Organizing Committee
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.

Image for "PDEs and Geometry: Numerical Aspects"

Confirmed Speakers & Participants

Talks will be presented virtually or in-person as indicated in the schedule below.

  • Speaker
  • Poster Presenter
  • Attendee
  • Virtual Attendee

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.
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    Semi-Supervised Learning with the p-Laplacian in Geometric Methods in Machine Learning and Data Analysis
    11th Floor Lecture Hall
    • Speaker
    • Nadejda Drenska, Louisiana State University
    • Session Chair
    • Maxim Olshanskiy, University of Houston
    Abstract
    The field of semi-supervised learning involves learning from both labeled and unlabeled data. By exploiting the structure of the unlabeled data, such as its geometric or topological properties, semi-supervised classifiers can obtain good performance with far fewer labels than are required in fully supervised learning (when classifiers learn only from labeled data). A semi-supervised approach is necessary when labels are very expensive to obtain, as is the case in a majority of classification applications, such as website classification, text recognition, protein sequencing, medical imaging, natural language processing. In this talk we apply p-Laplacian regularization to cases of very low labeling rate; in such applications this approach classifies properly when the standard Laplacian regularization does not. Using the two-player stochastic game interpretation of the p-Laplacian, we prove asymptotic consistency of p-Laplacian regularized semi-supervised learning, thus justifying the utility of the p-Laplacian.
    This is joint work with Jeff Calder.
  • 11:30 am - 12:15 pm EDT
    Solving PDEs on point clouds with applications to shape analysis
    11th Floor Lecture Hall
    • Speaker
    • Hongkai Zhao, Duke University
    • Session Chair
    • Maxim Olshanskiy, University of Houston
    Abstract
    Using point clouds is the most natural and ubiquitous way of representing geometry and data in 3D and higher. In this talk, I will present a framework of solving geometric PDEs directly on point clouds based on local tangent space parametrization. Then I will talk about some applications in shape analysis for point clouds. Unlike images, which have a canonical form of representation as functions defined on a uniform grid on a rectangular domain, surfaces and manifolds in 3D and higher are geometric objects that do not have a canonical or natural form of representation or global parametrization. Moreover, their embeddings in the ambient space are not intrinsic. We show how geometric PDEs can be used to “connect the dots” and extract intrinsic geometric information for the underlying point clouds for shape analysis.
  • 12:30 - 2:30 pm EDT
    Lunch/Free Time
  • 2:30 - 3:15 pm EDT
    Finite element methods for ill-posed interface problems
    11th Floor Lecture Hall
    • Speaker
    • Erik Burman, University College London
    • Session Chair
    • Maxim Olshanskiy, University of Houston
    Abstract
    In this talk we will consider recent advances on the approximation of second order elliptic problems with interfaces that have poor, non-standard stability, or are ill-posed. Such problems arise in a multitude of applications for example in seismic inversion problems or the design of meta materials. As a model problem we will consider the classical ill-posed problem of unique continuation in a heterogeneous environment. First we will discuss primal-dual stabilized finite elements for the homogeneous case and recall recent results on the accuracy and optimality of such methods. Then we will show how the method can be modified to handle internal interfaces using an unfitted finite element method. We will report error estimates for this method and discuss how to handle the destabilizing effect of error in the geometrical data. Finally we will show how the ideas can be applied to so-called sign changing materials, where the coefficient of the diffusion operator is of different sign in different subdomain. The accurate approximation of wave propagation in such materials are important for the design of meta-materials.
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:45 pm EDT
    Divergence preserving cut finite element methods
    11th Floor Lecture Hall
    • Speaker
    • Sara Zahedi, KTH Royal Institute of Technology
    • Session Chair
    • Maxim Olshanskiy, University of Houston
    Abstract
    I will give an introduction to Cut Finite Element Methods (CutFEM) for interface problems and present our recent development that results in pointwise divergence-free velocity approximations of incompressible flows.
Wednesday, March 13, 2024
  • 9:00 - 9:45 am EDT
    Navier-Stokes equations on surfaces: Analysis and numerical simulations
    11th Floor Lecture Hall
    • Speaker
    • Arnold Reusken, Aachen University
    • Session Chair
    • Ricardo Nochetto, University of Maryland
    Abstract
    In this presentation we consider a Navier-Stokes type system, posed on a smooth closed stationary or evolving two-dimensional surface embedded in three dimensional space. We briefly address modeling aspects related to this system. We introduce the so-called tangential surface Navier-Stokes equations and discuss a well-posed weak variational formulation of this PDE system that forms the basis for finite element discretization methods. Furthermore we explain the basic ideas of an unfitted finite element method, known as TraceFEM, that is used in our numerical simulation of the tangential surface Navier-Stokes system. Results of numerical experiments with this method are presented that illustrate how lateral flows are induced by smooth deformations of a material surface.
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    Nodal FEM for the surface Stokes problem
