Numerical Analysis Archives for Fall 2022 to Spring 2023
Nonconforming and Discontinuous Methods in the Numerical Approximation of Nonsmooth Variational Problems (Sayas Numerics Seminar)
When: Tue, September 7, 2021 - 3:30pmWhere: slides: https://sayasseminar.math.umd.edu/Bartels.pdf
Speaker: Soeren Bartels (University of Freiburg, Germany) -
Abstract: Nonconforming and discontinuous finite elements are attractive for discretising variational problems with limited regularity properties, in particular, when discontinuities may occur. The possible lack of differentiability of related functionals prohibits the use of classical arguments to derive error estimates. As an alternative we make use of discrete and continuous convex duality relations and of quasi-interpolation operators with suitable projection properties. For total variation regularised minimisation problems a quasi-optimal error estimate is derived which is not available for standard finite element methods. Our results use and extend recent ideas by Chambolle and Pock.
Vibrations with nonlinear frequency dependencies: Linearization for eigenvalue computations and time integration
When: Tue, September 14, 2021 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=c98aa5ab-291d-413e-8470-ada301550d6c
Speaker: Karl Meerbergen (KU Leuven, Belgium) -
Abstract: Finite element models for the analysis of vibrations typically have a quadratic dependency on the frequency. This makes the finite element method suitable for eigenvalue computations and time integration, by a formulation as a first order system, which we call a linearization.
The study of new damping materials often leads to nonlinear frequency dependencies, sometimes represented by rational functions but, often, by truly nonlinear functions. In classical analyses, vibrations are studied in the frequency domain. In the context of numerical algorithms for digital twins, time integration of mathematical models is required, which is not straightforward for models that are not linear or polynomial in the frequency.
We will discuss rational approximation and linearization of nonlinear frequency dependencies and their use for time integration. In particular, we use the (set-valued and weighted) AAA rational approximations and associated linearizations. We show how real valued matrices can be obtained. We experimentally show how the parameters can be tuned to obtain a stable linear model.
Krylov Subspace Regularization for Inverse Problems
When: Tue, September 21, 2021 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=9acd7eb3-6a15-4617-8ac7-adaa0156ba3f
Speaker: Jim Nagy (Emory University) -
Abstract: Inverse problems arise in a variety of applications: image processing, finance, mathematical biology, and more. Mathematical models for these applications may involve integral equations, partial differential equations, and dynamical systems, and solution schemes are formulated by applying algorithms that incorporate regularization techniques and/or statistical approaches. In most cases these solutions schemes involve the need to solve a large-scale ill-conditioned linear system that is corrupted by noise and other errors. In this talk we describe Krylov subspace-based regularization approaches to solve these linear systems that combine direct matrix factorization methods on small subproblems with iterative solvers. The methods are very efficient for large scale inverse problems, they have the advantage that various regularization approaches can be used, and they can also incorporate methods to automatically estimate regularization parameters.
Arbitrarily regular virtual element methods for elliptic partial differential equations
When: Tue, September 28, 2021 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=a8a92961-c9b3-47a1-a8f4-adb101597d33
Speaker: Marco Verani (Politecnico di Milano, Italy) -
Abstract: In recent years, there has been an intensive research on numerical approximations of partial differential equations on polygonal and polyhedral (polytopal, for short) meshes. Such research activity has led to the design of several families of numerical discretizations for PDEs, as, for example, the polygonal/polyhedral finite element method, the mimetic finite difference, the virtual element method, the discontinuous Galerkin method on polygonal/polyhedral grids, the hybrid discontinuous Galerkin method and the hybrid high-order method.
In this talk, we focus on the virtual element method (VEM) introduced in [Beirao da Veiga, Brezzi, Cangiani, Manzini, Marini, Russo 2013] which offers a great flexibility in designing approximation spaces featuring important properties other than just supporting polytopal meshes. The remarkable aspect that makes the VEM so appealing in this respect is that the formulation of arbitrarily regular approximations and their implementation are relatively straightforward. The crucial point here is that in the virtual element setting we do not need to know explicitly the shape functions spanning the virtual element space. The basis functions are uniquely defined by a set of values dubbed the degrees of freedom and these values are the only knowledge that are needed to formulate and implement the numerical scheme. During the talk, we show how this feature makes the construction of arbitrarily regular conforming virtual element approximations for linear elliptic equations of any order much simpler than, e.g., in the classical simplicial finite element context and almost immediate to implement. A priori error estimates in suitable norms and paradigmatic numerical examples will be also presented and discussed.
