Numerical Analysis Archives for Fall 2024 to Spring 2025
High order positivity-preserving entropy stable discontinuous Galerkin discretizations
When: Tue, September 12, 2023 - 3:30pmWhere: Video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=b2682da3-4e16-451e-93d2-b08501795c20&start=450
Speaker: Jesse Chan (Rice University) - https://profiles.rice.edu/faculty/jesse-chan
Abstract: High order discontinuous Galerkin (DG) methods provide high order accuracy and geometric flexibility, but are known to be unstable when applied to nonlinear conservation laws whose solutions exhibit shocks and under-resolved solution features. Entropy stable schemes improve robustness by ensuring that physically relevant solutions satisfy a semi-discrete cell entropy inequality independently of numerical resolution and solution regularization while retaining formal high order accuracy. In this talk, we will review the construction of entropy stable high order discontinuous Galerkin methods and describe approaches for enforcing that solutions are "physically relevant" (i.e., thermodynamic variables remain positive).
Fast and Accurate Boundary Integral Methods for Two-Phase Flow with Surfactant
When: Tue, September 26, 2023 - 3:30pmWhere: Video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=94e6c1a5-c63b-494c-88ea-b089017d03d0
Speaker: Michael Siegel (New Jersey Institute of Technology) - https://people.njit.edu/faculty/misieg
Abstract: We present accurate and efficient numerical methods to simulate the de- formation of drops and bubbles in Stokes flow with surfactant. The majority of the talk will focus on a ‘hybrid’ or multiscale numerical method devel- oped over several years to address difficulties in the numerical computation of fluid interfaces with soluble surfactant, which advects and diffuses in fluids and adsorbs/desorbs from interfaces. In the physically representative large Peclet number limit, a thin transition layer develops near an interface in which physical quantities rapidly vary, yet must be well resolved for accurate computation of interface dynamics. The hybrid method uses the slenderness of the layer to incorporate a separate analytical reduction of the layer’s dy- namics into a novel boundary integral formulation. We present several recent developments, including a fast mesh-free algorithm for resolving the transi- tion layer, and a method which captures the transfer of surfactant between the exterior and interior fluid via transport through the combined interface- transition layer structure.
A Low Rank Tensor Approach for Nonlinear Vlasov Simulations
When: Tue, October 3, 2023 - 3:30pmWhere: Video: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=eb06dce2-9eeb-4f6f-baf2-b0930115f592
Speaker: Jingmei Qiu (University of Delaware) - https://jingmeiqiu.github.io/
Abstract: In this work, we present a low-rank tensor approach for approximating solutions to the nonlinear Vlasov equation. Our method takes advantage of the tensor-friendly nature of the differential operators in the Vlasov equation to dynamically and adaptively construct a low-rank solution basis through the discretization of the equation and an SVD-type truncation procedure. We utilize finite difference WENO and discontinuous Galerkin spatial discretizations, along with a second-order strong stability preserving multi-step time discretization. To preserve conservation properties, we develop low-rank schemes with local mass, momentum, and energy conservation for the corresponding macroscopic equations. The mass and momentum are conserved using a conservative SVD truncation, while the energy is conserved by replacing the energy component of the kinetic solution with one obtained from a conservative scheme for the macroscopic energy equation. We employ hierarchical Tucker decomposition for high-dimensional problems, and demonstrate the high-order convergence, efficiency, and local conservation properties of our algorithm through a series of linear and nonlinear Vlasov examples.
