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exercise: switch to geometric multigrid
exercise: switch to high-order

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exercise: adaptive discretisation on L-shaped domain

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exercise: Crank-Nicolson for the heat equation
exercise: implicit RK discretisation of the heat equation

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exercise: non-Newtonian Stokes

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exercise: write a solver from scratch

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exercise: apply a multilevel solver

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exercise: recreate MATLAB logo

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exercise: Reynolds-robust solvers for Navier-Stokes

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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We will discuss some recent results on fluid-poroelastic structrure interaction between multilayered poroelastic media and viscous, incompressible fluids.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

• Video Available Soon

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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.

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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.

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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).

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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 Navier--Stokes 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 non-zero stationary flow of mass. We derive a shape equation that extends a classical Helfrich model with area constraint to membranes of non-negligible mass. We introduce a simple numerical method to compute solutions of this highly non-linear equation. The numerical method is then applied to find a diverse family of equilibrium configurations.

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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.

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 real-life 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 Q-tensor 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 energy-stable 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).

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).

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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.

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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).

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 Allen-Cahn equation and the Chan-Hilliard 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.

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We introduce the WaveHoltz iteration, for solving the Helmholtz equation. The method is inspired by recent work on exact controllability (EC) methods and as in EC methods we make use of time domain methods for wave equations to design frequency domain Helmholtz solvers, but unlike EC methods we do not require adjoint solves. We show that the WaveHoltz iteration is symmetric and positive definite (compared to the indefinite Helmholtz equation). We present numerical examples, using various discretization techniques, that show that our method can be used to solve problems with rather high wave numbers.

Abstract

The plate bending or Babuska paradox refers to the failure of convergence when a linear bending problem with simple support boundary conditions is approximated using polygonal domain approximations. We provide an explanation based on a variational viewpoint and identify sufficient conditions that avoid the paradox and which show that boundary conditions have to be suitably modified. We show that the paradox also matters in nonlinear thin-sheet folding problems and devise approximations that correctly converge to the original problem.

Abstract

Nonoverlapping Spectral Additive Schwarz Methods-NOSAS is a family of adaptive Schwarz preconditioners designed originally for solving elliptic problems with highly heterogeneous coefficients with prescribed rate of convergence. And later extended to the Helmholtz problem. In this talk I will give a short introduction on Domain Decomposition methods as a preconditioner and then discuss NOSAS. This is a joint work with Yu Yi and Maksymilian Dryja.

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We will discuss a posteriori error estimation techniques and adaptive algorithms for two popular, albeit conceptually different, FEM-based solution strategies for partial differential equations with uncertain or parameter-dependent inputs: one strategy is projection-based (stochastic Galerkin FEM) and the other is sampling-based (stochastic collocation FEM).

Abstract

The study of interfacial dynamics has become a key component to understand the behavior of a great variety of systems, in scientific, engineering and industrial applications. A very effective approach for representing interface problems is the diffuse interface/phase field approach, which describes the interfaces by layers of small thickness and whose structure is determined by a balance of molecular forces, in such a way that the tendencies for mixing and de-mixing compete through a non-local mixing energy. This approach uses a phase-field function that takes distinct values in the pure phases (for instance 0 in one phase and 1 in the other one) and varies smoothly in the interfacial regions. In particular, the Cahn- Hilliard equation was originally introduced to model the thermodynamic forces driving phase separation, arriving to a system with a gradient flow structure, that is, when there are no external forces applied to the system, the total free energy of the mixture is not increasing in time. In the first part of the presentation I will talk about the Cahn-Hilliard model and the main ideas behind the derivation of numerical schemes, showing the main advantages and disadvantages of each approach. The key point is to try to preserve the properties of the original model while the numerical schemes are efficient in time. On the second part I will present two new numerical schemes to approximate the Cahn-Hilliard equation with degenerate mobility. I will show that both schemes are energy stable and boundedness preserving, in the sense that the amount of the solution being outside of the interval [0, 1] goes to zero in terms of a truncation parameter. Finally, I will discuss how the ideas considered for designing numerical schemes to approximate phase-fields models can be extended to other energy based applications, like liquid crystals or mixture of fluids.

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Instructions for installing and running "PyNucleus" can be found here: https://sandialabs.github.io/PyNucleus/installation.html In order to make sure that learners can run the code locally on their machine on the day of the tutorial, I would recommend to follow the instructions for using the container image beforehand.

