Numerical PDEs: Analysis, Algorithms, and Data Challenges

Institute for Computational and Experimental Research in Mathematics (ICERM)

January 29, 2024 - May 3, 2024
Monday, January 29, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 9:00 am - 4:30 pm EST
    Check In
    11th Floor Collaborative Space
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Tuesday, January 30, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 9:30 - 10:00 am EST
    ICERM Director and Organizer Welcome
    Welcome - 11th Floor Lecture Hall
    • Johnny Guzman, Brown University
    • Brendan Hassett, ICERM/Brown University
    • Maxim Olshanskiy, University of Houston
    • Sara Pollock, University of Florida
    • Abner Salgado, University of Tennessee
    • Valeria Simoncini, Università di Bologna
  • 12:00 - 1:00 pm EST
    Postdoc/Graduate Student Meeting with ICERM Director
    Meeting - 11th Floor Conference Room
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Wednesday, January 31, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 10:00 am - 12:00 pm EST
    Postdoc/ Grad Introductions
    Lightning Talks - 11th Floor Lecture Hall
    • John Carter, Rensselaer Polytechnic Institute
    • Casey Cavanaugh, Louisiana State University
    • Tristan Goodwill, University of Chicago
    • Sijing Liu, Brown University
    • Marissa Masden, ICERM
    • Henry von Wahl, Friedrich Schiller University Jena
    • Christopher Wang, Cornell University
    • Yukun Yue, University of Wisconsin, Madison
  • 2:00 - 3:00 pm EST
    Informal Tea
    Coffee Break - 11th Floor Collaborative Space
Thursday, February 1, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 9:00 - 9:45 am EST
    Introduction and TensorFlow basics
    Tutorial - 11th Floor Lecture Hall
    • Ravi Patel, Sandia National Laboratories
  • 9:45 - 10:15 am EST
    Neural networks I
    Tutorial - 11th Floor Lecture Hall
    • Ravi Patel, Sandia National Laboratories
  • 10:15 - 10:30 am EST
    Break
    Coffee Break
  • 10:30 - 11:30 am EST
    Neural networks II
    Tutorial - 11th Floor Lecture Hall
    • Ravi Patel, Sandia National Laboratories
  • 11:30 am - 1:30 pm EST
    Lunch/Free Time
  • 1:30 - 4:00 pm EST
    Physics informed neural networks and inverse problems
    Tutorial - 11th Floor Lecture Hall
    • Ravi Patel, Sandia National Laboratories
  • 4:00 - 4:30 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 4:30 - 5:00 pm EST
    Bayesian inference and Gaussian Processes I
    Tutorial - 11th Floor Lecture Hall
    • Ravi Patel, Sandia National Laboratories
Friday, February 2, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 9:00 - 10:15 am EST
    Bayesian Inference and Gaussian Processes II
    Tutorial - 11th Floor Lecture Hall
    • Ravi Patel, Sandia National Laboratories
  • 10:15 - 10:30 am EST
    Break
    Coffee Break
  • 10:30 - 11:30 am EST
    Operator Learning I
    Tutorial - 11th Floor Lecture Hall
    • Ravi Patel, Sandia National Laboratories
  • 11:30 am - 1:30 pm EST
    Lunch/Free Time
  • 1:30 - 3:30 pm EST
    Operator Learning II
    Tutorial - 11th Floor Lecture Hall
    • Ravi Patel, Sandia National Laboratories
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:30 pm EST
    Advanced topics. Future directions. Reproducibility.
    Tutorial - 11th Floor Lecture Hall
  • 4:30 - 5:00 pm EST
    Open Problem Session
    Problem Session - 11th Floor Lecture Hall
    • Ravi Patel, Sandia National Laboratories
Monday, February 5, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 8:30 - 10:30 am EST
    Poisson equation with Dirichlet conditions
    Tutorial - 11th Floor Lecture Hall
    • Patrick Farrell, University of Oxford
    Abstract
    exercise: switch to geometric multigrid
    exercise: switch to high-order
  • 10:30 - 10:45 am EST
    Break
    Coffee Break - 11th Floor Collaborative Space
  • 10:45 am - 12:45 pm EST
    Meshes and meshing
    Tutorial - 11th Floor Lecture Hall
    • Patrick Farrell, University of Oxford
    Abstract
    exercise: adaptive discretisation on L-shaped domain
  • 12:45 - 2:30 pm EST
    Lunch/Free Time
  • 2:30 - 4:30 pm EST
    Poisson with Neumann and Robin conditions
    Tutorial - 11th Floor Lecture Hall
    • Patrick Farrell, University of Oxford
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Tuesday, February 6, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 8:30 - 10:30 am EST
    A foray into transient PDEs
    Tutorial - 11th Floor Lecture Hall
    • Patrick Farrell, University of Oxford
    Abstract
    exercise: Crank-Nicolson for the heat equation
    exercise: implicit RK discretisation of the heat equation
  • 10:30 - 10:45 am EST
    Break
    Coffee Break - 11th Floor Collaborative Space
  • 10:45 am - 12:45 pm EST
    Mixed formulations: the Stokes equations
    Tutorial - 11th Floor Lecture Hall
    • Patrick Farrell, University of Oxford
    Abstract
    exercise: non-Newtonian Stokes
  • 12:45 - 2:30 pm EST
    Lunch/Free Time
  • 1:00 - 2:00 pm EST
    Post Doc/Graduate Student Seminar
    11th Floor Conference Room
  • 2:30 - 4:30 pm EST
    Compressible hyperelasticity
    Tutorial - 11th Floor Lecture Hall
    • Patrick Farrell, University of Oxford
    Abstract
    exercise: write a solver from scratch
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Wednesday, February 7, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 8:30 - 10:30 am EST
    Variational inequalities: the obstacle problem
    Tutorial - 11th Floor Lecture Hall
    • Patrick Farrell, University of Oxford
    Abstract
    exercise: apply a multilevel solver
  • 10:30 - 10:45 am EST
    Break
    Coffee Break - 11th Floor Collaborative Space
  • 10:45 am - 12:45 pm EST
    Eigenvalue problems
    Tutorial - 11th Floor Lecture Hall
    • Patrick Farrell, University of Oxford
    Abstract
    exercise: recreate MATLAB logo
  • 12:45 - 2:30 pm EST
    Lunch/Free Time