Algebraic Multigrid for mixed finite element discretizations of the Stokes Equations
When: Tue, October 5, 2021 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=d1056954-1839-4dc7-86e4-adb8015839c6
Speaker: Ray Tuminaro (Sandia National Laboratories) -
Abstract: We discuss the development of monolithic algebraic multigrid (AMG) preconditioners for mixed finite-element discretizations of the Stokes Equations. Specifically, we focus on the well-known Q2-Q1 discretization where quadratic basis functions are used to discretize velocities while linear basis functions are used for pressures. In this case, the number of velocity unknowns is significantly greater than the number of pressure unknowns, and so it is clearly not possibly to apply AMG techniques that rely on the co-location of different types of unknowns at mesh nodes. We describe some of the challenges associated with applying AMG directly to the discretized Stokes system including the preservation of inf-sup stability for the coarse grid discretizations within the AMG hierarchy. To avoid some of these difficulties, we instead consider applying AMG to a stable low-order system based on a Q1-iso-Q2/Q1 discretization and using this to precondition the Q2-Q1 system. We show that AMG is more amenable to the Q1-iso-Q2/Q1 system and how the Q1-iso-Q2/Q1 solver can be used to precondtion the Q2-Q1 system. Numerical results are given to to demonstrate the overall effectiveness of the approach.
Efficiently computing the effect of small model perturbations on the statistics/long-term averages of a chaotic dynamical model
When: Tue, October 12, 2021 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=3368bdba-653c-49c2-8873-adbf015b790b
Speaker: Nisha Chandramoorthy (MIT) -
Abstract: Linear response is the derivative of statistics or long-time averages of a dynamical system with respect to its input parameters. In many ergodic chaotic systems, such as certain turbulent fluid flows, detailed climate models etc., linear response exists but has been notoriously difficult to compute exactly, especially in these high-dimensional problems. The difficulty may be attributed to a fundamental aspect of chaotic systems that obey linear response: infinitesimal parameter perturbations along a given orbit grow in norm exponentially, and yet, long-time averages or statistical averages over an ensemble of orbits respond smoothly to small parameter changes. Recently, several conceptually different methods including blended response, fast linear response algorithm and shadowing-based methods have been developed to avoid the unstable evolution of infinitesimal linear perturbations to obtain a bounded value for linear response. In this work, we present a new alternative for linear response computation called the space-split sensitivity (S3) algorithm. S3 is a provably convergent and efficient way to compute Ruelle’s linear response formula, which holds rigorously in uniformly hyperbolic systems. In this talk, the key ideas in deriving and implementing S3 will be presented. In particular, Ruelle’s formula is decomposed in a specific manner to allow the individual components to be computed recursively along a numerical solution of the dynamics, and with a precision that improves exponentially with the length of the solution. We will restrict ourselves to uniformly hyperbolic systems with one-dimensional unstable manifolds. This exact computation of linear response is useful for fundamental understanding of statistical response of chaotic dynamics to small parameter changes, and further, also for design optimization and uncertainty quantification problems.
An introduction to linear poroelasticity and its transition to nonlinear implicit elastic and related models
When: Tue, October 19, 2021 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=3be92d5d-a40f-46bb-96c0-adc6015878cd
Speaker: Vivette Girault (Sorbonne Universite CNRS, Laboratoire Jacques-Louis Lions, Paris, France) -
Abstract: Poroelasticity has many applications in energy, environmental engineering, and even biomedicine. When linear, it simulates the transient flow of a single-phase fluid in a deformable linear elastic porous medium. However, when the medium is brittle and fractured, linear elasticity is not applicable, but new nonlinear implicit models of elasticity lead to a good description of the phenomenon. I shall introduce for beginners the simplest model of poroelasticity, then describe some new implicit nonlinear models and explain their mathematical and numerical issues.
Collaborators: Tameem Almani, Andrea Bonito, Saumik Dana, Benjamin Ganis, Maria Gonzalez Taboada, Diane Guignard, Frederic Hecht, Kundan Kumar, Xueying Lu, Marc Mear, Kumbakonam Rajagopal, Gurpreet Singh, Endre Suli, Mary Wheeler.
