondays at 4:30 pm in Room 119 Physics Building unless otherwise noted.
Tea and refreshments are served before the seminars at 4:00 pm in Physics 101.
Past speakers in the Duke Applied Mathematics seminars
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