Riemannian optimization and Riemannian Langevin Monte Carlo for PSD fixed rank constraints
When: Tue, October 10, 2023 - 3:30pmWhere: Video: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=ed687d23-01aa-43b9-99fa-b09b009076b6
Speaker: Xiangxiong Zhang (Purdue University) - https://www.math.purdue.edu/~zhan1966/
Abstract: Positive semi-definite fixed rank constraint arises in certain machine learning and data science applications, e.g., it is also used for approximating the PSD constraint. For optimization under such constraints, we study and compare three methodologies for minimizing f(X) with X being a Hermitian PSD fixed rank matrix. The first approach is the simplest factor-based Burer-Monteiro method, in which a PSD fixed rank matrix X is replaced by its low-rank decomposition YY^* thus an unconstrained minimization of f(YY^*) can be solved instead. The second approach is to regard the set of Hermitian PSD fixed rank matrices as an embedded manifold in the Euclidean space and consider the Riemannian optimization over the embedded manifold. The third approach is to regard it as a quotient manifold and consider the Riemannian optimization over the quotient manifold. For simplicity, we only consider the nonlinear conjugate gradient (CG) algorithm, which is an efficient algorithm in these methods. We show that CG in the first two methodologies is equivalent to CG on the quotient manifold with suitably chosen metrics, retractions, and vector transports. In particular, the simple Bure-Monteiro approach corresponds to the Bures-Wasserstein metric. We also analyze the condition number of the Riemannian Hessian under these different metrics. The difference in the condition number of the Riemannian Hessian under different metrics is consistent with the difference in the numerical performance of three methodologies for problems including matrix completion, phase retrieval, and interferometry recovery. This part is based on joint work with Shixin Zheng at Purdue, Wen Huang at Xiamen University and Bart Vandereycken at University of Geneva.
It is also natural to consider sampling schemes under the same constraints. For the manifold of positive semi-definite matrices of fixed rank, we construct two sampling schemes on the manifold, which can generate samples from the Gibbs distribution. The two sampling schemes correspond to the Euler-Maruyama scheme for the Stochastic Differential Equation on the manifold under the embedded geometry and the Bures-Wasserstein metric. This can be regarded as Monte Carlo sampling on the manifold. There are many potential applications. For integrating a nice function on the manifold, one can simply take the Arithmetic mean of the samples, thus it would be a very simple quadrature rule on the manifold since all geometric information are encoded in the samples. The sampling schemes are constructed by adding white noise and a proper geometric correction term to the gradient descent algorithms on the manifold, thus the computational complexity of each iteration is comparable to the Riemannian gradient decent methods. Preliminary numerical results will be shown. This is based on an ongoing joint work with Tianmin Yu and Govind Menon at Brown University, Jianfeng Lu at Duke University, and Shixin Zheng at Purdue University.
Hydrodynamics of Liquid Crystals on Curved Thin Films: Modeling and Numerics
When: Tue, October 17, 2023 - 3:30pmWhere: Video: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=76bd642a-c58d-42aa-9a61-b0a601108db9
Speaker: Lucas Bouck (Carnegie Mellon University) - https://lbouck.github.io/
Abstract: Liquid crystals (LC) on curved thin films has numerous applications in active matter and self assembly of materials. In this talk, we present a hydrodynamic model of LC on a curved thin film using the Q-tensor description of an LC. We derive the model using Onsager's principle, which ensures that the model has an energy law and is thermodynamically consistent. To discretize the system, we employ a Trace Finite Element Method (TraceFEM) and implement a first order splitting scheme for the time discretization. The resulting numerical method is unconditionally stable and satisfies an energy law that mimics the continuous problem. We briefly discuss partial results towards proving convergence of the method, which highlights the importance of stabilization of TraceFEM for parabolic problems. Our computations show the influence of curvature on escape to the third dimension of LC and the influence of surface anchoring on defect configurations, which match theoretical and experimental results. The work on hydrodynamics of LC is joint with Ricardo H. Nochetto (UMD) and Vladimir Yushutin (Clemson), and the work on stabilization of TraceFEM is also joint with Mansur Shakipov (UMD).
Dynamical low-rank methods for high-dimensional collisional kinetic equations
When: Tue, October 24, 2023 - 3:30pmWhere: Video: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=434a41d5-a296-4b51-a454-b0a601108db8
Speaker: Jingwei Hu (University of Washington) - https://jingweihu-math.github.io/webpage/
Abstract: Kinetic equations describe the nonequilibrium dynamics of a complex particle system using a probability density function. Despite of their important role in multiscale modeling to bridge microscopic and macroscopic scales, numerically solving kinetic equations is computationally demanding as they lie in the six-dimensional phase space. Dynamical low-rank method is a dimension-reduction technique that has recently been applied to kinetic theory, yet most of the endeavor is devoted to linear or collisionless problems. In this talk, we introduce efficient dynamical low-rank methods for Boltzmann type collisional kinetic equations, building on certain prior knowledge about the low-rank structure of the solution.
