PDEs and Geometry: Numerical Aspects
Institute for Computational and Experimental Research in Mathematics (ICERM)
March 11, 2024  March 15, 2024
Your device timezone is . Do you want to view schedules in or choose a custom timezone?
Monday, March 11, 2024

8:30  8:50 am EDTCheck In11th Floor Collaborative Space

8:50  9:00 am EDTWelcome11th Floor Lecture Hall
 Brendan Hassett, ICERM/Brown University

9:00  9:45 am EDTA nonlinear leastsquares convexity enforcing finite element method for the MongeAmpere equation11th Floor Lecture Hall
 Speaker
 Susanne Brenner, Louisiana State University
 Session Chair
 Michael Neilan, University of Pittsburgh
Abstract
We present a nonlinear leastsquares finite element method for computing the smooth convex solutions of the Dirichlet boundary value problem of the MongeAmpere 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 EDTCoffee Break11th Floor Collaborative Space

10:30  11:15 am EDTThe Second Boundary Value Problem for a Discrete Monge–Ampere Equation11th 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 EDTA Volumetric Approach to Monge's Optimal Transport on Surfaces11th 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 MongeAmpere (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 EDTLunch/Free Time

2:30  3:15 pm EDTDiscretizations of anisotropic PDEs using Voronoi's reduction of quadratic forms.11th Floor Lecture Hall
 Speaker
 JeanMarie Mirebeau, ENS ParisSaclay, CNRS, Université ParisSaclay
 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 HamiltonJacobiBellman PDEs, as well as a recent extensions to the elastic wave equation in a fully general anisotropic medium.

3:30  4:00 pm EDTCoffee Break11th Floor Collaborative Space

4:00  4:45 pm EDTControlling growth and form: mineral, vegetable and animal11th 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 nonliving 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 pandisciplinary problems provide a plethora of questions in mathematics, physics and biology, with potential implications for technology.

5:00  6:30 pm EDTWelcome ReceptionReception  11th Floor Collaborative Space
Tuesday, March 12, 2024

9:00  9:45 am EDTComputational Meanfield Games: From Conventional Methods to Deep Generative Models11th 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 lowdimensional setting, employing conventional discretization and optimization methods, delving into the convergence results of our proposed approach. Afterwards, I will extend my discussion to highdimensional problems by bridging the trajectory representation of MFG with a special type of deep generative model—normalizing flows. This connection not only helps solve highdimensional 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 lowdimensional and higherdimensional setting, respectively.

10:00  10:30 am EDTCoffee Break11th Floor Collaborative Space

10:30  11:15 am EDTSemiSupervised Learning with the pLaplacian in Geometric Methods in Machine Learning and Data Analysis11th Floor Lecture Hall
 Speaker
 Nadejda Drenska, Louisiana State University
 Session Chair
 Maxim Olshanskiy, University of Houston
Abstract
The field of semisupervised learning involves learning from both labeled and unlabeled data. By exploiting the structure of the unlabeled data, such as its geometric or topological properties, semisupervised classifiers can obtain good performance with far fewer labels than are required in fully supervised learning (when classifiers learn only from labeled data). A semisupervised 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 pLaplacian regularization to cases of very low labeling rate; in such applications this approach classifies properly when the standard Laplacian regularization does not. Using the twoplayer stochastic game interpretation of the pLaplacian, we prove asymptotic consistency of pLaplacian regularized semisupervised learning, thus justifying the utility of the pLaplacian.
This is joint work with Jeff Calder. 
11:30 am  12:15 pm EDTSolving PDEs on point clouds with applications to shape analysis11th 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 EDTLunch/Free Time

2:30  3:15 pm EDTFinite element methods for illposed interface problems11th 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, nonstandard stability, or are illposed. 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 illposed problem of unique continuation in a heterogeneous environment. First we will discuss primaldual 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 socalled 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 metamaterials.

3:30  4:00 pm EDTCoffee Break11th Floor Collaborative Space

4:00  4:45 pm EDTDivergence preserving cut finite element methods11th 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 divergencefree velocity approximations of incompressible flows.
Wednesday, March 13, 2024

9:00  9:45 am EDTNavierStokes equations on surfaces: Analysis and numerical simulations11th Floor Lecture Hall
 Speaker
 Arnold Reusken, Aachen University
 Session Chair
 Ricardo Nochetto, University of Maryland
Abstract
In this presentation we consider a NavierStokes type system, posed on a smooth closed stationary or evolving twodimensional surface embedded in three dimensional space. We briefly address modeling aspects related to this system. We introduce the socalled tangential surface NavierStokes equations and discuss a wellposed 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 NavierStokes 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 EDTCoffee Break11th Floor Collaborative Space

10:30  11:15 am EDTNodal FEM for the surface Stokes problem11th Floor Lecture Hall
 Speaker
 Alan Demlow, Texas A&M University
 Session Chair
 Ricardo Nochetto, University of Maryland
Abstract
The Stokes and NavierStokes 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 (H1conforming) 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 TaylorHood elements. This is joint work with Michael Neilan.