Variational Physics informed neural networks: the role of quadratures and test functions
When: Tue, October 26, 2021 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=f4aba6c3-3451-42b3-abca-adcd01587afa
Speaker: Claudio Canuto, Politecnico di Torino, Turin, Italy
Abstract: We look at Variational Physics Informed Neural Networks (VPINN) from a finite element perspective, with the aim of solving elliptic boundary-value problems. In particular, we analyze how Gaussian or Newton-Cotes quadrature rules of different precisions, as well as piecewise polynomial test functions of different degrees, affect the convergence rate of VPINN with respect to mesh refinement. Using a Petrov-Galerkin framework relying on an inf-sup condition, we derive an a priori error estimate in the energy norm between the exact solution and a suitable high-order piecewise interpolant of a computed neural network. Numerical experiments confirm the theoretical predictions, and also indicate that the error decay follows the same behavior when the neural network is not interpolated. Our results suggest, perhaps counterintuitively, that for smooth solutions the best strategy to achieve a high decay rate of the error consists in choosing test functions of the lowest polynomial degree, while using quadrature formulas of sufficiently high precision.
This is a joint work with Stefano Berrone and Moreno Pintore.
Hybrid analytic-numerical compact models for radiation induced photocurrent effects
When: Tue, November 2, 2021 - 3:30pmWhere: ONLINE (ter.ps/joinNAS to get Zoom link)
Speaker: Pavel Bochev (Sandia Labs) -
Abstract: Compact semiconductor device models are essential for efficiently designing and analyzing large circuits. However, traditional compact model development requires a large amount of manual effort and can span many years. Moreover, inclusion of new physics such as radiation induced photocurrent effects into an existing model is not trivial and may require redevelopment from scratch. Data-driven approaches have the potential to automate and significantly speed up the development of compact models. In this talk we focus on the demonstration of this approach for the development of a hybrid numerical-analytical compact photocurrent model.Compact photocurrent models are generally formulated by separating the total photocurrent into prompt and delayed components. The former is treated by invoking the depletion approximation, which reduces the Drift-Diffusion Equations in the depletion region to a Poisson equation that can be solved analytically. The delayed component is handled with the charge balance assumption, under which the excess carrier dynamics can be modeled by the Ambipolar Diffusion Equation (ADE). However, the ADE is a nonlinear, time-dependent PDE that cannot be solved analytically. Compact analytic models apply further physical approximations and assumptions that render the ADE solvable in closed form but may reduce model's accuracy.In this talk we present a hybrid analytic-numerical approach to replace analytic solutions of the governing equations by numerical ones obtained from synthetic and/or experimental data by using a hierarchy of data-driven and machine-learning approaches. This obviates the need for additional approximations and yields a hierarchy of accurate and computationally efficient compact photocurrent models. We demonstrate these models by comparing their predictions with those of state-of-the-art analytic models using synthetic data and photocurrent measurements obtained at the Little Mountain Test Facility at Hill AFB, Utah.We will also briefly review Xyce PyMi, which is a new Python interface that enables execution of data-driven compact device models from Sandia's massively parallel production circuit simulator Xyce.This is joint work with J. Hanson, B. Paskaleva, E. Keiter, C. Hembree, P. Kuberry. Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy's National Nuclear Security Administration contract number DE-NA0003525. This work describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government.
Spectral calculations for magic angles in a simple model of twisted bilayer graphene
When: Tue, November 9, 2021 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=2919dc02-3c16-4110-ac13-addb01660897
Speaker: Mark Embree (Virginia Tech) -
Abstract: In 2019, Tarnopolsky, Kruchkov, and Vishwanath proposed a simplified model of twisted bilayer graphene based on a Hamiltonian that is periodic with respect to a special two-dimensional lattice. The spectral properties of this model can be understood by investigating a certain non-self-adjoint operator D(a); the parameter a describes the angle of orientation of the two graphene sheets. For most values of a, the spectrum of D(a) is a constant discrete set in the complex plane; however, for special "magic" values of a, the spectrum is the entire complex plane. We illustrate this behavior by studying the pseudospectra of D(a) and conducting a variety of other spectral calculations that investigate the separation of the "magic" values of a, the dependence of this behavior on the potential, and the accuracy of the Fourier spectral methods that enable these computations. (This talk describes a collaboration with Simon Becker, Jens Wittsten, and Maciej Zworski.)