Numerical methods for hydrodynamics at small scales
When: Tue, October 31, 2023 - 3:30pmWhere: Video: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=e4f592ef-ae46-4bb6-ad3c-b0c30155d802
Speaker: Sean Carney (George Mason University ) - https://math.gmu.edu/~scarney6/
Abstract: Biological cells, battery science, and microfluidic devices all feature sub-micron ($10^{-6}$ m) scale fluid dynamics. At these scales, the effects of viscosity, chemical reactions, surface tension, and electrostatic interactions can be dominant, while inertial forces that drive typical atmospheric and aerodynamic flows are relatively unimportant. At even smaller, nanometer ($10^{-9}$ m) length scales, it becomes critical to additionally account for thermal fluctuations that arise from the discrete, atomistic nature of fluid mixtures. It is a significant challenge to develop mathematical models that faithfully capture the physics of such small-scale fluid systems without resorting to fully discrete simulations (such as molecular dynamics) that are computationally intractable. The resulting systems of equations are usually stiff, nonlinearly coupled, and stochastic, and hence developing accurate and efficient numerical methods for their simulation is a great challenge as well. In this talk I will describe my previous work in this area, as well as ongoing work to develop a machine learning surrogate to accelerate hybrid particle-continuum models of complex fluids.
Low-rank PINNs for model reduction of nonlinear hyperbolic conservation laws
When: Tue, November 7, 2023 - 3:30pmWhere: Video: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=1e3026f5-d13e-4e64-8aa5-b0c30155d807
Speaker: Donsub Rim (Washington University in St. Louis) - https://dsrim.github.io/
Abstract: Model reduction for hyperbolic PDEs using classical techniques is difficult due to the slow decay in the Kolmogorov n-width, making it necessary to explore new forms of approximation. We will discuss a new approach using deep neural networks endowed with a particular low-rank structure, which we call low-rank Physics-Informed Neural Networks (LR-PINNs). LR-PINNs are a form of implicit neural representation in which the weights and biases belong to linear spaces of small dimensions. We will show that entropy solutions to scalar conservation laws can be represented efficiently by such a representation. Numerical examples illustrating the efficacy of the neural network will be shown, and we will also discuss applications of LR-PINNs regarding the so-called failure modes of PINNs.
This talk is based on joint works with Woojin Cho, Kookjin Lee, Randall J. LeVeque, Noseong Park, and Gerrit Welper.
Optimization on Matrix Manifolds, with Applications in Data Science
When: Tue, November 14, 2023 - 3:30pmWhere: Kirwan Hall 3206
Speaker: Pierre-Antoine Absil (University of Louvain) - https://sites.uclouvain.be/absil/
Abstract: This talks gives an introduction to the area of optimization on manifolds - also termed Riemannian optimization - and its applications in engineering and the sciences. Such applications arise when the optimization problem can be formulated as finding an optimum of a real-valued cost function defined on a smooth nonlinear search space. Oftentimes, the search space is a "matrix manifold", in the sense that its points admit natural representations in the form of matrices. In most cases, the matrix manifold structure is due either to the presence of nonlinear constraints (such as orthogonality or rank constraints), or to invariance properties in the cost function that need to be factored out in order to obtain a nondegenerate optimization problem. Manifolds that come up in applications include the rotation group SO(3) (e.g., for the generation of rigid body motions from sample points), the set of fixed-rank matrices (appearing for example in low-rank models for recommender systems), the set of 3x3 symmetric positive-definite matrices (e.g., for the interpolation and denoising of diffusion tensors in brain imaging), and the shape manifold (involved notably in morphing tasks).
In the recent years, the practical importance of optimization problems on manifolds has stimulated the development of geometric optimization algorithms that exploit the differential structure of the manifold search space. In this talk, we give an overview of geometric optimization algorithms and their applications, with an emphasis on the underlying geometric concepts and on the numerical efficiency of the algorithm implementations.