  • 2:30 - 4:30 pm EST
    Block preconditioners: Stokes and Navier-Stokes
    Tutorial - 11th Floor Lecture Hall
    • Patrick Farrell, University of Oxford
    Abstract
    exercise: Reynolds-robust solvers for Navier-Stokes
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Thursday, February 8, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:00 - 3:30 pm EST
    Informal discussion on computational linear algebra
    Discussion - 11th Floor Lecture Hall
    • Valeria Simoncini, Università di Bologna
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Friday, February 9, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:30 pm EST
    Discontinuous Galerkin for Moist Air with Implicit Condensation
    11th Floor Lecture Hall
    • Henry von Wahl, Friedrich Schiller University Jena
Monday, February 12, 2024
  • 8:30 - 8:50 am EST
    Check In
    11th Floor Collaborative Space
  • 8:50 - 9:00 am EST
    Welcome
    11th Floor Lecture Hall
    • Brendan Hassett, ICERM/Brown University
  • 9:00 - 9:45 am EST
    Numerical modeling of fluid-structure interaction
    11th Floor Lecture Hall
    • Speaker
    • Martina Bukač, University of Notre Dame
    • Session Chair
    • Sara Pollock, University of Florida
    Abstract
    Fluid-structure interaction problems arise in many applications. In biomedicine, such models are used to describe the interaction between blood and arterial walls. Other applications include geomechanics and aerodynamics. When a deformable structure is porous and allows flow through it, poroelastic models are commonly used to describe its behavior. The numerical simulation of fluid-elastic/poroelastic structure interaction problems has received considerable attention, but still remains a significant challenge in the mathematical and computational sciences. Main difficulties stem from the the intricate multiphysics nature of the problem, and strong nonlinearities. In this talk, we will present some recent advances in numerical modeling of fluid-structure interaction problems, including adaptive, partitioned methods where the domain movement is handled using an Arbitrary Lagragian-Eulerian approach, and a fixed mesh scheme based on the diffuse interface method. We will also present an application of solvers for fluid-structure interaction in the design of a bioartifical pancreas.
  • 10:00 - 10:30 am EST
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EST
    On the design and analysis of property-preserving finite element schemes for hyperbolic problems
    11th Floor Lecture Hall
    • Speaker
    • Dmitri Kuzmin, Technische Universität Dortmund
    • Session Chair
    • Sara Pollock, University of Florida
    Abstract
    This talk presents a family of algebraically constrained finite element schemes for hyperbolic conservation laws. The validity of (generalized) discrete maximum principles is enforced using monolithic convex limiting (MCL), a new flux correction procedure based on representation of spatial semi-discretizations in terms of admissible intermediate states. Semi-discrete entropy stability is enforced using a limiter-based fix. Time integration is performed using explicit or implicit Runge-Kutta methods, which can also be equipped with property-preserving flux limiters. In MCL schemes for nonlinear systems, problem-dependent inequality constraints are imposed on scalar functions of conserved variables to ensure physical and numerical admissibility of approximate solutions. After explaining the design philosophy behind our flux-corrected finite element approximations and showing some numerical examples, we turn to the analysis of consistency and convergence. In particular, we prove a Lax-Wendroff-type theorem for the inequality-constrained semi-discrete problem. A key component of our analysis is the use of a weak estimate on bounded variation, which follows from the semi-discrete entropy stability property of the method under investigation. For the Euler equations of gas dynamics, we prove weak convergence to a dissipative weak solution. The convergence analysis to be presented in this talk is joint work with Maria Lukáčová-Medvid’ová and Philipp Öffner.
  • 11:30 am - 12:15 pm EST
    Coupling mechanics with biochemistry to understand single and collective cell migration: A geometric bulk-surface partial differential equation approach
    11th Floor Lecture Hall
    • Speaker
    • Anotida Madzvamuse, The University of British Columbia
    • Session Chair
    • Sara Pollock, University of Florida
    Abstract
    In this talk I will present a geometric bulk-surface partial differential equations (GBS-PDEs) approach for coupling mechanics with biochemistry to understand mechanisms for single and collective cell migration. The GBS-PDEs are solved efficiently using tailor-made numerical methods depending on the properties of the mathematical models; two novel numerical methods will be presented: (i) the evolving bulk-surface finite element method suitable for solving GBS-PDEs on evolving domains and manifolds for sharp-interface formulations and (ii) the geometric multigrid method suitable for solving PDEs on evolving domains and manifolds using diffuse interface formulations. Experimentally driven inspired applications will be presented, demonstrating the novelty, applicability and generality of this mechanobiochemical modelling approach to studying single and collective cell migration. Cell migration is essential for many physiological and pathological processes. It plays a central role in the development and maintenance of multicellular organisms. Tissue formation during embryonic development, wound healing and immune responses as well as the formation of cancer, all require the orchestrated movement of cells in particular directions to specific locations. Hence, understanding single cell dynamics during movement is critically important in development, biomedicine and biomedical engineering.
