Numerical Analysis Archives for Fall 2023 to Spring 2024
Time fractional gradient flows: Theory and numerics
When: Tue, July 19, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=216f47af-ab58-4775-85cd-aed7015ab404
Speaker: Wenbo Li (University of Tennessee, Knoxville) -
Abstract: We consider a so-called fractional gradient flow: an evolution equation aimed at the minimization of a convex and lower semicontinuous energy, but where the evolution has memory effects. This memory is characterized by the fact that the negative of the (sub)gradient of the energy equals the so-called Caputo derivative of the state. We introduce a notion of “energy solutions” for which we refine the proofs of existence, uniqueness, and certain regularizing effects. We generalize the "deconvolution" scheme for Caputo derivative to non-uniform time steps and obtain properties that allow us to develop a “fractional minimizing movements” scheme for the gradient flow problem. We derive an a posteriori error estimate and show its reliability. We also obtain a priori error estimates under different assumptions. All our analysis could be extended to the case where there are Lipschitz perturbations of the convex energy.
Finite Element Approximation of a Membrane Model for Liquid Crystal Polymeric Networks
When: Tue, September 13, 2022 - 3:30pmWhere: MTH3206 and ONLINE (ter.ps/joinNAS to get Zoom link)
Speaker: Lucas Bouck (University of Maryland, College Park) -
Abstract: Liquid crystal polymeric networks are materials where a nematic liquid crystal is coupled with a rubbery material. When actuated with heat or light, the interaction of the liquid crystal with the rubber creates complex shapes. Starting from the classical 3D trace formula energy of Bladon, Warner and Terentjev (1994), we derive a 2D membrane energy as the formal asymptotic limit of the 3D energy. The derivation is similar to derivations in Ozenda, Sonnet, and Virga (2020) and Cirak et. al. (2014). We characterize the zero energy deformations and prove that the energy lacks certain convexity properties. We propose a finite element method to discretize the problem. To address the lack of convexity of the membrane energy, we regularize with a term that mimics a higher order bending energy. We prove that minimizers of the discrete energy converge to minimizers of the continuous energy. For minimizing the discrete problem, we employ a nonlinear gradient flow scheme, which is energy stable. Additionally, we present computations showing the geometric effects that arise from liquid crystal defects. Computations of configurations from nonisometric origami are also presented.
Approximation of fractional Operators and fractional PDEs using a sinc-basis
When: Tue, September 20, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=355d161d-1df9-4615-9af0-af1601540d8d
Speaker: Ludwig Striet (University of Freiburg / George Mason University) -
Abstract: We introduce a spectral method to approximate PDEs involving the fractional Laplacian with zero exterior condition. Our approach is based on interpolation by tensor products of sinc-functions, which combine a simple representation in Fourier-space with fast enough decay to suitably approximate the bounded support of solutions to the Dirichlet problem. This yields a numerical complexity of O(N logN) for the application of the operator to a discretization with N degrees of freedom. Iterative methods can then be employed to solve the fractional partial differential equations with exterior Dirichlet condition. We show a number of example applications and establish rates of convergence that are in line with rates for finite element based approaches.
Surrogate Approximation for the Grad-Shafranov Free BoundaryProblem using Stochastic Collocation and Multilevel Monte Carlo Methods
When: Tue, October 4, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=981fa1f5-9c89-490c-8d58-af24015792a6
Speaker: Jiaxing Liang (University of Maryland, College Park) -
Abstract: In magnetic confinement fusion devices, the equilibrium configuration of a plasma is determined by the balance between the hydrostatic pressure in the fluid and the magnetic forces generated by an array of external coils and the plasma itself. The location of the plasma is not known a priori and must be obtained as the solution to a free boundary problem. The partial differential equation that determines the behavior of the combined magnetic field depends on a set of physical parameters (location of the coils, intensity of the electric currents going through them, magnetic permeability, etc.) that are subject to uncertainty and variability. The confinement region is in turn a function of these stochastic parameters as well. In this work, we consider variations on the current intensities running through the external coils as the dominant source of uncertainty. This leads to a parameter space of dimension equal to the number of coils in the reactor. With the aid of a surrogate function built on a sparse grid in parameter space, a Monte Carlo strategy is used to explore the effect that stochasticity in the parameters has on important features of the plasma boundary such as the location of the x-point, the strike points, and shaping attributes such as triangularity and elongation. The use of the surrogate function reduces the time required for the Monte Carlo simulations by factors that range between 7 and over 30. In addition, as a remedy for the large computational cost of Monte Carlo simulation, we will also investigate the Multilevel Monte Carlo, using both a set of hierarchical uniform grids and adaptive grid refinement.