Jaywalking at the Intersection of Machine Learning and Interesting Math
When: Tue, November 21, 2023 - 3:30pmWhere: Video: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=c3071a68-22fa-4e5c-b86d-b0c30155d802
Speaker: Michael Puthawala (South Dakota State University) - https://www.sdstate.edu/directory/michael-puthawala
Abstract: Machine learning (ML) is a powerful tool for solving problems in fields ranging from robotics, medicine, materials science, cosmology and beyond. I’ll try and convince you that it’s mathematically interesting too. No prior knowledge of machine learning or deep learning is needed, but a general distrust of flashy claims on artificial intelligence may help. I will first give an overview of ML and some of its applications. Next, I will present two vignettes showing how ‘practical’ problems in ML yield mathematical problems with satisfying and interesting answers. First, we’ll consider how ‘one simple computational trick’ discovered to improve the speed of convergence of training makes a statement about knot theory. Second, I will show how a desire to build ML models to shortcut PDE simulations yield the development of neural operators that ‘properly’ approximating integral operators on a Hilbert space.
Towards efficient deep operator learning for forward and inverse PDEs: theory and algorithms
When: Tue, November 28, 2023 - 4:30pmWhere: Brin Math Center Colloquium Room
Speaker: Ke Chen (University of Maryland College Park) - https://math.umd.edu/~kechen/
Abstract: Deep neural networks (DNNs) have been a successful model across diverse machine learning tasks, increasingly capturing the interest for their potential in engineering problems where PDEs have long been the dominant model. This talk delves into efficient training for PDE operator learning in both the forward and the inverse problems setting. Firstly, we address the curse of dimensionality in PDE operator learning, demonstrating that certain PDE structures require fewer training samples through an analysis of learning error estimates. Secondly, we introduce an innovative DNN, the pseudo-differential auto-encoder integral network (pd-IAE net), designed for solving multiple inverse problems.
The Obstacle Problem: Pointwise Adaptive Finite Element Method and Optimal Control
When: Tue, December 5, 2023 - 3:30pmWhere: Kirwan Hall 3206
Speaker: Rohit Khandelwal (George Mason University) - https://rohitedu.github.io/
Abstract: Free boundary problems governed by physical principles are ubiquitous in science and engineering. In particular variational inequalities of firstkind, such as obstacle problem, arise in elasticity, fluid filtration in porous me-dia and finance. Special attention has been given to study the obstacle problemwhich acts as a model problem in many of these applications. In this talk, wefocus on the obstacle problem.
In the first part, we discuss the pointwise adaptive finite element method for theobstacle problem using quadratic conforming finite element method. The mainproof hinges on the regularized Green’s function and the construction of upperand lower barrier functions corresponding to continuous solution. We haveexploited the property that midpoint quadrature rules are exact for quadraticpolynomials and later, derive the key sign property of the discrete Lagrangemultiplier. Theory is confirmed by several numerical examples.Second part of the talk focuses on the a priori error estimates for an optimalcontrol problem constrained by an elliptic obstacle problem where the finiteelement discretization is carried out using the symmetric interior penalty dis-continuous Galerkin method. The main proofs are based on the improved L2-error estimates for the obstacle problem, the discrete maximum principle, anda well-known quadratic growth property.
Waveform Inversion via Reduced Order Modeling
When: Tue, January 30, 2024 - 3:30pmWhere: Video: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=d13c17ba-7eaf-42a5-80ac-b12a013c35ee
Speaker: Liliana Borcea (University of Michigan) - https://websites.umich.edu/~borcea/
Abstract: This talk is concerned with the following inverse problem for the wave equation: Determine the variable wave speed from data gathered by a collection of sensors, which emit probing signals and measure the generated backscattered waves. Inverse backscattering is an interdis-ciplinary field driven by applications in geophysical exploration, radar imaging, non-destructive evaluation of materials, etc. There are two types of methods:
(1) Qualitative (imaging) methods, which address the simpler problem of locating reflective
structures in a known host medium.