  • 12:30 - 2:30 pm EST
    Lunch/Free Time
  • 2:30 - 3:15 pm EST
    Convergence of numerical methods for coupled Cahn-Hilliard and Navier-Stokes equations
    11th Floor Lecture Hall
    • Speaker
    • Beatrice Riviere, Rice University
    • Session Chair
    • Sara Pollock, University of Florida
    Abstract
    Modeling multicomponent flows in porous media is important for many applications relevant to energy and environment. Advances in pore-scale imaging, increasing availability of computational resources, and developments in numerical algorithms have started rendering direct pore-scale numerical simulations of multiphase flow in pore structures feasible. This talk presents recent advances in the discretization of phase-field models for systems of two-phase flows. Spatial discretization is based on the interior penalty discontinuous Galerkin methods. Time discretization utilizes a decoupled splitting approach. Both theory and application of the proposed methods to model flows in porous structures are discussed.
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:45 pm EST
    New optimized Robin-Robin domain decomposition methods using Krylov solvers for the Stokes-Darcy system
    11th Floor Lecture Hall
    • Speaker
    • Xiaoming He, Missouri University of Science and Technology
    • Session Chair
    • Sara Pollock, University of Florida
    Abstract
    In this presentation, we design optimized Schwarz domain decomposition algorithms to accelerate the Krylov type solution for the Stokes-Darcy system. We use particular solutions of this system on a circular geometry to analyze the iteration operator mode by mode. We introduce a new optimization strategy of the so-called Robin parameters based on a specific linear relation between these parameters, using the min-max and the expectation minimization approaches. Moreover, we use a Krylov solver to deal with the iteration operator and accelerate this new optimized domain decomposition algorithm. Several numerical experiments are provided to validate the effectiveness of this new method.
  • 5:00 - 6:30 pm EST
    Welcome Reception
    Reception - 11th Floor Collaborative Space
Tuesday, February 13, 2024
  • 9:00 - 9:45 am EST
    Immersed methods for fluid-structure interaction
    Virtual
    • Virtual Speaker
    • Boyce Griffith, University of North Carolina at Chapel Hill
    • Session Chair
    • Amnon Meir, Southern Methodist University
    Abstract
    The immersed boundary (IB) method is a framework for modeling systems in which an elastic structure interacts with a viscous incompressible fluid. The fundamental feature of the IB approach to such fluid-structure interaction (FSI) problems is its combination of an Eulerian formulation of the momentum equation and incompressibility constraint with a Lagrangian description of the structural deformations and resultant forces. In conventional IB methods, Eulerian and Lagrangian variables are linked through integral equations with Dirac delta function kernels, and these singular kernels are replaced by regularized delta functions when the equations are discretized for computer simulation. This talk will focus on three related extensions of the IB method. I first detail an IB approach to structural models that use the framework of large-deformation nonlinear elasticity. I will focus on efficient numerical methods that enable finite element structural models in large-scale simulations, with examples focusing on models of the heart and its valves. Next, I will describe an extension of the IB framework to simulate soft material failure using peridynamics, which is a nonlocal structural mechanics formulation. Numerical examples demonstrate constitutive correspondence with classical mechanics for non-failure cases along with essentially grid-independent predictions of fluid-driven soft material failure. Finally, I will introduce a reformulation of the IB large-deformation elasticity framework that enables accurate and efficient fluid-structure coupling through a version of the immersed interface method, which is a sharp-interface IB-type method. Computational examples demonstrate the ability of this methodology to simulate a broad range of fluid-structure mass density ratios without suffering from artificial added mass instabilities, and to facilitate subgrid contact models. I will also present biomedical applications of the methodology, including models of clot capture by inferior vena cava filters.
  • 10:00 - 10:30 am EST
    Coffee Break
    Virtual
  • 10:30 - 11:15 am EST
    AAPicard-Newton solver for Navier-Stokes and related problems
    Virtual
    • Virtual Speaker
    • Leo Rebholz, Clemson University
    • Session Chair
    • Amnon Meir, Southern Methodist University
    Abstract
    We consider the composition of AA-Picard fixed point iteration with Newton iteration, to create a more robust and stable yet still quadratically convergent solver. We analyze for Navier-Stokes in cases of small data (sufficient for uniqueness) and large data. Tests for NSE and other PDEs with this solver show a remarkable ability to converge for larger Reynolds number (NSE), Rayleigh number (Boussinesq), Kerr coefficient and refraction index (nonlinear Helmholtz), and so on.