Some Nonsmooth Function Classes and Their Optimization
When: Tue, November 16, 2021 - 3:30pmWhere: video https://coursemedia.gmu.edu/media/SAYAS+NUMERICS+SEMINAR/1_itslk1m2
Speaker: Jong-Shi Pang (University of Southern California) -
Abstract: Optimization problems with coupled nonsmoothness and nonconvexity are pervasive in statistical learning and many engineering areas. They are very challenging to analyze and solve. In particular, since the computation of their minimizers, both local and global, is generally intractable, one should settle for computable solutions with guaranteed properties and practical significance. In the case when these problems arise from empirical risk minimization in statistical estimation, inferences should be applied to the computed solutions to bridge the gap between statistical analysis and computational results.This talk gives an overview of several nonsmooth function classes and their connections and sketches an iterative surrogation-based algorithm for the minimization of one particular class of non-Clarke regular composite optimization problems. We will also very briefly touch on the general surrogation approach supplemented by exact penalization to handle challenging constraints.This talk is based on the monograph titled “Modern Nonconvex Nondifferentiable Optimization” joint with Ying Cui at the University of Minnesota, to be published in mid-November 2021.
Optimality in Learning
When: Tue, December 7, 2021 - 3:30pmWhere: video https://coursemedia.gmu.edu/media/SAYAS+NUMERICS+SEMINAR/1_164ivk1f
Speaker: Ronald DeVore (Texas A&M University) -
Abstract: Learning an unknown function $f$ from data observations arises in
a myriad of application domains. This talk will present the mathematical view
of learning. It has two main ingredients: (i) a model class assumption which summarizes the totality of information we have about $f$ in addition to the data observations, (ii) a metric which describes how we measure performance of a learning algorithm.
We first mathematically describe optimal recovery which is the best possible
performance for a learning algorithm. While optimal recovery provides the ideal
benchmark on which to evaluate performance of a numerical learning algorithm, it does not, in and of itself, provide a numerical recipe for learning. We then turn to
the construction of discrete optimization problems whose solution provides a
provably near optimal solution to the learning problem. We compare these discretizations with what is typically done in machine learning and in particular
explain some of the reasons why machine learning prefers over parameterized neural networks for numerically resolving the corresponding discrete optimization problems. The main obstacle in machine learning is to produce a numerical algorithm to solve the correct discretization problem and prove its convergence to a minimizer. We close with a few remarks on this bottleneck.
Convergent evolving surface finite element algorithms for geometric evolution equations
When: Tue, February 1, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=01fb6c32-dd15-4a4d-8f23-ae2f0167e71c
Speaker: Christian Lubich (University of Tuebingen, Germany) -
Abstract: Geometric flows of closed surfaces are important in a variety of
applications, ranging from the diffusion-driven motion of the surface
of a crystal to models for biomembranes and tumor growth. Basic
geometric flows are mean curvature flow (described by a spatially
second-order evolution equation) and Willmore flow and the closely
related surface diffusion flow (described by spatially fourth-order
evolution equations).
Devising provably convergent surface finite element algorithms for
such geometric flows of closed two-dimensional surfaces has long
remained an open problem, going back to pioneering work by Dziuk in
1988. Recently, Balázs Kovács, Buyang Li and I arrived at a first
solution to this problem for various geometric flows including those
mentioned above. The proposed algorithms discretize evolution
equations for geometric quantities along the flow, in our cases the
normal vector and mean curvature, and use these evolving geometric
quantities in the velocity law interpolated to the finite element
space. This numerical approach admits a convergence analysis in the
case of continuous finite elements of polynomial degree at least two.
The error analysis combines stability estimates and consistency
estimates to yield optimal-order H^1-norm error bounds for the
computed surface position, velocity, normal vector and mean curvature.
The stability analysis is based on the matrix-vector formulation of
the finite element method and does not use geometric arguments. The
geometry only enters into the consistency estimates.
The talk is based on joint work with Balázs Kovács and Buyang Li.
