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.
To join the applied mathematics mailing list visit: subscribe applied-math-seminar
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.
- Wednesday, October 17, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
Deep Learning-Based Numerical Methods for High-Dimensional Parabolic PDEs and Forward-Backward SDEs
Jiequn Han (Princeton University)
- Developing algorithms for solving high-dimensional partial differential equations (PDEs) and forward-backward stochastic differential equations (FBSDEs) has been an exceedingly difficult task for a long time, due to the notorious difficulty known as the curse of dimensionality. In this talk we introduce a new type of algorithms, called "deep BSDE method", to solve general high-dimensional parabolic PDEs and FBSDEs. Starting from the BSDE formulation, we approximate the unknown Z-component by neural networks and design a least-squares objective function for parameter optimization. Numerical results of a variety of examples demonstrate that the proposed algorithm is quite effective in high-dimensions, in terms of both accuracy and speed. We furthermore provide a theoretical error analysis to illustrate the validity and property of the designed objective function.
- Wednesday, October 24, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
Almut Burchard (U of Toronto)
- Monday, October 29, 2018, 12:00pm, TBA, Applied Math And Analysis Seminar
Anna Mazzucato (Pennsylvania State University)
- 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)
- Friday, November 9, 2018, 12:00pm, TBA, Applied Math And Analysis Seminar
Non-Equilibrium Steady States for Networks of Oscillators
NoÚ Cuneo (UniversitÚ Paris VI, Mathematics)
- Non-equilibrium steady states for chains of oscillators interacting with stochastic heat baths at different temperatures have been the subject of several studies. In this talk I will discuss how to generalize these results to multidimensional networks of oscillators. I will first introduce the model and motivate it from a physical point of view. Then, I will present conditions on the topology of the network and on the interaction potentials which imply the existence and uniqueness of the non-equilibrium steady state, as well as exponential convergence to it. The two main ingredients of the proof are (1) a controllability argument using H÷rmander's bracket criterion and (2) a careful study of the high-energy dynamics which leads to a Lyapunov-type condition. I will also mention cases where the non-equilibrium steady state is not unique, and cases where its existence is an open problem. This is joint work with J.-P. Eckmann, M. Hairer and L. Rey-Bellet, Electronic Journal of Probability 23(55): 1-28, 2018 (arXiv:1712.09413).
- Wednesday, November 14, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
Finite Difference Methods for Boundary Value Problems: Using Interface Problems
Tom Beale (Duke University)
- Finite difference methods are awkward for solving boundary value problems, such as the Dirichlet problem, with general boundaries, but they are well suited for interface problems, which have prescribed jumps in an unknown across a general interface or boundary. The two problems can be connected through potential theory: The Dirichlet boundary value problem is converted to an integral equation on the boundary, and the integrals can be thought of as solutions to interface problems. Wenjun Ying et al. have developed a practical method for solving the Dirichlet problem, and more general ones, by solving interface problems with finite difference methods and iterating to mimic the solution of the integral equation. We will describe some analysis which proves that a
simplified version of Ying's method works. A recent view of classical potential theory leads to a finite difference version of the theory in which, remarkably, the crude discrete operators have much of the structure of the exact operators. This simplified method produces the Shortley-Weller solution of the Dirichlet problem. Details can be found
at arxiv.org/abs/1803.08532 .
- Friday, November 30, 2018, 12:00pm, Physics 119, Applied Math And Analysis Seminar
Elina Robeva (MIT)
- Wednesday, March 27, 2019, 12:00pm, Physics 119, Applied Math And Analysis Seminar
Alina Chertock (NC State)
- Wednesday, April 24, 2019, 12:00pm, Physics 119, Applied Math And Analysis Seminar
Stefan Steinerberger (Yale University)
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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