  • 11:30 am - 12:15 pm EST
    On Reliability and Economics
    Virtual
    • Virtual Speaker
    • Jed Brown, University of Colorado Boulder
    • Session Chair
    • Amnon Meir, Southern Methodist University
    Abstract
    Data structures and algorithms have changed the relative costs, but few production pipelines have internalized the new economics of simulation. For example, matrix-free invert the marginal cost of high order discretizations, enabling large speed-ups for engineering workflows. Meanwhile, there is frequent dispute among practitioners of when to apply linearizations and assumptions on physical regime, when to use structured vs unstructured meshes, and many other important design choices in a simulation tool. For single-physics problems, it is perhaps tractable for analysts to determine (usually a posteriori) when these approximations are valid, but that is more difficult and error-prone for multi-physics problems. We reflect on the computational cost, robustness, and user interface consequences of "simplifying" this decision landscape by embracing fully nonlinear formulations with unstructured meshes and implicit solvers. Via case studies in fluid and structural mechanics, we observe that solvers for more general regimes can have low overhead relative to regime-specific solvers, yet come equipped with diagnostics that provide effective a posteriori indicators of physical regime.
  • 12:30 - 2:30 pm EST
    Lunch/Free Time
  • 2:30 - 3:15 pm EST
    Fluid-poroelastic structure interaction
    Virtual
    • Speaker
    • Suncica Canic, University of California, Berkeley
    • Session Chair
    • Amnon Meir, Southern Methodist University
    Abstract
    We will discuss some recent results on fluid-poroelastic structrure interaction between multilayered poroelastic media and viscous, incompressible fluids.
Wednesday, February 14, 2024
  • 9:00 - 9:45 am EST
    Multiscale/Multiphysics Modeling and Simulation in Ophthalmology
    11th Floor Lecture Hall
    • Virtual Speaker
    • Riccardo Sacco, Politecnico di Milano
    • Session Chair
    • Hyesuk Lee, Clemson University
    Abstract
    Glaucoma is a multifactorial ocular neuropathology representing the second major cause of irreversible blindness. Elevated intraocular pressure (IOP) is an established risk factor of glaucoma determined by aqueous humor dynamics (AHDyn), the balance among production (Pr), diffusion (Diff) and drainage (Dr) of aqueous humor (AH), a watery transparent fluid including electrolytes and low protein concentration. Reducing AH-Pr and/or increasing AH-Dr are possible approaches to reduce IOP. In this talk we illustrate a multiscale/multiphysics approach to develop a computational virtual laboratory (CVL) for the simulation of AHDyn based on a 3D-to-0D reduction of (1) 3D Velocity-Extended Poisson-Nernst-Planck PDE system to model AH-Pr; (2) 3D diffusion bulk flow to model AH-Diff; (3) 3D poroelastic PDE system to model AH-Dr. Model variables are the compartment values of electric potential, ion molar densities and fluid pressure and are numerically determined by a fixed-point iteration which transforms AHDyn simulation into the successive solution of two nonlinear systems of algebraic equations, representing mass balance in AH-Pr and in AH-Diff + AH-Dr, respectively. Computational tests suggest that Na+/K+ pump and TM/Uv hydraulic facilities are the main biomarkers of a pathological increase in IOP. These results support the potential use of a CVL to assist and optimize the design of IOP lowering medications.
  • 10:00 - 10:30 am EST
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EST
    Multiphysics at multiple scales for coupled [TpHM] processes in permafrost soils
    11th Floor Lecture Hall
    • Speaker
    • Malgorzata Peszynska, Oregon State University
    • Session Chair
    • Hyesuk Lee, Clemson University
    Abstract
    In this joint work with many students and collaborators we describe robust and conservative computational schemes for coupled process of flow, deformation, and energy with phase change (TpHM) in the soils in permafrost regions. The models present challenges due to the free boundary of freezing/thawing, strong dependence of constitutive parameters on the micro-physics of TpHM, disparate time scales, and micro- and macro heterogeneity. We also discuss how to get the data for the Darcy scale [TpH] ad [HM] models from the models at the pore-scale by computational upscaling.
  • 11:30 am - 12:15 pm EST
    Multiphysics problems related to brain clearance, sleep and dementia
    11th Floor Lecture Hall
    • Speaker
    • Kent-Andre Mardal, University of Oslo
    • Session Chair
    • Hyesuk Lee, Clemson University
    Abstract
    Recent theories suggest that a fundamental reason for sleep is simply clearance of metabolic waste produced during the activities of the day. In this talk we will present multi-physics problems and numerical schemes that target these applications. In particular, we will be lead from basic applications of neuroscience into multi-physics problems involving Stokes, Biot and fractional solvers at the brain-fluid interface.
  • 12:25 - 12:30 pm EST
    Group Photo (Immediately After Talk)
    11th Floor Lecture Hall
  • 12:30 - 2:30 pm EST
    Poster Session Lunch
    Poster Session - 10th Floor Collaborative Space
  • 2:30 - 3:15 pm EST
    Direct van der Waals Simulation (DVS): Towards Predictive Simulations of Cavitation and Boiling
    11th Floor Lecture Hall
    • Speaker
    • Hector Gomez, Purdue University
    • Session Chair
    • Hyesuk Lee, Clemson University
    Abstract
    Cavitating flows are ubiquitous in engineering and science. Despite their significance, a number of fundamental problems remain open; and our ability to make quantitative predictions is very limited. The Navier-Stokes-Korteweg equations constitute a fundamental model of cavitation, which has potential for predictive computations of liquid-vapor flows, including cavitation inception —one of the most elusive aspects of cavitation. However, numerical simulation of the Navier-Stokes-Korteweg equations is very challenging, and state of the art simulations are limited to very small Reynolds numbers, open flows (no walls), and in most cases, micrometer length scales. The computational challenges emerge from, at least, (a) the dispersive nature of the solutions to the equations, (b) a complicated eigenstructure of the isentropic form of the equations, which limits the use of standard CFD techniques, and (c) the need to resolve the liquid-vapor interface, which without special treatment, has a thickness in the order of nanometers. Here, we present Direct van der Waals simulation (DVS), a new approach that permits, for the first time as far as we are aware, large-scale simulations of wall-bounded flows with large Reynolds numbers. The proposed discretization scheme is a residual-based approach that emanates from the dispersive nature of the equations and outperforms standard stabilization schemes for advection-dominated problems. We feel that this work opens possibilities for predictive simulations of cavitation.