Flow of liquid crystals in curved thin films: a new model and an unfitted method
When: Tue, February 8, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=63969fae-4081-4431-a0c6-ae36016782b9
Speaker: Vladimir Yushutin (Dept. of Math., University of Maryland, College Park) -
Abstract: Consider a surface $\Gamma\subset\mathbb{R}^3$ representing a thin film
of a liquid crystal material. The aim of the first part of the talk is
to present a novel model in which a general three-dimensional Q-tensor
may undergo a two dimensional flow along $\Gamma$. The main difficulty
is to define a proper notion of Q-tensor transport on surfaces. A
thermodynamically consistent coupling of the momentum transport and the
Landau-de Gennes dynamics is derived from the generalized Onsager
principle.
The second part is devoted to the numerical treatment of the new model
on an arbitrary $\Gamma$. The model is formulated via the matrix algebra
in $\mathbb{R}^3$ so it is possible to use the unfitted finite element
method on a bulk, regular mesh intersected by $\Gamma$. The main
challenge is to supplement the $\Gamma$-based weak forms with some
stabilizations to have a non-degenerate formulation in $\mathbb{R}^3$
while still retaining the energy structure on the fully discrete level.
Interesting simulations including evolution of defects are also presented.
Programming and predicting the effective shapes of origami and kirigami surfaces
When: Tue, February 15, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=e9ea914a-13e0-4454-b0f7-ae3d017b675a
Speaker: Paul Plucinsky (Aerospace and Mechanical Engineering, University of Southern California Viterby) -
Abstract: Shape-morphing finds widespread utility, from the deployment of small stents and large
solar sails to actuation and propulsion in soft robotics. Origami and kirigami – patterns of cuts and
folds in a sheet – are versatile platforms for shape-morphing, inspiring the design of many morphing
structures and devices. However, it remains a challenge to design patterns that morph into a
specified surface on demand, and to predict their response to a broad range of loads and stimuli.
This talk explores general design and modeling principles for origami and kirigami structures.
In the first part of the talk, we develop an efficient algorithm that explicitly characterizes the
designs and deformations of a large class of easily deployable origami. We then employ this algorithm
in an inverse design framework to approximate a targeted surface. In the second part of the
talk, we describe a coarse-graining procedure to determine all the slighty stressed (soft) modes
of deformation of a large class of periodic and planar kirigami. The procedure gives a system of
nonlinear partial differential equations (PDE) expressing geometric compatibility of angle functions
related to the motion of individual slits. Leveraging known solutions of the PDE, we present
excellent agreement between simulations and experiments across kirigami designs. Our results reveal
a surprising nonlinear wave-type response persisting even at large boundary loads, the existence of
which is determined completely by the Poisson’s ratio of the unit cell.
Proximal gradient methods for control problems with non-smooth and non-convex control cost
When: Tue, February 22, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=d0366b8a-42e1-4daa-863d-ae44016551c3
Speaker: Daniel Wachsmuth (University of Wuerzburg) -
Abstract: We investigate the convergence of the proximal gradient method applied to
control problems with non-smooth and non-convex control cost. Here, we
focus on control cost functionals that promote sparsity, which includes
functionals of $L^p$-type for $p \in [0, 1)$. We prove stationarity
properties of weak limit points of the method. These properties are weaker
than those provided by Pontryagin’s maximum principle and weaker than
L-stationarity.
BayesCG: A probabilistic numeric linear solver
When: Tue, March 1, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=ee2e2959-795f-47bb-a650-ae4b016786bb
Speaker: Ilse C.F. Ipsen (North Carolina State University) -
Abstract: We present the probabilistic numeric solver BayesCG, for solving linear systems with real symmetric positive definite coefficient matrices. BayesCG is an uncertainty aware extension of the conjugate gradient (CG) method that performs solution-based inference with Gaussian distributions to capture the uncertainty in the solution due to early termination. Under a structure exploiting Krylov prior, BayesCG produces the same iterates as CG. The Krylov posterior covariances have low rank, and are maintained in factored form to preserve symmetry and positive semi-definiteness. This allows efficient generation of accurate samples to probe uncertainty in subsequent computations.