11:30 am  12:15 pm EDTFluid flow on surfaces11th 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 wellposedness 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 twodimensional, 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 EDTGroup Photo (Immediately After Talk)11th Floor Lecture Hall

12:30  2:30 pm EDTMentoring 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 EDTTwophase flows on deformable surfaces11th 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 twophase flows. The equations are derived by a Larganged'Alembert principle and solved by surface finite elements. We demonstrate the huge possibilities of shape evolutions resulting from the strong interplay of phasedependent bending properties, the line tension and the surface viscosity.

3:30  5:00 pm EDT
Thursday, March 14, 2024

9:00  9:45 am EDTFinite Element Methods For Curvature11th 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 HellanHerrmannJohnson (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 postprocessing 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 nontrivial 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 EDTCoffee Break11th Floor Collaborative Space

10:30  11:15 am EDTFinding equilibrium states of fluid membranes11th Floor Lecture Hall
 Speaker
 Maxim Olshanskiy, University of Houston
 Session Chair
 Axel Voigt, Institute of Scientific Computing  Technische Universitat Dresden
Abstract
We are interested in finding equilibrium configurations of inextensible elastic membranes exhibiting lateral fluidity. Differential equations governing the mechanical equilibrium are derived using a continuum description of the membrane motions given by the surface NavierStokes equations with bending forces. Equilibrium conditions that are found appear to be independent of lateral viscosity and relate tension, pressure and tangential velocity of the fluid. These conditions yield that only surfaces with Killing vector fields, such as axisymmetric shapes, can support nonzero stationary flow of mass. We derive a shape equation that extends a classical Helfrich model with area constraint to membranes of nonnegligible mass. We introduce a simple numerical method to compute solutions of this highly nonlinear equation. The numerical method is then applied to find a diverse family of equilibrium configurations.

11:30 am  12:15 pm EDTLiquid Crystal Variational Problems11th 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 EDTLunch/Free Time

2:30  3:15 pm EDTA finite element scheme for the Qtensor model of liquid crystals subjected to an electric field11th Floor Lecture Hall
 Virtual Speaker
 Franziska Weber, Carnegie Mellon University
 Session Chair
 Axel Voigt, Institute of Scientific Computing  Technische Universitat Dresden
Abstract
Liquid crystal is an intermediate state of matter between the liquid and the solid phase, where the elongated molecules of the material are in partial alignment. Due to this, materials exhibiting a liquid crystal phase have unique physical properties that are used in various reallife applications, such as monitors (LCDs), smart glasses, navigation systems, shampoos, and others. Various mathematical models are available to describe their dynamics, one commonly used one is the Qtensor model by Landau and de Gennes, in which the alignment of the liquid crystal molecules and its variation over time are described through systems of nonlinear PDEs. In this talk, I will describe this model for the case where the liquid crystal molecules are subject to an electric field and present an energystable numerical discretization for it. Furthermore, within a particular range of material parameters, the convergence of the scheme can be shown to a weak solution of the system of PDEs. This is a joint work with Max Hirsch (UC Berkeley).

3:30  4:00 pm EDT"Pi Day" Coffee BreakCoffee Break  11th Floor Collaborative Space

4:00  4:45 pm EDTNumerical Approximation of the Stochastic AllenCahn Equation11th 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 AllenCahn 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 AllenCahn equation. This is joint work with M. Ondrejat (Prague, CZ) and A. Prohl (Tuebingen, DE).
Friday, March 15, 2024

9:00  9:45 am EDTPDE: spectra, geometry and spectral geometry11th 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 highaccuracy discretizations are needed. This is particularly true of the Steklov problems. In this talk we'll present strategies for computing SteklovLaplace and SteklovHelmholtz 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 SteklovMaxwell problem.

10:00  10:30 am EDTCoffee Break11th Floor Collaborative Space

10:30  11:15 am EDTNumerical analysis of an evolving bulk–surface model of tumour growth11th 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 bulksurface 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 bulksurface 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 bulksurface coupling is a major concern. The error analysis combines stability estimates and consistency estimates to yield optimalorder H^1norm 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 EDTPhase Field Models and Continuous Data Assimilation11th 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 this talk, we will briefly discuss two popular models for phase separation: the AllenCahn equation and the ChanHilliard equation. We will then discuss finite element methods that incorporate 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 will demonstrate the effectiveness of our methods via several numerical experiments.

12:30  2:30 pm EDTLunch/Free Time

3:30  4:00 pm EDTCoffee Break11th Floor Collaborative Space
All event times are listed in ICERM local time in Providence, RI (Eastern Daylight Time / UTC4).
All event times are listed in .
ICERM local time in Providence, RI is Eastern Daylight Time (UTC4). Would you like to switch back to ICERM time or choose a different custom timezone?
Schedule Timezone Updated