(2) Quantitative methods, also known as velocity estimation. Typically, velocity estimation is
formulated as a PDE constrained optimization, where the data are fit in the least squares sense by
the wave computed at the search wave speed. The increase in computing power has lead to growing interest in this approach, but there is a fundamental impediment, which manifests especially for high frequency data: The objective function is not convex and has numerous local minima even in the absence of noise. The main goal of the talk is to introduce a novel approach to velocity estimation, based on a reduced order model (ROM) of the wave operator. The ROM is called data driven because it is obtained from the measurements made at the sensors. The mapping between these measurements and the ROM is nonlinear, and yet the ROM can be computed efficiently using methods from numerical linear algebra. More importantly, the ROM can be used to define a better objective function for velocity estimation, so that gradient based optimization can succeed even for a poor initial guess.
(Joint work with Josselin Garnier, Alexander Mamonov and John Zimmerling)
Error control in a diffusion map-based PDE solver
When: Tue, February 6, 2024 - 3:30pmWhere: Video: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=47566412-eba1-4592-9459-b12a013c35d5
Speaker: Maria Cameron (University of Maryland) - https://www.math.umd.edu/~mariakc/
Abstract: Diffusion maps allow us to approximate differential operators on point clouds and therefore can be used as PDE solvers in high dimensions. The target measure diffusion map (TMDmap) (Banisch et al., 2020), a variant of diffusion maps that admits input point clouds sampled from an arbitrary density, is a variant of diffusion maps particularly suited to solving the committor PDE arising in quantifying rare transitions in metastable systems. We have derived sharp error estimates for the bias and variance errors for the approximation of the generator of the overdamped Langevin dynamics via TMDmap and obtained an error bound for the numerical solution to the committor problem. Our error formulas reveal the dependence of numerical errors on the parameters and settings of TMDmap and the required scaling between the number of data points and the bandwidth of TMDmap. The benefit of a spatially quasi-uniform input point cloud is demonstrated on a 12-dimensional example with a butane molecule.
This is a joint work with S. Sule and L. Evans.
Coarsening and mean field control of volatile droplets
When: Tue, February 13, 2024 - 3:30pmWhere: Kirwan Hall 3206
Speaker: Hangjie Ji (North Carolina State University) - https://hji5.math.ncsu.edu/
Abstract: A lubrication model can be used to describe the dynamics of a weakly volatile viscous fluid layer on a hydrophobic substrate. Thin layers of the fluid are unstable to perturbations and break up into slowly evolving interacting droplets. In this talk, we will first present a reduced-order dynamical system derived from the lubrication model based on the nearest-neighbour droplet interactions in the weak condensation limit. Dynamics for periodic arrays of identical drops and pairwise droplet interactions are investigated which provide insights to the coarsening dynamics of a large droplet system. Weak condensation is shown to be a singular perturbation, fundamentally changing the long-time coarsening dynamics for the droplets and the overall mass of the fluid in two additional regimes of long-time dynamics. For the second part of the talk, I will briefly discuss our recent results on a mean field control formulation for droplet dynamics. Numerical examples with high-order finite element computations for droplet formation, transport, merging, and splitting demonstrate the effectiveness of the proposed mean field control. This talk is based on joint works with Thomas Witelski, Guosheng Fu, and Wuchen Li.
Nonlinear scientific computing in machine learning and applications
When: Tue, February 20, 2024 - 3:30pmWhere: Video: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=a2bf1a52-28f2-4724-8579-b12a013c35d5
Speaker: Wenrui Hao (Pennsylvania State University) - https://sites.psu.edu/whao/
Abstract: Machine learning has seen remarkable success in various fields such as image classification, speech recognition, and medical diagnosis. However, this success has also raised intriguing mathematical questions about optimizing algorithms more efficiently and applying machine-learning techniques to address complex mathematical problems. In this talk, I will discuss the neural network model from a nonlinear scientific computing perspective and present recent work on developing a homotopy training algorithm to train neural networks layer-by-layer and node-by-node. I will also showcase the use of neural network discretization for solving nonlinear partial differential equations. Finally, I will demonstrate how machine learning can be used to learn a mathematical model from clinical data in cases where the pathophysiology of a disease, such as Alzheimer's, is not well understood.