Abstract

Peridynamics is a popular nonlocal formulation of continuum mechanics which has been shown to be highly effective for fracture modeling. A meshfree discretization is commonly used in peridynamics for engineering computations due to its implementation simplicity and its ability to easily handle large deformation and material separation. The PDMATLAB2D code is a meshfree peridynamics implementation in MATLAB suitable for simulation of two-dimensional fracture problems. PDMATLAB2D provides an entry-level peridynamics computational tool for both research and education. This short course will provide an overview of peridynamics and its meshfree discretization followed by a description of the PDMATLAB2D code and a hands-on session with illustrative examples.
Materials: PDMATLAB2D code: The code can be downloaded from GitHub: https://github.com/ORNL/PDMATLAB2D/ PDMATLAB2D paper: The following paper describes the code: P. Seleson, M. Pasetto, Y. John, J. Trageser, S. T. Reeve. PDMATLAB2D: A Peridynamics MATLAB Two-dimensional Code. J Peridyn Nonlocal Model (2024). https://doi.org/10.1007/s42102-023-00104-w
The paper can be accessed by clicking here.
Instructions:
Although it is not required, it is recommended that short-course attendees read the paper and download and run the code before the short course. The code can be directly downloaded by selecting the green Code button on the main repository page. Examples can be run (from the top-level directory) in MATLAB using either of the following commands: PDMATLAB2D('WavePropagation') PDMATLAB2D('CrackBranching')

Abstract

PDMATLAB2D: A Meshfree Peridynamics MATLAB Code for 2D Fracture Computations
Peridynamics is a popular nonlocal formulation of continuum mechanics which has been shown to be highly effective for fracture modeling. A meshfree discretization is commonly used in peridynamics for engineering computations due to its implementation simplicity and its ability to easily handle large deformation and material separation. The PDMATLAB2D code is a meshfree peridynamics implementation in MATLAB suitable for simulation of two-dimensional fracture problems. PDMATLAB2D provides an entry-level peridynamics computational tool for both research and education. This short course will provide an overview of peridynamics and its meshfree discretization followed by a description of the PDMATLAB2D code and a hands-on session with illustrative examples.
Materials: PDMATLAB2D code: The code can be downloaded from GitHub: https://github.com/ORNL/PDMATLAB2D/ PDMATLAB2D paper: The following paper describes the code: P. Seleson, M. Pasetto, Y. John, J. Trageser, S. T. Reeve. PDMATLAB2D: A Peridynamics MATLAB Two-dimensional Code. J Peridyn Nonlocal Model (2024). https://doi.org/10.1007/s42102-023-00104-w <br< The paper can be accessed by clicking here. Instructions: Although it is not required, it is recommended that short-course attendees read the paper and download and run the code before the short course. The code can be directly downloaded by selecting the green Code button on the main repository page. Examples can be run (from the top-level directory) in MATLAB using either of the following commands: PDMATLAB2D('WavePropagation') PDMATLAB2D('CrackBranching')

Abstract

There has been a growing interest in studying nonlocal models as more general and sometimes more realistic alternatives to conventional PDE models. In this tutorial, we will introduce nonlocal models. In particular, we will focus on the nonlocal models with a finite range of nonlocal interactions, which connect the classical PDEs, nonlocal discrete models, and fractional differential equations. This talk will cover nonlocal modeling, nonlocal vector calculus, and numerical analysis for the nonlocal models.

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We will discuss several approaches to extend the notion of a derivative to a fractional order, their meaning, limitations, and possible applications. For some of these, we will study the existence, uniqueness, and regularity of solutions to initial boundary value problems with these operators, and their implications for numerical approximations

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Peridynamics is a powerful nonlocal reformulation of classical continuum mechanics, suitable for material failure and damage simulation, which has become very popular in recent years. In contrast to classical constitutive relations that employ spatial differential operators, peridynamic models use spatial integral operators which do not require spatial differentiability assumptions of displacement fields. This enables a natural representation of material discontinuities such as cracks. Peridynamic models possess length scales, making them also suitable for multiscale modeling. This tutorial will provide an overview of peridynamics, including some of its mathematical, computational, and modeling aspects.