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:45 pm EST
    A data-driven exterior calculus
    11th Floor Lecture Hall
    • Speaker
    • Nathaniel Trask, Sandia National Laboratory
    • Session Chair
    • Hyesuk Lee, Clemson University
    Abstract
    Despite the recent flurry of work employing machine learning to develop surrogate models to accelerate scientific computation, the "black-box" underpinnings of current techniques fail to provide the verification and validation guarantees provided by modern finite element methods. In this talk we present a data-driven finite element exterior calculus for developing reduced-order models of multiphysics systems when the governing equations are either unknown or require closure. The framework employs deep learning architectures typically used for logistic classification to construct a trainable partition of unity which provides notions of control volumes with associated boundary operators. This alternative to a traditional finite element mesh is fully differentiable and allows construction of a discrete de Rham complex with a corresponding Hodge theory. We demonstrate how models may be obtained with the same robustness guarantees as traditional mixed finite element discretization, with deep connections to contemporary techniques in graph neural networks. For applications developing digital twins where surrogates are intended to support real time data assimilation and optimal control, we further develop the framework to support Bayesian optimization of unknown physics on the underlying adjacency matrices of the chain complex. By framing the learning of fluxes via an optimal recovery problem with a computationally tractable posterior distribution, we are able to develop models with intrinsic representations of epistemic uncertainty.
Thursday, February 15, 2024
  • 9:00 - 9:45 am EST
    The Role of Multiphysics Modeling in the Design of Coronary Stents
    11th Floor Lecture Hall
    • Speaker
    • Alessandro Veneziani, Emory University
    • Session Chair
    • Noel Walkington, Carnegie Mellon University
    Abstract
    Since their introduction in the Eighties, coronary stents have undergone significant design improvements, making them a critical tool for treating severe obstructions. From original Bare-Metal Stents (BMS) to Drug Eluting Stents (DES) to the most recent experience of Bioresorbable Stents, the design of these scaffolds was minimally supported by mathematical tools. The patient-specific quantitative analysis of stented coronaries is a difficult task for the variety of complex morphologies left by the stent deployment. Therefore, this type of analysis was limited to a minimal number of patients, not compatible with clinical trials. On the other hand, the development and the failure of Brioresorbable Stents clearly pointed out the importance of rigorous quantitative tools in the design of next-generation scaffolds. In this talk, we will present recent results in investigating coronary stents based on Applied Mathematics (as opposed to traditional animal models). We will consider in detail (i) the modeling of the elution in a multidomain problem solved by iterative substructuring methods involving simultaneously the lumen, the wall, and the struts of the stents; (2) the impact of the struts on the wall shear stress of a significant number of patients; (3) the consequent role of shape optimization and model order reduction in the design of scaffolds. This journey through a sophisticated combination of data and models will pinpoint the critical role of applied mathematics and scientific computing not only for a basic understanding of the biomechanics of stents but also for the clinical routine and the design of more performing prostheses.
  • 10:00 - 10:30 am EST
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EST
    Domain Agnostic Neural Operators for Multiphysics Problems
    11th Floor Lecture Hall
    • Speaker
    • Yue Yu, Lehigh University
    • Session Chair
    • Noel Walkington, Carnegie Mellon University
    Abstract
    Over the past several decades, physics-based Partial Differential Equations (PDEs) have been the cornerstone for modeling multiphysics problems. Traditional numerical methods have been employed to solve these PDEs and various approaches have been proposed to capture the multiphysics interfaces. However, their accuracy and computational feasibility can be compromised when dealing with unknown governing laws or complex interface geometries, such as in the crack propagation, fluid—structure interaction, and heterogeneous material design problems. In this talk, we develop to use data-driven modeling approaches to learn the hidden physics, capture irregular geometries, and provide accelerated predictions. In particular, we introduce domain agnostic Fourier neural operator (DAFNO), which learns the surrogate mapping between loading conditions and the corresponding physical responses with irregular geometries and evolving domains. The key idea is to incorporate a smoothed characteristic function in the integral layer architecture of neural operators, and leverage FFT to achieve rapid computations for evaluating these integrals, in such a way that the geometric information is explicitly encoded in the architecture. Once trained, DAFNO can provide efficient predictions for physical problems under unseen loading scenarios and evolving domain geometries, which makes it especially suitable to handle the complex interfacial problems in multiphysics modeling. To illustrate the applicability of DAFNO in multiphysics problems, we show three examples. Firstly, we consider a brittle material crack propagation problem which features complex domains with topology changes. Then, in the second example we further consider the corrosion induced cracking in reinforced concrete, which is a multiphysics system involving the interactions between diffusion, chemical reaction, mechanical strain, and crack fields. Last but not least, we show that DAFNO can act as an efficient surrogate for the inverse microstructure design of multifunctional metamaterials. These examples highlight the features of DAFNOs in its generalizability, flexibility, and efficiency.