Quantum algorithms for linear differential equations
When: Tue, February 27, 2024 - 3:30pmWhere: Video: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=5597b928-6293-43f2-b864-b12a013c35f0
Speaker: Dong An (University of Maryland College Park) - https://dong-an.github.io/
Abstract: Quantum computers are expected to simulate unitary dynamics (i.e., Hamiltonian simulation) much faster than classical computers. However, most applications in scientific computing involve non-unitary dynamics and processes. In this talk, we will discuss a recently proposed quantum algorithm for solving general linear differential equations. The idea of the algorithm is to reduce general differential equations to a linear combination of Hamiltonian simulation (LCHS) problems. For the first time, this approach allows quantum algorithms to solve linear differential equations with near-optimal dependence on all parameters. Additionally, we will discuss a hybrid quantum-classical differential equation algorithm based on LCHS, which may be more feasible on near-term quantum devices.
(This talk assumes no prior knowledge in quantum computation and information, and is based on [arXiv:2303.01029, arXiv:2312.03916])
Data-adaptive RKHS regularization for learning kernels in operators
When: Tue, March 5, 2024 - 3:30pmWhere: Kirwan Hall 3206
Speaker: Fei Lu (Johns Hopkins University) - https://math.jhu.edu/~feilu/
Abstract: Kernels are efficient in representing nonlocal dependence and are widely used to design operators between function spaces or high-dimensional data. Thus, learning kernels in operators from data is an inverse problem of general interest. Due to the nonlocal dependence, the inverse problem is often severely ill-posed with a data-dependent operator that is nearly singular. Therefore, regularization is necessary. However, little information is available to select a proper regularization norm. We tackle this issue by introducing a data-adaptive RKHS for regularization, penalizing small singular values. It leads to convergent estimators that are robust to noise, outperforming the widely used L2- or l2-regularizers. We will discuss both direct and iterative methods.
Approximation of differential operators on unknown manifolds and applications
When: Tue, March 12, 2024 - 3:30pmWhere: Kirwan Hall 3206
Speaker: John Harlim (Pennsylvania State University) - https://jharlim.github.io/myhomepage/
Abstract: I will discuss the approximation of differential operators on unknown manifolds where the manifolds are identified by a finite sample of point cloud data. While our formulation is general, we will focus on Laplacian operators whose spectral properties are relevant to manifold learning. I will report the spectral convergence results of these formulations with Radial Basis Functions approximation and their strengths/weaknesses in practice. Supporting numerical examples, involving the spectral estimation of various vector Laplacians such as the Bochner, Hodge, and Lichnerowicz Laplacians will be demonstrated. Application to solve elliptic PDEs will be discussed. If time permits, I will discuss also spectral convergence of the graph-Laplacian approach to approximate the Laplace-Beltrami operator with Dirichlet boundary condition.
Numerically stable methods for electromagnetic scattering in layered media
When: Tue, March 26, 2024 - 3:30pmWhere: Kirwan Hall 3206
Speaker: Mike O'Neil (New York University) - https://cims.nyu.edu/~oneil/
Abstract: It is well known that many of the standard integral formulations for time-harmonic electromagnetic scattering break down, both numerically and analytically, as the frequency tends to zero. In many instances, the breakdown is directly due to the use of electric current as the fundamental unknown (as opposed to both current and charge, or other non-physical variables). It is somewhat less well known that similar instabilities arise for scattering problems in perfectly conducting half-spaces or piecewise layered media. In this talk we show a direct extension of the classic Lorenz-Debye-Mie scattering theory from spheres to half-spaces and layered media, and introduce a 'generalized Debye' formulation which is immune from low-frequency breakdown and gracefully decouples the non-physical unknowns in the limit. Each of these formulations is based on the classic Sommerfeld formulation of half-space scattering, which is equivalent to transverse Fourier transforms of the underlying Green's function.