Abstract

Curvature is a fundamental concept in physics and it plays a crucial role in various areas such as: classical mechanics, general relativity, optics, and fluid dynamics. In particular, the curvature of surfaces can affect the mechanical, electrical, and optical properties of materials, so curvature effects need to be taken into account when designing and analyzing new materials. The recently introduced concept of nonlocal curvatures provide a frameworks for measuring the “bend” of a surface under little or no smoothness assumptions, while connecting to classical curvature as the horizon of interaction converges to zero. In this talk I will focus on two distinct problems: of constant nonlocal curvature and of ordered curvature and show how they relate to their classical counterparts.

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Fractional diffusion in bounded domains is notorious for the lack of boundary regularity of solutions regardless of the smoothness of domain boundary. We explore this matter for the homogeneous Dirichlet problem for fractional-order quasi-linear operators with variable coefficients in Lipschitz domains and any dimensions; this includes fractional p-Laplacians and operators with finite horizon. We prove lift theorems in Besov norms which are consistent with the boundary behavior of solutions in smooth domains. The proof exploits the underlying variational structure and uses a new and flexible local translation operator. We further apply these regularity estimates to derive novel error estimates for finite element approximations of fractional p-Laplacians and present several simulations that reveal the boundary behavior of solutions.

Abstract

The problem of learning an unknown function from given data, i.e., construct an approximation to the function that predicts its values away from the data is ubiquitous in modern science. There are numerous settings for this learning problem depending on what additional information is provided about the unknown function, how the accuracy is measured, what is known about the data and data sites, and whether the data observations are polluted by noise. We consider the specific case where the learning problem consists of determining the solution to a second order elliptic partial differential equation (PDE) without information on its boundary condition. The lack of sufficient information necessary to uniquely determine the targeted function is alleviated by given finitely many linear noiseless measurements. The recovery performance is measured in energy norm and for the recovery problem to be tractable, we assume that the function to be recovered belongs to a compact subset of the energy space. The latter is referred to as the model class assumption. Among all functions satisfying the measurements, the PDE, and the model class assumption, the proposed algorithm constructs an approximation of the representative with minimal norm. For each measurement, this requires the approximation of a fractional diffusion problem on the boundary of the computational domain. We present and discuss the performances of the Dunford-Taylor method employed for the space discretization. In addition, we show how the resulting recovery algorithm is near-optimal and asymptotically optimal in the limit of the vanishing space discretization error.

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Point cloud data (PCD) represents 3D spatial information as a set of points in a 3D coordinate system, typically obtained from various sensors such as LiDAR (Light Detection And Ranging). 3D PCD generated by LiDAR is accurate and precise; however, it can be massive, requiring significant storage and computational resources for extracting meaningful information from PCD sets. Also, the sparse and irregular nature of the data renders the compression process a challenging task. Furthermore, the raw data inevitably may contain outliers or noise in real-world situations. Majority of the current methods for compression of 3D PCD usually use sampling approaches to select points from the original point clouds for conducting local feature learning. A major drawback of the existing methods is that they are highly data-specific, challenge-specific, or approach-specific. This study presents an approach for optimized manipulation of 3D PCD to achieve high fidelity reconstruction of 3D LiDAR data by employing the Peridynamic Differential Operator (PDDO). It is a single mathematical framework for optimum and accurate data compression and decompression in the presence of irregularities and scatter in a multi-dimensional space. The capability of this approach is demonstrated by considering a 3D LiDAR PCD for adaptive data compression and recovery.

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We consider theoretical and computational aspects of models coupling local and nonlocal operators with variable diffusivity. We discuss the well-posedness and strong formulation of such problems, as well as the regularity of weak solutions. We also focus on the approximation of solutions by a conforming finite element method.

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The naive discretization of nonlocal operators leads to matrices with significant density, as compared to classical PDEs. This makes the efficient solution of nonlocal models a challenging task. In this presentation, we will discuss ongoing research into efficient hierarchical matrix assembly and geometric and algebraic multigrid preconditioners that are suitable for nonlocal models.