    11th Floor Lecture Hall
    • Speaker
    • Alan Demlow, Texas A&M University
    • Session Chair
    • Ricardo Nochetto, University of Maryland
    Abstract
    The Stokes and Navier-Stokes problems formulated on surfaces present a number of challenges distinct from those encountered for the corresponding Euclidean equations. In the context of numerical methods, these include the inability to formulate standard surface finite element velocity fields which are simultaneously continuous (H1-conforming) and tangential to the surface. In this talk we will give an overview of various finite element methods that have been derived for the surface Stokes problem, along with their advantages and drawbacks. We will then present a surface counterpart to the Euclidean MINI element which is the first FEM for the surface Stokes problem which does not require any penalization. Finally, we will briefly discuss extension to other nodal Stokes FEM such as Taylor-Hood elements. This is joint work with Michael Neilan.
  • 11:30 am - 12:15 pm EDT
    Fluid flow on surfaces
    11th Floor Lecture Hall
    • Speaker
    • Gieri Simonett, Vanderbilt University
    • Session Chair
    • Ricardo Nochetto, University of Maryland
    Abstract
    I will consider the motion of an incompressible viscous fluid on compact manifolds (with or without boundary). Local in time well-posedness is established in the framework of $L_p$-$L_q$ maximal regularity for initial values in critical spaces. It will be shown that the set of equilibria consists exactly of the Killing vector fields. Each equilibrium is stable and any solution starting close to an equilibrium converges at an exponential rate to a (possibly different) equilibrium. In case the surface is two-dimensional, it will be shown that any solution with divergence free initial value in $L_2$ exists globally and converges to an equilibrium.
  • 12:25 - 12:30 pm EDT
    Group Photo (Immediately After Talk)
    11th Floor Lecture Hall
  • 12:30 - 2:30 pm EDT
    Mentoring Discussion for Early Career Researchers and Students (Organized by Susanne Brenner, Sara Pollock, Michael Neilan)
    Lunch/Free Time - 11th Floor Collaborative Space
  • 2:30 - 3:15 pm EDT
    Two-phase flows on deformable surfaces
    11th Floor Lecture Hall
    • Speaker
    • Axel Voigt, Institute of Scientific Computing - Technische Universitat Dresden
    • Session Chair
    • Ricardo Nochetto, University of Maryland
    Abstract
    We extend the concept of fluid deformable surfaces to two-phase flows. The equations are derived by a Largange-d'Alembert principle and solved by surface finite elements. We demonstrate the huge possibilities of shape evolutions resulting from the strong interplay of phase-dependent bending properties, the line tension and the surface viscosity.
  • 3:30 - 5:00 pm EDT
    Poster Session/ Coffee Break
    Poster Session - 10th Floor Collaborative Space
Thursday, March 14, 2024
  • 9:00 - 9:45 am EDT
    Finite Element Methods For Curvature
    11th Floor Lecture Hall
    • Speaker
    • Shawn Walker, Louisiana State University
    • Session Chair
    • Axel Voigt, Institute of Scientific Computing - Technische Universitat Dresden
    Abstract
    This talk presents some recent advances in extending the classic Hellan--Herrmann--Johnson (HHJ) finite element to surfaces for approximation of bending problems and computing curvature. We give a review of the surface version of the HHJ method which leads to a convergent method to solve the surface Kirchhoff plate problem on surfaces embedded in three dimensions, along with numerical examples. We also describe a post-processing technique for approximating the surface Hessian of a scalar function from discrete data. We show how this scheme is easily extended to give convergent approximations of the *full shape operator* of the underlying surface, even for piecewise linear triangulations. Several numerical examples are given on non-trivial surfaces to illustrate the method. We then show how the surface HHJ finite element can also be used in computing Willmore flow, which is a gradient flow for the bending energy. In particular, we present key identities for the derivation of the method and discuss its stability. Several numerical examples show the efficacy of the method.
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    Accurate close interactions in Stokes flow using coarse grids
    11th Floor Lecture Hall
    • Speaker
    • Anna-Karin Tornberg, Kungliga Tekniska Hogskolan
    • Session Chair
    • Axel Voigt, Institute of Scientific Computing - Technische Universitat Dresden
    Abstract
    Near-contacts of rigid particles in viscous flows are notoriously difficult and typically computationally expensive to accurately resolve. With the aim of controlling the accuracy with a computationally cheap method, we introduce a singularity-free technique that combines the method of fundamental solutions and the method of images. Sources on inner proxi boundaries are complemented as needed by a small set of extra singularities at locations obtained by considering an image system. Appropriate preconditioning of the resulting linear system is a key ingredient of the method. Results are compared to a well-resolved boundary integral equation method.
  • 11:30 am - 12:15 pm EDT
    Liquid Crystal Variational Problems
    11th Floor Lecture Hall
    • Speaker
    • Ricardo Nochetto, University of Maryland
    • Session Chair
    • Axel Voigt, Institute of Scientific Computing - Technische Universitat Dresden
    Abstract