Differentiable physics for turbulence closure modeling from data
When: Tue, April 2, 2024 - 3:30pmWhere: Kirwan Hall 3206
Speaker: Romit Maulik (Pennsylvania State University) - https://romit-maulik.github.io/
Abstract: Deep learning is increasingly becoming a promising pathway to improving the accuracy of sub-grid scale (SGS) turbulence closure models for large eddy simulations (LES). We leverage the concept of differentiable turbulence, whereby an end-to-end differentiable solver is used in combination with physics-inspired choices of deep learning architectures to learn highly effective and versatile SGS models for two-dimensional turbulent flow. We perform an in-depth analysis of the inductive biases in the chosen architectures, finding that the inclusion of small-scale non-local features is most critical to effective SGS modeling, while large-scale features can improve pointwise accuracy of the a-posteriori solution field. The filtered velocity gradient tensor can be mapped directly to the SGS stress via decomposition of the inputs and outputs into isotropic, deviatoric, and anti-symmetric components. We see that the model can generalize to a variety of flow configurations, including higher and lower Reynolds numbers and different forcing conditions. We show that the differentiable physics paradigm is more successful than offline, a-priori learning, and that hybrid solver-in-the-loop approaches to deep learning offer an ideal balance between computational efficiency, accuracy, and generalization. Our experiments provide physics-based recommendations for deep-learning based SGS modeling for generalizable closure modeling of turbulence.
An effective discretization scheme for singular integral operators on surfaces
When: Tue, April 9, 2024 - 3:30pmWhere: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=c74250af-2974-4f7e-bd72-b14d015a80b7
Speaker: James Bremer (University of Toronto) - https://www.math.ucdavis.edu/~bremer/
Abstract: Integral equation methods are a natural approach to solving many linear elliptic boundary value problems. However, there are a number of significant challenges in developing fast and robust integral equations solvers. Prominent among these challenges is the difficulty in discretizing
the singular integral operators which arise when linear elliptic boundary value problems are
reformulated as integral equations. Singular integral operators given on surfaces are
particularly challenging as both the geometry of the surface and the singularity in the
integrand pose problems. I will discuss a highly effective method for the discretization of
such operators. It achieves high accuracy in a wide array of contexts and and is considerably
more efficient than competing algorithms. This scheme dates to 2012, but it has only recently become widely used and is rapidly become the standard approach in the field.
A Domain Decomposition Method for Solution of a PDE-Constrained Generalized Nash Equilibrium Model of Biofilm Community Metabolism
When: Tue, April 16, 2024 - 3:30pmWhere: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=5bb20ee2-7d2f-4fe2-8735-b154015a9f28
Speaker: Daniel B. Szyld (Temple University) - https://www.math.temple.edu/~szyld/
Abstract: In this talk we propose a domain-decomposition-based method for the solution of a PDE-coupled generalized Nash equilibrium problem. Our motivation is a problem for a multispecies biofilm community. Microbes are able to deploy different strategies in response to, and depending upon, local environmental conditions. In the setting of a microbial community, this property induces a Nash equilibrium problem because access to environmental resources is bounded. If microbes are also distributed in space, then those resources are subject to transport limitations (encoded in a PDE) and so microbial strategies at one location influence resources and, hence, microbial strategies, at another. The domain decomposition method we propose here uses overlapping subdomains, and this feature is important in the numerical solution of the PDEs.
We illustrate this in particular for a 1D problem, where the resulting ODE is stiff, but the proposed method is shown to overcome this difficulty. An example consisting of a model with two microbial species biofilm with 33 externally transported chemical concentrations is also presented.
Joint work with:
Isaac Klapper, Tianyu Zhang, Karsten Zengler, Cristal Z'{u}\~{n}iga, and Xinli Yu
Solution of Forward and Inverse Problems for Extreme-Scale 1-km-Resolution Earth Mantle Models
When: Tue, April 23, 2024 - 3:30pmWhere: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=a2add021-4c05-45db-bd36-b15b015deeb4
Speaker: Johann Rudi (Virginia Tech) - https://math.vt.edu/people/faculty/rudi-johann.html
Abstract: We target large-scale computational methods and parallel algorithms centered
around a challenging application: global-scale high-resolution mantle
convection.