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Nonlocal models associated with a finite horizon parameter that characterizes the effective range of nonlocal interactions have demonstrated potential as effective alternatives to local partial differential equations (PDEs) in various applications, such as the study of fracture and damage using peridynamics, and traffic flow of autonomous vehicles using nonlocal traffic models. However, to enhance the predictive capabilities and broaden the applicability of such models, new challenges arise in their numerical simulations. We provide illustrative examples, encompassing both theoretical analysis and practical implementation. These include choices regarding discretization methods, approximations of interaction neighborhoods, treatment of interfaces and boundaries, learning of nonlocal interactions from data, a priori and a posteriori error estimations, adaptive computation techniques, and the development of fast and scalable solvers.

Abstract

Population and vegetation models often use nonlocal forms of dispersal to describe the spread of individuals and plants. When these long-range effects are modeled by spatially extended convolution kernels, the mathematical analysis of solutions can be simplified by posing the relevant equations on unbounded domains. However, in order to numerically validate these results, these same equations then need to be restricted to bounded sets. Thus, it becomes important to understand what effects, if any, do the different boundary constraints have on the solution. To address this question we present a quadrature method valid for convolution kernels with finite second moments. This scheme is designed to approximate at the same time the convolution operator together with the prescribed nonlocal boundary constraints, which can be Dirichlet, Neumann, or what we refer to as free boundary constraints. We then apply this scheme to study pulse solutions of an abstract 1-d nonlocal Gray-Scott model as a case study. We consider convolution kernels with exponential and with algebraic decay. We find that, surprisingly, boundary effects can be more prominent when using exponentially decaying kernels.

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Over the past decade, there has been a growing interest in evolution equations of parabolic type that involve fractional-order derivatives in time of order in (0, 1). Such equations, also called subdiffusion equations, arise in various applications in engineering, physics, biology and finance. Hence, it is quite important to develop efficient and reliable computational tools for their numerical solution. For such problems, I will present a simple and general numerical-stability analysis using barrier functions, which yields sharp pointwise-in-time error bounds. Furthermore, pointwise-in-time a posteriori error bounds will be given, which lead to an adaptive time stepping algorithm with a local time step criterion.

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During the last 20 years there has been a lot of progress in applying neural networks (NNs) to many machine learning tasks. However, their employment in scientific machine learning with the purpose of learning complex responses of physical systems from experimental measurements has been explored much less. Unlike classical machine learning tasks, such as computer vision and natural language processing where a large amount of unstructured data are available, physics-based machine learning tasks often feature scarce and structured measurements. In this talk, we will consider learning of heterogeneous material responses as an exemplar problem to investigate automated physical model discovery from experimental data. In particular, we propose to parameterize the mapping between excitation and corresponding system responses in the form of nonlocal neural operators, and infer the neural network parameters from experimental measurements. As such, the model is built as mappings between infinite-dimensional function spaces, and the learnt network parameters are resolution-agnostic: measurements with different resolutions can be integrated to train the same model. Moreover, the nonlocal operator architecture also allows the incorporation of fundamental mathematical and physics knowledge, which further improves the learning efficacy and robustness from scarce measurements. To demonstrate the applicability of our nonlocal operator learning framework, three typical scenarios in physics-based machine learning will be discussed: (1) learning of a material-specific constitutive law, (2) learning of an efficient PDE solution operator, and (3) development of a foundation constitutive law across multiple materials. As an application, we learn material models directly from digital image correlation (DIC) displacement tracking measurements on a porcine tricuspid valve leaflet tissue, and we will show that the learnt model substantially outperforms conventional constitutive models.

Abstract

A novel method to compute nonlocal transport is presented [1]. The models of interest are Fokker-Planck type equations in which local (e.g., diffusive) transport is represented by differential operators and nonlocal transport by integrodifferential operators (e.g., non-diffusive turbulent transport closures and fractional diffusion). Computational approaches for these problems can be roughly classified as deterministic methods (e.g., finite-difference) and stochastic methods (e.g., Monte-Carlo). Although extensively used, continuum deterministic methods face stability and scalability challenges specially in the case of nonlocal operators that result in dense (non-sparse) matrices. On the other hand, particle-based stochastic methods face poor convergence due to statistical sampling. Here we present an alternative approach based on the Feynman-Kac theory that establishes a link between the Fokker-Planck equation and the stochastic differential equation of the underlying stochastic process. In local transport, the SDE are driven by Brownian motion and in nonlocal transport, depending on the nonlocal kernel, by compound Poisson processes or alpha-stable processes. The proposed method reduces the computation to the evaluation of expectations (bypassing the need of sampling stochastic trajectories) and it is unconditionally stable and parallelizable. We present applications to exit time and initial value problems for local and nonlocal transport in fluid dynamics and plasma physics. [1] M. Yang, G. Zhang, D. del-Castillo-Negrete, and Y.Cao, “A probabilistic scheme for semilinear nonlocal diffusion equations with volume constrains.” SIAM Journal of Numerical Analysis 61, (6), 2718-2743 (2023).