The Helmholtz boundary element method does not suffer from the pollution effect
When: Tue, March 8, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=eedf33f8-1815-425c-b841-ae5201665e28
Speaker: Euan Spence (University of Bath, UK) -
Abstract: In d dimensions, approximating an arbitrary function oscillating with frequency \leq k requires \sim k^d degrees of freedom. A numerical method for solving the Helmholtz equation (with wavenumber k and in d dimensions) suffers from the pollution effect if, as k increases, the total number of degrees of freedom needed to maintain accuracy grows faster than this natural threshold (i.e., faster than k^d for domain-based formulations, such as finite element methods, and k^{d-1} for boundary-based formulations, such as boundary element methods).
It is well known that the h-version of the finite element method (FEM) suffers from the pollution effect. In contrast, at least empirically, the h-version of the boundary element method (BEM) does not suffer from the pollution effect, but this has not been proved up till now.
In this talk, I will discuss recent results showing that the h-BEM does not suffer from the pollution effect in certain common situations.
A posteriori error analysis and adaptivity with general meshes
When: Tue, March 15, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=fe804cfe-1e5e-4442-b7f2-ae590158165d
Speaker: Andrea Cangiani (SISSA, Italy) -
Abstract: Recent years have witnessed a surge in research into flexible FEM-like Galerkin methods (discontinuous Galerkin, virtual elements, CutFEM, MFD, etc.) on meshes consisting of general (possibly curved) polygonal and polyhedral (“polytopic") elements. An obvious motivation of polytopic meshes is in their application within adaptive simulations, which are now accepted as the key technology for automatic computational complexity reduction. However, little has been done so far to exploit the endless possibilities offered by polytopic meshes, specifically in the context of mesh adaptive algorithms driven by reliable a posteriori error estimators.
I will present recent background work on a posteriori analysis and mesh adaptivity for two representative classes of polytopic methods: the discontinuous Galerkin method and the Virtual Element Method. I will show how in some cases well established analysis techniques can be extended to cover the general mesh case, while in others new ideas are required. Further, I will discuss the approach’s computational complexity reduction potential through some numerical examples tackling a range of problems, such as elliptic interface problems with flux-balancing interface conditions and free boundary evolution problems.
Balancing and Reduction of Dynamical Systems Based on Data
When: Tue, March 29, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=a7b52640-8265-4d20-9947-ae67015832f8
Speaker: Christopher Beattie (Virginia Polytechnic Institute) -
Abstract: Balanced truncation is a classical approach to model
reduction that has long been the "gold standard" for high fidelity
reduced order modeling of large scale linear dynamical systems. I will
discuss a novel data-driven reformulation of this approach that does not
require intrusive access to internal system dynamics, that is, knowledge
of an original system realization is unnecessary. Instead, observations
are accumulated of the system response - either sampling the transfer
function evaluated at complex driving frequencies, or sampling (in time)
the system's impulse response. Notably, we do not require access to
state-space trajectory snapshots as would be found in POD-type methods,
nor do we approximate any system Grammians. A variety of numerical
experiments are provided that verify the effectiveness of this approach.
Discrete Solvability of Helmholtz Problems
When: Tue, April 5, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=9b2bf319-9c3c-4275-978d-ae6e015725a6
Speaker: Celine Torres (University of Maryland, College Park) -
Abstract: We study the unique solvability of the discrete Helmholz problem with impedance boundary conditions using a conforming Galerkin hp-finite element method. Typically well-posedness of the problem is proved by using a compact perturbation argument to the continuous problem with the assumption that the discrete space is "sufficiently rich".
In this talk, we present an alternative approach to the asymptotic perturbation argument by mimicking the tools for proving well-posedness of the continuous problem directly on the discrete level. By using this, we prove new existence and uniqueness results of the hp-FEM for the Helmholtz problem. On the other hand, we present an example of a mesh in 2d where the resulting matrix is singular for some wave number. We introduce a simple algorithm which checks whether a given triangulation leads to a regular matrix and propose strategies to fix well-posedness if the mesh is "critical". The algorithm is independent of the mesh width and therefore is an alternative to the perturbation argument to prove well-posedness of the discrete problem.
A variational method for generating cross fields using higher order Q-tensors
When: Tue, April 12, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=06bd9faa-e4d0-4e22-9895-ae75015c12a1
Speaker: Dmitry Golovaty (The University of Akron) -
Abstract: A cross field is a locally defined orthogonal coordinate system invariant with respect to the cubic symmetry group. These objects are of interest in a variety of settings, from mesh generation to materials science.