Estimating parameters in mantle convection and plate tectonics models from
surface observations results in an optimization problem. The forward problem
for mantle flow is governed by highly nonlinear, heterogeneous, and
incompressible Stokes equations. Solving these governing equations is already a
major challenge at extreme computing scales. Adding an outer loop for parameter
estimation adds another dimension of solver challenges.
The computational methods for the forward problem rely heavily on adaptive
meshes for local 1-km-resolution of the globe. The methods further include
inexact Newton-Krylov with a combination of "BFBT" and multigrid
preconditioning for the saddle point linear systems in each Newton step.
Scalable parameter estimation is enabled by analytically derived adjoint Stokes
equations (i.e., optimize-then-discretize) within a Newton's method for
optimizing in the parameter space.
Uncertainties of parameters are revealed by local approximations (of the
Bayesian posterior) at the MAP point by computing a Gauss-Newton approximation
of the Hessian. We show inference on cross-sectional models of the Pacific and
the first global inference results for 1-km-resolving models.
An exact and efficient algorithm for Basis Pursuit Denoising via differential inclusions
When: Tue, April 30, 2024 - 3:30pmWhere: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=2fae8c78-3dcf-4a5c-b2d2-b1620155a97e
Speaker: Gabriel Provencher Langlois (Courant Institute of Mathematical Sciences, NYU) - https://gabrielpl.com/
Abstract: Basis Pursuit Denoising (BPDN) is a cornerstone of compressive sensing, statistics and machine learning. Its applicability to high-dimensional signal reconstruction, feature selection, and regression problems has motivated much research and effort to develop algorithms for performing BPDN effectively, yielding state-of-the-art algorithms via first-order optimization, coordinate descent, or homotopy methods. Recent work, however, has questioned the efficiency, robustness and accuracy of these state-of-the-art algorithms for BPDN. For example, the glmnet package for BPDN, which is state-of-the-art due to its claimed efficiency, lacks robustness and can yield inaccurate solutions that lead to many so-called false discoveries. Another example is existing homotopy methods for BPDN; most require technical assumptions that may not hold in practice to compute exact solution paths. Without an exact robust and efficient algorithm, these shortcomings will continue to hinder BPDN for high-dimensional applications. In this talk, I will present a novel homotopy algorithm based on differential inclusions that efficiently and robustly computes a solution to BPDN exactly up to machine precision. I will present some numerical experiments to illustrate the efficiency of our algorithm and discuss various theoretical implications of our algorithm.
Stabilization with extended bubbles of convection-diffusion problems
When: Tue, May 7, 2024 - 2:30pmWhere: MATH 1310
Speaker: Pedro Morin (Universidad Nacional del Litoral and CONICET) - https://www.fiq.unl.edu.ar/depto-mate/pmorin/
Abstract: We explore the idea of stabilizing convection-dominated convection-diffusion problems with bubbles. Besides the element bubbles proposed by Brezzi et.al. in the seminal paper of 1999, we consider bubbles associated with two-element bubbles. We will discuss the advantages and disadvantages of this proposal, its implementation with recursivity to compute the bubbles, and will present some numerical experiments, illustrating the performance of the method.
A structure-preserving parametric finite element method for geometric PDEs and applications
When: Tue, May 14, 2024 - 3:30pmWhere: https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=8b93239a-f3e0-488f-a5cc-b170015fda47
Speaker: Weizhu Bao (National University of Singapore) - https://blog.nus.edu.sg/matbwz/
Abstract: In this talk, I begin with a review of different geometric flows (PDEs) including mean curvature (curve shortening) flow, surface diffusion flow, Willmore flow, etc., which arise from materials science, interface dynamics in multi-phase flows, biology membrane, computer graphics, geometry, etc. Different mathematical formulations and numerical methods for mean curvature flow are then discussed. In particular, an energy-stable linearly implicit parametric finite element method (PFEM) is presented in details. Then the PFEM is extended to surface diffusion flow and anisotropic surface diffusion flow, and a structure-preserving implicit PFEM is proposed. Finally, sharp interface models and their PFEM approximations are presented for solid-state dewetting. This talk is based on joint works with Harald Garcke, Wei Jiang, Yifei Li, Robert Nuernberg, Yan Wang and Quan Zhao.