Abstract

In any nonlocal model of a physical phenomenon, one has to decide the extent of the nonlocality. In peridynamics (PD), nonlocality is defined by the “horizon region”, with its radius (when taken as a circular disk in 2D, or a sphere in 3D), the PD horizon size. In certain cases, particular material length scales are prominent and measurable. For example, the size of inclusions in a matrix leads to specific wave scattering effects. Material properties (elastic moduli, fracture toughness) and loading conditions also lead to identifiable length scales. One can then select the PD horizon (and the PD kernel) of a homogenized model of heterogeneous material to be in the same range as that particular length scale. A more complex scenario is when several length scales are present in a material, and not all of them are easily quantifiable, or when they combine in some complex fashion. I will discuss the options for selecting a “proper” size for the peridynamic horizon for a particular model. While for generic isotropic and homogeneous elastic materials subject to homogeneous deformations one can select an arbitrary horizon size (the response will be the same, in the bulk), when deformations are not homogeneous, as they happen to be near holes or notches, the horizon has to be selected in relation to the dimensions of these geometrical details. When a material has such a multitude of holes that we have to “rename” it as a porous material, homogenization can be used to allow using a much larger horizon size than the minute size of the pores. In fracture problems, regular homogenization for such materials may not be able to capture the observed failure modes, and an “intermediately-homogenized” PD model may be needed to be able to predict fracture and damage in such material systems at a lower cost than models that represent the detailed, pore-level microstructure. Distinguishing experiments that create thresholds in terms of crack behavior can also be used to decide on the appropriate size of the peridynamic horizon. An example from thermally-induced fracture in glass will be shown. I will conclude with progress in speeding up peridynamic computations that use the fast convolution-based method (FCBM, see PeriFast codes on GitHub).

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Phase-field modeling is widely used in material science to describe dynamics of substances with multiple phases, and it is indispensable to model solidification of materials in additive manufacturing. The interface between two substances, which is usually sharp, is replaced by a diffuse interface phase-field model that avoids an explicit tracking of the moving boundary. Resolving very thin interfaces in existing phase-field models requires very fine meshes or adding more structural complexity to the model in order to accurately capture the sharp interface. As a remedy, we propose a novel non-isothermal model with coupled local-nonlocal dynamics. This provably allows to provide sharper interfaces in the solution up to the mesh resolution and can help to alleviate the limitations of resolving thin diffuse interfaces. Additionally, we design spatial and temporal discretization schemes that allow for efficient practical realizations of those models. We propose first and second-order time-stepping schemes, which guarantee energy stability in the isothermal case, and combine them with the finite element or spectral approximations in space. A particular structure of the model and nonlocal kernel allows for the characterization of the solution in terms of a projection formula. This provides an efficient way to evaluate the solution, and in some cases avoids solving a coupled nonlinear and nonlocal system.

Abstract

In this talk, we introduce corrosion degradation fields defined nonlocally. Emphasizing the challenge of defining boundary conditions for the nonlocal generation of current density maps, we explore the application of a simplified nonlocal Finite Element (FE) fractional-Laplacian solver for modeling corrosion degradation, with a particular focus on aerospace environments. We discuss the involvement of undergraduate researchers and the interdisciplinary validation of theoretical models through experimental data obtained using a Scanning Reference Electrode Technique (SRET) apparatus, with a specific focus on the aerospace context and the consideration of Microbiologically Induced Corrosion (MIC).

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The variable-order fractional Laplacian plays an important role in the study of heterogeneous systems. However, current numerical methods for computing the variable-order fractional Laplacian still remain limited. Compared to its constant-order counterpart, the combination of nonlocality and heterogeneity in variable-order fractional Laplacian introduces significant storage and computational challenges. Consequently, many numerical methods developed for the constant-order fractional Laplacian become ineffective for computing the variable-order cases. In this talk, we introduce two numerical methods for the the variable-order Laplacian as well as their fast implementation. Some application of the variable-order fractional Laplacian in diffusion and wave propagation in heterogeneous media will also be discussed.