  • 11:30 am - 12:15 pm EST
    The Shifted Boundary Method: An embedded approach for computational mechanics
    11th Floor Lecture Hall
    • Speaker
    • Guglielmo Scovazzi, Duke University
    • Session Chair
    • Noel Walkington, Carnegie Mellon University
    Abstract
    Embedded/immersed/unfitted boundary methods obviate the need for continual re-meshing in many applications involving rapid prototyping and design. Unfortunately, many finite element embedded boundary methods are also difficult to implement due to the need to perform complex cell cutting operations at boundaries, and the consequences that these operations may have on the overall conditioning of the ensuing algebraic problems. We present a new, stable, and simple embedded boundary method, named Shifted Boundary Method (SBM), which eliminates the need to perform cell cutting. Boundary conditions are imposed on a surrogate discrete boundary, lying on the interior of the true boundary interface. We then construct appropriate field extension operators by way of Taylor expansions, with the purpose of preserving accuracy when imposing the boundary conditions. We demonstrate the SBM on large-scale solid and fracture mechanics problems; thermoelasticity problems; porous media flow problems; incompressible flow problems governed by the Navier-Stokes equations (also including free surfaces); and problems governed by hyperbolic conservation laws.
  • 12:30 - 2:30 pm EST
    Lunch: Classical theory vs machine learning in education
    Working Lunch - 11th Floor Collaborative Space
  • 2:30 - 3:15 pm EST
    A FEM for a phase-field model of two-phase incompressible surface flow with electrostatic interaction
    11th Floor Lecture Hall
    • Speaker
    • Annalisa Quaini, University of Houston
    • Session Chair
    • Noel Walkington, Carnegie Mellon University
    Abstract
    We consider a thermodynamically consistent phase-field model of a two-phase flow of incompressible viscous fluids with electrostatic interaction. The model allows for a nonlinear dependence of the fluid density on the phase-field order parameter. Driven by applications in biomembrane studies, the model is written for tangential flows of fluids constrained to a surface and consists of (surface) Navier–Stokes–Cahn–Hilliard type equations. We apply an unfitted finite element method to discretize the system and introduce a fully discrete time-stepping scheme with the following properties: (i) the scheme decouples the fluid and phase-field equation solvers at each time step, (ii) the resulting two algebraic systems are linear, and (iii) the numerical solution satisfies the same stability bound as the solution of the original system under some restrictions on the discretization parameters. We provide numerical examples to demonstrate the stability, accuracy, and overall efficiency of the approach and provide validation against experimental data.
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
  • 4:00 - 4:45 pm EST
    Positivity-preserving discretisations in general meshes
    11th Floor Lecture Hall
    • Speaker
    • Gabriel Barrenechea, University of Strathclyde
    • Session Chair
    • Noel Walkington, Carnegie Mellon University
    Abstract
    In this talk I will present a method that enforces bound-preservation (at the degrees of freedom) of the discrete solution (recently presented in [1]). The method is built by first defining an algebraic projection onto the convex closed set of finite element functions that satisfy the bounds given by the solution of the PDE. Then, this projection is hardwired into the definition of the method by writing a discrete problem posed for this projected part of the solution. Since this process is done independently of the shape of the basis functions, and no result on the resulting finite element matrix is used, this process guarantees bound-preservation independently of the underlying mesh. The core of the talk will be devoted to explaining the main idea in the context of linear (and nonlinear) reaction-diffusion equations. Then, I will explain the main difficulties encountered when extending this method to convection-diffusion equations, and to a finite element method defined in polytopal meshes. The results in this talk have been carried out in collaboration with Abdolreza Amiri (Strathclyde, UK), Emmanuil Geourgoulis (Heriot-Watt, UK and Athens, Greece), Tristan Pryer (Bath, UK), and Andreas Veeser (Milan, Italy). References 1. G.R. Barrenechea, E. Georgoulis, T. Pryer, and A. Veeser, A nodally bound-preserving finite element method. arXiv:2304.01067, IMA Journal on Numerical Analysis, to appear.
Friday, February 16, 2024
  • 9:00 - 9:45 am EST
    Quantum Digital Twins - a numerical methodist’s adventure in the land of quantum computers
    11th Floor Lecture Hall
    • Speaker
    • Daniel Appelö, Virginia Tech
    • Session Chair
    • Valeria Simoncini, Università di Bologna
    Abstract
    In this talk I will introducing the most basic concepts in quantum computing and describe one type of quantum computing hardware (a transmon) and how it is modeled. We will then outline the computationally challenging tasks that are needed for making a quantum computer run and introduce numerical methods tailored especially for these tasks. Time permitting I will take you on a comprehensive journey through a real-world example involving characterization, control, and experimental validation, showcasing our experiences with a qutrit device within the Lawrence Livermore QUDIT testbed.
  • 10:00 - 10:30 am EST
    Coffee Break
    11th Floor Collaborative Space
  • 10:30 - 11:15 am EST
    Monolithic and Partitioned FEM for FSI: ALE divergence-free HDG fluid solver + TDNNS structural solver
    11th Floor Lecture Hall
    • Speaker
    • Guosheng Fu, University of Notre Dame
    • Session Chair
    • Valeria Simoncini, Università di Bologna
    Abstract
    We present novel (high-order) finite element schemes for the fluid-structure interaction (FSI) problem based on an arbitrary Lagrangian-Eulerian divergence-free hybridizable discontinuous Gakerkin (ALE divergence-free HDG) incompressible flow solver, a Tangential-DisplacementNormal-Normal-Stress (TDNNS) nonlinear elasticity solver, and a generalized Robin interface condition treatment. Temporal discretization is performed using the high-order backward difference formulas (BDFs). Both monolithic and strongly coupled partitioned fully discrete schemes are obtained. Numerical convergence studies are performed for the flow and elasticity solvers, and the coupled FSI solver, which verify the high-order space-time convergence of the proposed schemes. Numerical results on classical two dimensional benchmark problems also showed good performance of our proposed methods.