    We discuss modeling, numerical analysis and computation of liquid crystal networks (LCNs). These materials couple a nematic liquid crystal with a rubbery material. When actuated with heat or light, the interaction of the liquid crystal with the rubber creates complex shapes. Thin bodies of LCNs are natural candidates for soft robotics applications. We start from the classical 3D trace energy formula and derive a reduced 2D membrane energy as the formal asymptotic limit of vanishing thickness and characterize the zero energy deformations. We design a sound numerical method and prove its Gamma convergence despite the strong nonlinearity and lack of convexity properties of the membrane energy. We present computations showing the geometric effects that arise from liquid crystal defects as well as computations of nonisometric origami within and beyond the theory. This work is joint with L. Bouck and S. Yang.
  • 12:30 - 2:30 pm EDT
    Lunch/Free Time
  • 2:30 - 3:15 pm EDT
    tba
    11th Floor Lecture Hall
    • Virtual Speaker
    • Franziska Weber, Carnegie Mellon University
    • Session Chair
    • Axel Voigt, Institute of Scientific Computing - Technische Universitat Dresden
  • 3:30 - 4:00 pm EDT
    "Pi Day" Coffee Break
    Coffee Break - 11th Floor Collaborative Space
  • 4:00 - 4:45 pm EDT
    Numerical Approximation of the Stochastic Allen-Cahn Equation
    11th Floor Lecture Hall
    • Speaker
    • Noel Walkington, Carnegie Mellon University
    • Session Chair
    • Axel Voigt, Institute of Scientific Computing - Technische Universitat Dresden
    Abstract
    Convergence theory for numerical approximations of the stochastic Allen-Cahn equation will be reviewed. This talk will illustrate how structural properties of the partial differential operator(s) and probabilistic methods can be combined to establish stability and convergence of numerical schemes to approximate martingale solutions of the Allen-Cahn equation. This is joint work with M. Ondrejat (Prague, CZ) and A. Prohl (Tuebingen, DE).
Friday, March 15, 2024
  • 9:00 - 9:45 am EDT
    PDE: spectra, geometry and spectral geometry
    11th Floor Lecture Hall
    • Speaker
    • Nilima Nigam, Simon Fraser University
    • Session Chair
    • Noel Walkington, Carnegie Mellon University
    Abstract
    The spectra of elliptic operators are intricately connected to the geometrical properties of the spatial domains on which the operators are defined. Numerical computations are invaluable in studying this interplay, and high-accuracy discretizations are needed. This is particularly true of the Steklov problems. In this talk we'll present strategies for computing Steklov-Laplace and Steklov-Helmholtz spectra based on integral operators, and their efficacy in solving questions on the impact of geometry on spectral asymptotics. If time permits, we'll also present work in progress on a (modification of) the Steklov-Maxwell problem.
  • 10:00 - 10:30 am EDT
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EDT
    Numerical analysis of an evolving bulk–surface model of tumour growth
    11th Floor Lecture Hall
    • Speaker
    • Balázs Kovács, University of Paderborn
    • Session Chair
    • Noel Walkington, Carnegie Mellon University
    Abstract
    In this talk we will discuss an evolving bulk--surface finite element method for a model of tissue growth, which is a modification of the model of Eyles, King and Styles (2019). The model couples a Poisson equation on the domain with a forced mean curvature flow of the free boundary, with nontrivial bulk-surface coupling in both the velocity law of the evolving surface and the boundary condition of the Poisson equation. The numerical method discretizes evolution equations for the mean curvature and the outer normal and it uses a harmonic extension of the surface velocity into the bulk. The discretization admits a convergence analysis in the case of continuous finite elements of polynomial degree at least two. The stability of the discretized bulk-surface coupling is a major concern. The error analysis combines stability estimates and consistency estimates to yield optimal-order H^1-norm error bounds for the computed tissue pressure and for the surface position, velocity, normal vector and mean curvature. We will present some numerical experiments illustrating and complementing our theoretical results. The talk is based on joint work with D. Edelmann and Ch. Lubich (Tuebingen).
  • 11:30 am - 12:15 pm EDT
    Phase Field Models and Continuous Data Assimilation
    11th Floor Lecture Hall
    • Speaker
    • Amanda Diegel, Mississippi State University
    • Session Chair
    • Noel Walkington, Carnegie Mellon University
    Abstract
    Phase field models have become popular tools that help capture important features of a variety of physical processes. In part one of this talk, we consider a relatively new class of phase field models known as phase field crystal models that have emerged as a way to simulate physical processes in which atomic- and micro-scales are tightly coupled. Specifically, we will present a novel finite element method that approximates solutions to the phase field crystal equation. We will then show that the numerical method is unconditionally energy stable and convergent and support our conclusions with a few numerical experiments. In part two of this talk, we propose a numerical approximation method for the Cahn-Hilliard equation that incorporates continuous data assimilation in order to achieve long time accuracy and stability for arbitrarily inaccurate initial conditions provided enough data measurements are incorporated into the simulation. We conclude with a demonstration of the effectiveness of our method via several numerical experiments.
  • 12:30 - 2:30 pm EDT
    Lunch/Free Time
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space

All event times are listed in ICERM local time in Providence, RI (Eastern Standard Time / UTC-5).

All event times are listed in .

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Associated Semester Workshops

Numerical Analysis of Multiphysics Problems
Image for "Numerical Analysis of Multiphysics Problems"