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In this talk, we first introduce convolutional neural networks designed to predict and analyze damage patterns on a disk resulting from molecular dynamic collision simulations. We propose the use of neural network approximations to complement peridynamic simulations by providing quick estimates which maintain much of the accuracy of the full simulations while reducing simulation times by a factor of 1500. We propose two distinct convolutional neural networks: one trained to perform the forward problem of predicting the damage pattern on a disk provided the location of a colliding object’s impact, and another trained to solve the inverse problem of identifying the collision location, angle, velocity, and size given the resulting damage pattern. Second, we introduce a Physics-informed DeepONet model to solve the multi-term time-fractional mixed diffusion equations. Deep Operator networks such as DeepONet can approximate nonlinear operators between infinite dimensional Banach spaces. PINN-DeepONet employs deep neural operator networks with the PDEs explicitly encoded into the Neural Networks using automatic differentiation to impose the underlying physical laws via soft penalty constraints during model training. Here we extend PINN-DeepONet to fractional physics-informed DeepONets to solve multi-term time-fractional mixed diffusion-wave equations. We demonstrate that the proposed method can not only yield a significant improvement in predictive accuracy but also reduce the need for large training data sets.

Abstract

Coarse-graining or model reduction of dynamical systems is a modelling tool used to extend the time-scale of simulations in a range of fields. When applied in molecular dynamics with moderate time-scale separations, standard coarse-graining approaches seek to approximate the potential of mean force, and use this to drive an effective model. Meanwhile, there is no free lunch: fewer degrees of freedom necessarily means lower accuracy of the resulting model. Here, I will discuss work with Dr Thomas Hudson from the University of Warwick, UK, in which we derived a hierarchy of models to coarse-grain simple systems and analysed the resulting models. It is shown that while the standard recipe for model reduction accurately captures equilibrium statistics, it is possible to derive an easy-to-implement Markovian effective model to better capture dynamical statistics such as the mean-squared displacement. Our results focus particularly on the temporal accuracy of coarse-grained models. Both analytical and numerical evidence for the efficacy of the new approach is provided.

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Meshfree methods, such as the Reproducing Kernel Particle Method (RKPM) are entering their fourth decade of development. While a powerful concept, the ability of meshfree methods like RKPM to realize their full potential hinges on overcoming several technological challenges. One of them, and perhaps the thorniest of all, is the development of numerical integration or quadrature techniques that are stable, efficient, and accurate. In FEM, or in IGA, the decomposition of the problem domain into elements allows the analyst to develop quadrature techniques that are element-based. Because approximation functions are infinitely smooth over the element interiors, Gaussian quadrature presents a provably accurate solution that hinges on its ability to optimally integrate smooth functions. However, the meshfree nature of methods like RKPM makes the definition of integration zones somewhat ambiguous. In addition, the infinite smoothness property inside the integration zones is also lost. As a result, traditional quadrature methods do not present a good practical solution for RKPM and similar techniques.

Numerical quadrature based on Taylor-series expansion approaches was introduced in the mid-80s for FEM to develop a parameter-free approach for hourglass control. It laid dormant for a while, and, much later, in the mid 2010s, it resurfaced in the context of RKPM to develop a so-called Natural Stabilization approach, which is arguably the most important recent breakthrough in RKPM that brought the necessary added robustness for a wide range of nonlinear solid mechanics applications. In this talk, I will present a general framework of Taylor-expansion-based methods in computational solid mechanics and its broad applicability to meshfree methods and beyond. I will demonstrate: i. How to develop Taylor-series-expansion-based formulations that are accurate and stable for nearly incompressible deformations; ii. How to stabilize correspondence-based Peridynamics without resorting to costly bond-associated approaches; and iii. How to develop general-purpose large-deformation meshfree thin shells.

Abstract

Two-phase flows in porous media is known as the Muskat problem. The Muskat problem can be ill-posed. In this talk we introduce a quasi-incompressible Cahn-Hilliard-Darcy model as a relaxation of the Muskat problem. We show global existence of weak solution to the model. We then present a high order accurate bound-preserving and unconditionally stable numerical method for solving the equations.