  • 11:30 am - 12:15 pm EST
    Mixed methods for the coupled Stokes/Poisson-Nernst-Planck equations in Banach spaces
    11th Floor Lecture Hall
    • Speaker
    • Ricardo Ruiz-Baier, Monash University
    • Session Chair
    • Valeria Simoncini, Università di Bologna
    Abstract
    I will discuss a Banach spaces-based framework and new mixed finite element methods for the numerical solution of the coupled Stokes and Poisson-Nerns-Planck equations (a nonlinear model describing the dynamics of electrically charged incompressible fluids). The pseudostress tensor, the electric field (rescaled gradient of the potential) and total ionic fluxes are used as new mixed unknowns. The resulting fully mixed variational formulation consists of two saddle-point problems, each one with nonlinear source terms depending on the remaining unknowns, and a perturbed saddle-point problem with linear source terms, which is in turn additionally perturbed by a bilinear form. The well-posedness of the continuous formulation is a consequence of a fixed-point strategy in combination with the Banach theorem, the Babuska-Brezzi theory, the solvability of abstract perturbed saddle-point problems, and the Banach-Necas-Babuska theorem. An analogous approach (but using now both the Brouwer and Banach theorems and stability conditions on arbitrary FE subspaces) is employed at the discrete level. A priori error estimates are derived, and examples of discrete spaces that fit the theory, include, e.g., Raviart--Thomas elements along with piecewise polynomials. Finally, several numerical experiments confirm the theoretical error bounds and illustrate the balance-preserving properties and applicability of the proposed family of methods. This talk is based on joint work with Claudio I. Correa and Gabriel N. Gatica (from CI2MA, Concepcion).
  • 12:30 - 2:30 pm EST
    Lunch: Networking
    Working Lunch - 11th Floor Collaborative Space
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Tuesday, February 20, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 12:00 - 12:30 pm EST
    de Rham complex
    Post Doc/Graduate Student Seminar - 11th Floor Conference Room
    • Casey Cavanaugh, Louisiana State University
  • 12:30 - 1:00 pm EST
    Potential theory
    Post Doc/Graduate Student Seminar - 11th Floor Conference Room
    • Tristan Goodwill, University of Chicago
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Wednesday, February 21, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 11:00 am - 12:00 pm EST
    Data-driven computation of the interior solutions of LTI PDE problems with unknown coefficients via network realizations of reduced order models
    11th Floor Lecture Hall
    • Vladimir Druskin, Worcester Polytechnic Institute
    Abstract
    We consider computation of the solutions of linear time-invariant hyperbolic PDEs with unknown coefficients from the measurements of their multi-input/multi-output (MIMO) transfer functions with limited number of inputs and outputs. Such problems are paramount in remote sensing and other “noninvasive problems”, e.g., radar imaging, seismic exploration and medical applications, where measurements are not available in the interior problem. We first compute an equivalent network approximation using the interpretation on the Lanczos algorithm via Stieltjes continued fraction, and then compute the state solution by embedding this approximation in the underlying PDE using the finite-difference Gaussian quadratures. We show application of this approach to nondestructive acoustic testing and SAR (Synthetic Aperture Radar) imaging . Contributors to different stages of this long-term project have been Jorn Zimmerling, Mikhail Zaslavsky, Shari Moskow, Alexander Mamonov, Liliana Borcea, Elena Cherkaev and Justin Baker.
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Thursday, February 22, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 12:00 - 1:00 pm EST
    Advances in Flight Simulation and Flow Instability
    11th Floor Lecture Hall
    • Ridgway Scott, University of Chicago
    Abstract
    A new era in flight is emerging that requires a moreeffective simulation strategy. Many modes of transportation are being developed industrially, including air-taxi drones and ground-effect transport. We describe an approach to simulating flight that is based on instabilities in flow and provides a new view of turbulence based on chaotic dynamics of computed flow profiles. The method we use is the Reynolds-Orr definition of instability that is more general than what is commonly used to define flow instability. We show that our results correlate well with what can be observed by both experiment and direct numerical simulation.