In this talk, I will discuss the problem of constructing an arbitrary cross field that satisfies some prescribed boundary conditions by using a fourth-order Q-tensor theory that is build upon tensor products of projection matrices. We use a Ginzburg-Landau-type relaxation to formulate an appropriate variational problem that allows us to reliably produce cross fields on arbitrary Lipschitz domains. The relaxed framework provides us with tools to study the behavior of the singular set, i.e. the set on which the domain fails to be a cross field. In particular we can use the classical Ginzburg-Landau theory to study singularities of the associated energy.
This is a joint work with Dan Spirn and Alberto Montero.
Models for liquid crystals on surfaces
When: Tue, April 19, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=9e707ce9-1ee3-42db-b54b-ae7c015d806f
Speaker: Axel Voigt (Dresden University, Germany) -
Abstract: While continuous models for liquid crystals are well established in 2D and 3D, their
formulation on curved surfaces introduces new phenomena. We consider model for polar, nematic and smectic liquid crystals on curved surfaces, where we ensure a tangential alignment of the director field.
This leads to a strong coupling of the unknowns with topological and geometrical properties of the surface. Numerical investigations by surface finite elements for the resulting vector- and tensor-valued surface partial differential equations are used to explore these couplings. If in addition the surface is allowed to relax, this leads to unexpected equilibrium shapes. They are determined by the texture of the liquid crystal and strongly depend on the assumptions made in the derivation of the surface models.
An algorithmic approach towards relaxation - analytical background, numerical challenges, and examples from the theory of nonlinear elasticity
When: Tue, April 26, 2022 - 3:30pmWhere: ONLINE (ter.ps/joinNAS to get Zoom link)
Speaker: Georg Dolzmann (University of Regensburg, Germany) -
Abstract: Variational models in the framework of nonlinear elasticity theory for solid materials undergoing solid to solid phase transformations are often ill-posed in the sense that minimizing sequences do not converge to minimizers in classical Sobolev spaces. In this lecture, examples of such situations will be described including shape memory materials and elasto-plasticity. After a concise discussion of an analytical approach via relaxation a numerical scheme for the computation of relaxed energies will be presented. The validation of the numerical scheme will be discussed as well as perspectives for future research. This is joint work with Sergio Conti (Bonn).
Immersed Virtual Element Methods for Maxwell Interface Problems in Three Dimensions
When: Tue, May 3, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=b988d4e8-8c82-4a7a-bde1-ae8a01570dc7
Speaker: Long Chen (University of California at Irvine) -
Abstract: Finite element methods for Maxwell's equations are highly sensitive to the conformity of approximation spaces, and non-conforming methods may cause loss of convergence. This fact leads to an essential obstacle for almost all the interface-unfitted mesh methods in the literature regarding the application to Maxwell interface problems. In this talk, a novel immersed virtual element method for solving a 3D Maxwell interface problems is developed, and the motivation is to combine the conformity of virtual element spaces and robust approximation capabilities of immersed finite element spaces. The proposed method is able to achieve optimal convergence for a class of 3D Maxwell interface problem. To develop a systematic framework, a de Rham complex for interface problems will be established based on which the HX preconditioner can be adapted to develop a fast solver for the Maxwell interface problem. An efficient polyhedral mesh generator is also provided to generate a polyhedral mesh with an interface fitted boundary triangulation.
This is a joint work with Shuhao Cao and Ruchi Guo.
Approximation properties of shallow neural networks and applications to solving elliptic PDEs
When: Tue, May 10, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=c133e40b-bf13-4367-a0fd-ae9101585e0d
Speaker: Jonathan Siegel (Penn State University) -
Abstract: A natural space of functions which can be efficiently approximated by shallow neural networks is the variation space corresponding to the dictionary of single neuron outputs. We will precisely define this space and study its approximation properties. Specifically, we develop techniques for bounding the metric entropy and n-widths of the unit ball in this variation space. These are fundamental quantities in approximation theory that control the limits of linear and non-linear approximation. Consequences of these results include: the optimal approximation rates which can be attained for shallow neural networks, that shallow neural networks dramatically outperform linear methods of approximation, and indeed that shallow neural networks outperform all stable methods of approximation on the associated variation space. Then, we introduce a class of greedy algorithms and show that they construct asymptotically optimal shallow neural network approximations. Finally, we use these results to solve elliptic PDEs using neural networks.