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Organizing Committee
University of Warwick
New Jersey Institute of Technology
University of Pittsburgh
University of Houston
Institute of Scientific Computing - Technische Universitat Dresden
Abstract
The development and analysis of numerical methods for PDEs whose formulation or interpretation is derived from an underlying geometry is a persistent challenge in numerical analysis. Examples include PDEs posed on complicated manifolds or graphs, PDEs that describe interactions across complex interfaces, and equations derived from intrinsically geometric concepts such as curvature-driven flows or highly nonlinear Monge-Ampere equations arising in optimal transport. In recent years, these PDEs have gained significance in diverse areas such as machine learning, optical design problems, meteorology, medical imaging, and beyond. Hence, the development of numerical methods for this class of PDEs is poised to lead to breakthroughs for a wide range of timely problems. However, designing methods to accurately and efficiently solve these PDEs requires careful consideration of the interactions between discretization methods, the PDE operators, and the underlying geometric properties.
This workshop aims to foster new interactions and collaborations between researchers in PDEs related to geometry. The expertise of the participants will span the analysis, computational implementation, and application of these problems. This collaborative effort will facilitate the identification of key problems in the field and the development of novel discretizations that respect both the underlying geometry of the problem and the needs of current applications.
This workshop aims to foster new interactions and collaborations between researchers in PDEs related to geometry. The expertise of the participants will span the analysis, computational implementation, and application of these problems. This collaborative effort will facilitate the identification of key problems in the field and the development of novel discretizations that respect both the underlying geometry of the problem and the needs of current applications.
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Workshop Schedule
Monday, March 11, 2024
8:30 - 8:50 AM EDT
Check In
11th Floor Collaborative Space
8:50 - 9:00 AM EDT
Welcome
11th Floor Lecture Hall
Brendan Hassett, ICERM/Brown University
9:00 - 9:45 AM EDT
A nonlinear least-squares convexity enforcing finite element method for the Monge-Ampere equation
11th Floor Lecture Hall
Speaker
Susanne Brenner, Louisiana State University
Session Chair
Michael Neilan, University of Pittsburgh
Abstract
We present a nonlinear least-squares finite element method
for computing the smooth convex solutions of the Dirichlet
boundary value problem of the Monge-Ampere equation on smooth
strictly convex planar domains. It is based on an isoparametric
finite element space with exotic degrees of freedom that can
enforce the convexity of the approximate solutions.
10:00 - 10:30 AM EDT
Coffee Break
11th Floor Collaborative Space
10:30 - 11:15 AM EDT
The Second Boundary Value Problem for a Discrete Monge–Ampere Equation
11th Floor Lecture Hall
Speaker
Gerard Awanou, University of Illinois, Chicago
Session Chair
Michael Neilan, University of Pittsburgh
Abstract
We propose a discretization of the second boundary condition for the Monge–Ampere equation arising in geometric optics and optimal transport. The discretization we propose is the natural generalization of the popular Oliker–Prussner method proposed in 1988. For the discretization of the differential operator, we use a discrete analogue of the subdifferential. Existence and stability of the solutions to the discrete problem are established. Convergence results to the continuous problem are given.
11:30 AM - 12:15 PM EDT
A Volumetric Approach to Monge's Optimal Transport on Surfaces
11th Floor Lecture Hall
Speaker
Richard Tsai, University of Texas
Session Chair
Michael Neilan, University of Pittsburgh
Abstract
In this talk, we present a novel approach for solving the Monge-Ampere (MA) equation defined on a sphere. Specifically, we extend the MA equation on a sphere to a narrowband around the sphere by formulating an equivalent optimal transport problem. We demonstrate that the extended MA equation can be solved using existing algorithms developed for the MA equation on Euclidean space, making the resulting algorithm simple and easy to implement. Our approach provides a useful tool for solving problems that involve the MA equation defined on or near a sphere, which has a wide range of applications in fields such as computer graphics, image processing, and fluid dynamics.
12:30 - 2:30 PM EDT
Lunch/Free Time
2:30 - 3:15 PM EDT
Discretizations of anisotropic PDEs using Voronoi's reduction of quadratic forms.
11th Floor Lecture Hall
Speaker
Jean-Marie Mirebeau, ENS Paris-Saclay, CNRS, Université Paris-Saclay
Session Chair
Michael Neilan, University of Pittsburgh
Abstract
Anisotropy, which refers to the existence of preferred direction in a domain, is a source of difficulty in the discretization of partial differential equations (PDEs). For instance, monotone discretization schemes for anisotropic PDEs cannot be strictly local, but need to use wide stencils.
When the PDE is discretized over a Cartesian grid domain, one can often leverage a matrix decomposition technique known as Voronoi's first reduction, which helps in finding the best possible compromises in the design of anisotropic finite difference schemes. I will describe this tool and its application to monotone discretizations of Hamilton-Jacobi-Bellman PDEs, as well as a recent extensions to the elastic wave equation in a fully general anisotropic medium.
3:30 - 4:00 PM EDT
Coffee Break
11th Floor Collaborative Space
4:00 - 4:45 PM EDT
Controlling growth and form: mineral, vegetable and animal
11th Floor Lecture Hall
Speaker
L Mahadevan, Harvard University
Session Chair
Michael Neilan, University of Pittsburgh
Abstract
Shape enables and constrains function across scales, in living and non-living systems. Following a brief introduction to morphogenesis in biology that rapidly touches on how stems, leaves, flowers, bodies, guts, beaks and brains get their shape, I will switch to the inverse problem of how to program and design shape using 3 examples: chemical precipitation, 4d printing and origami/kirigami. Along the way, I will indicate how these pan-disciplinary problems provide a plethora of questions in mathematics, physics and biology, with potential implications for technology.
5:00 - 6:30 PM EDT
Welcome Reception
Reception - 11th Floor Collaborative Space
Tuesday, March 12, 2024
9:00 - 9:45 AM EDT
Computational Mean-field Games: From Conventional Methods to Deep Generative Models
11th Floor Lecture Hall
Speaker
Rongjie Lai, Purdue University
Session Chair
Maxim Olshanskiy, University of Houston
Abstract
Mean field game (MFG) problems study how a large number of similar rational agents make strategic movements to minimize their costs. They have recently gained great attention due to their connection to various problems, including optimal transport, gradient flow, deep generative models, as well as reinforcement learning. In this talk, I will elaborate our recent computational efforts on MFGs. I will start with a low-dimensional setting, employing conventional discretization and optimization methods, delving into the convergence results of our proposed approach. Afterwards, I will extend my discussion to high-dimensional problems by bridging the trajectory representation of MFG with a special type of deep generative model—normalizing flows. This connection not only helps solve high-dimensional MFGs but also provides a way to improve the robustness of normalizing flows. If time permits, I will further address the extension of these methods to Riemannian manifolds in low-dimensional and higher-dimensional setting, respectively.