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Friday, February 23, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 9:30 - 10:30 am EST
    Ethics I
    Professional Development - 11th Floor Lecture Hall
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Monday, February 26, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Tuesday, February 27, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 12:00 - 1:00 pm EST
    Convection-dominated equations
    Post Doc/Graduate Student Seminar - 11th Floor Conference Room
    • Casey Cavanaugh, Louisiana State University
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Wednesday, February 28, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 12:00 - 1:00 pm EST
    Robust Implicit Adaptive Low Rank Time-Stepping Methods for Matrix Differential Equations
    11th Floor Lecture Hall
    • Yingda Cheng, Virginia Tech
    Abstract
    In this talk, we present a new class of implicit rank-adaptive schemes for time-dependent matrix differential equations. The dynamic low rank approximation (DLRA) is a well-known technique to capture the dynamic low rank structure based on Dirac–Frenkel time-dependent variational principle. In recent years, it has attracted a lot of attention due to its wide applicability. Our schemes are inspired by the three-step procedure used in the rank adaptive version of the unconventional robust integrator (the so called BUG integrator) for DLRA. First, a prediction (basis update) step is made computing the approximate column and row spaces at the next time level. Second, a Galerkin evolution step is invoked using a base implicit solve for the small core matrix. Finally, a truncation is made according to a prescribed error threshold. Since the DLRA is evolving the differential equation projected on to the tangent space of the low rank manifold, the error estimate of the BUG integrator contains the tangent projection (modeling) error which cannot be easily controlled by mesh refinement. This can cause convergence issue for equations with cross terms. To address this issue, we propose a simple modification, consisting of merging the row and column spaces from the explicit step truncation method together with the BUG spaces in the prediction step. In addition, we propose an adaptive strategy where the BUG spaces are only computed if the residual for the solution obtained from the prediction space by explicit step truncation method, is too large. We prove stability and estimate the local truncation error of the schemes under assumptions. We benchmark the schemes in several tests, such as anisotropic diffusion, solid body rotation and the combination of the two, to show robust convergence properties.
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Thursday, February 29, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 12:00 - 1:00 pm EST
    Fundamentals of (simplicial) Mesh Generation
    11th Floor Lecture Hall
    • Noel Walkington, Carnegie Mellon University
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Friday, March 1, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 9:30 - 10:30 am EST
    Ethics II
    Professional Development - 11th Floor Lecture Hall
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Monday, March 4, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Tuesday, March 5, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 12:00 - 12:30 pm EST
    Discontinuous Galerkin method
    Post Doc/Graduate Student Seminar - 11th Floor Conference Room
    • Henry von Wahl, Friedrich Schiller University Jena
  • 12:30 - 1:00 pm EST
    Surface PDEs
    Post Doc/Graduate Student Seminar - 11th Floor Conference Room
    • Tristan Goodwill, University of Chicago
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Wednesday, March 6, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Thursday, March 7, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 11:00 am - 12:00 pm EST
    Runge--Kutta discontinuous Galerkin methods beyond the method of lines
    11th Floor Lecture Hall
    • Zheng Sun, The University of Alabama
    Abstract
    In the common practice of the method-of-lines approach for discretizing a time-dependent partial differential equation (PDE), people first apply spatial discretization to convert the PDE into an ordinary differential equation system. Subsequently, a time integrator is used to discretize the time variable. When a multi-stage Runge-Kutta (RK) method is used for time integration, by default, the same spatial operator is used at all RK stages. But what if one allows different spatial operators at different stages? In this talk, we present two of our recent explorations on blending different stage operators in RK discontinuous Galerkin (DG) methods for solving hyperbolic conservation laws. In our first work, we mix the DG operator with the local derivative operator, yielding an RKDG method featuring compact stencils and simple boundary treatment. In our second work, we mix the DG operators with polynomials of degrees k and k-1, and the resulting method may allow larger time step sizes and fewer floating-point operations per time step.
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
Friday, March 8, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 9:30 - 10:30 am EST
    Job Applications
    Professional Development - 11th Floor Lecture Hall
  • 3:30 - 4:00 pm EST
    Coffee Break
    11th Floor Collaborative Space
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
Monday, March 18, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Tuesday, March 19, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 12:00 - 12:30 pm EDT
    Hybridizable discontinuous Galerkin method
    Post Doc/Graduate Student Seminar - 11th Floor Conference Room
    • Yukun Yue, University of Wisconsin, Madison
  • 12:30 - 1:00 pm EDT
    Long short-term memory (LSTM) neural networks
    Post Doc/Graduate Student Seminar - 11th Floor Conference Room
    • John Carter, Rensselaer Polytechnic Institute
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, March 20, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, March 21, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Friday, March 22, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Monday, March 25, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Tuesday, March 26, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 12:00 - 1:00 pm EDT
    Post Doc/Graduate Student Seminar
    11th Floor Conference Room
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, March 27, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, March 28, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Friday, March 29, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 9:30 - 10:30 am EDT
    Hiring
    Professional Development - 11th Floor Lecture Hall
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Monday, April 1, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Tuesday, April 2, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 12:00 - 1:00 pm EDT
    Post Doc/Graduate Student Seminar
    11th Floor Conference Room
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, April 3, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, April 4, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Friday, April 5, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 9:30 - 10:30 am EDT
    Papers
    Professional Development - 11th Floor Lecture Hall
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Monday, April 8, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Tuesday, April 9, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 12:00 - 1:00 pm EDT
    Post Doc/Graduate Student Seminar
    11th Floor Conference Room
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, April 10, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, April 11, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Friday, April 12, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 9:30 - 10:30 am EDT
    Grants
    Professional Development - 11th Floor Lecture Hall
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Monday, April 22, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Tuesday, April 23, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 12:00 - 1:00 pm EDT
    Post Doc/Graduate Student Seminar
    11th Floor Conference Room
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, April 24, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, April 25, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Friday, April 26, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Monday, April 29, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Tuesday, April 30, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 12:00 - 1:00 pm EDT
    Post Doc/Graduate Student Seminar
    11th Floor Conference Room
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Wednesday, May 1, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Thursday, May 2, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 3:30 - 4:00 pm EDT
    Coffee Break
    11th Floor Collaborative Space
Friday, May 3, 2024
Numerical PDEs: Analysis, Algorithms, and Data Challenges
  • 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 .