Analysis and Perturbation of Non-diffusive Variational Problems with Distributional and Weak Gradient Constraints
When: Tue, October 11, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=ca12a94f-ff06-42a5-b351-af2b01533b92
Speaker: Carlos Rautenberg (George Mason University) -
Abstract: We consider non-diffusive variational problems with mixed boundary conditions and with distributional or weak gradient constraints. This class of problems arises naturally in the modelling of the growth of non-homogeneous sand piles and the problem is posed in spaces of functions of bounded variation or Sobolev spaces. In this setting, the upper bound in the constraint is either a function or a Borel measure which allows the pile growth to observe discontinuities. We address existence and uniqueness of the model under low regularity assumptions, and rigorously identify its Fenchel pre-dual problem. The latter in some cases is posed on a non-standard space of Borel measures with square integrable divergences. Further, we investigate the perturbation of solutions to the problem by means of perturbation of the upper bound of the gradient constraint and establish stability results. We conclude the talk by introducing a mixed finite-element method and several numerical tests.
Diffusion Maps for solving the Backward Kolmogorov PDEs in moderately high dimensions
When: Tue, October 18, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=64c727d0-2013-4356-b6d2-af3201572594
Speaker: Maria Cameron (University of Maryland, College Park) -
Abstract: The diffusion map algorithm introduced by Coifman and Lafon in 2006 as a nonlinear dimensional reduction tool with proven theoretical guarantees has an important ability to approximate differential operators on point clouds. We show that by changing the kernel function inherent in diffusion maps and using renormalizations one can approximate the Backward Kolmogorov Operator for the stochastic differential equation governing the dynamics of biomolecules or atomic clusters described in collective variables: time-reversible dynamics with position-dependent and anisotropic diffusion. Moreover, the point cloud used as an input does not need to be sampled from the invariant density but can be generated by any standard enhanced sampling algorithm. Using the solution to the Backward Kolmogorov PDE on point cloud with appropriate boundary conditions one can identify reaction channels and calculate the transition rate between metastable states of interest. An application to alanine dipeptide in four collective variables whose configurational space is the four-dimensional torus will be discussed.
This is joint work with Luke Evans and Pratyush Tiwary.
Efficient solvers for Bayesian inverse problems and Gaussian random fields
When: Tue, October 25, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=e40eac4a-bf4c-463c-9742-af39015e70ed
Speaker: Arvind K. Saibaba (North Carolina State University) -
Abstract: A commonly used prior distribution for the infinite-dimensional Bayesian formulation is the Whittle-Mat\’ern prior, which involves a covariance operator based on a fractional elliptic operator. Due to computational considerations, the approach is computationally challenging for non-integer exponents. A similar computational bottleneck arises in generating samples using the SPDE approach to Gaussian random fields. We present two different solvers for efficiently applying the discretized covariance operator (and its square root) for all admissible values of the exponent: the first uses a multipreconditioned Krylov subspace method, and the second exploits a certain low-rank structure in the solution. We also discuss how the multipreconditioned solver can be used to accelerate the solution of linear Bayesian inverse problems to obtain the maximum a posteriori estimate and the approximate posterior variance. Numerical experiments demonstrate the performance and scalability of the solvers and their applicability to model and real-data inverse problems in tomography and a time-dependent heat equation. This is joint work with Harbir Antil (George Mason) and Hussam Al Daas (Rutherford Appleton Laboratory).
Mathematical modeling of cardiac valve dynamics by a resistive method
When: Mon, October 31, 2022 - 2:00pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=b3aa0a43-ff8f-49d6-9d5b-af3f01413985
Speaker: Ivan Fumagalli (Politecnico di Milano, Italy) -
Abstract: In the computational modeling of cardiac hemodynamics, complex deformations and topological changes of the fluid domain must be accounted for, particularly due to the opening/closing of the cardiac valves. In this framework, several unfitted methods for Computational Fluid Dynamics (CFD) have been proposed. Among them, the Resistive Immersed Implicit Surface (RIIS) method proved its effectiveness and computational efficiency in different clinical applications.
After introducing the RIIS method, recent developments will be presented, concerning two main aspects of its use for modeling cardiac valves. First, we discuss the issue of properly defining the ventricular pressure during the isovolumetric phases of the heartbeat, when all valves are closed. To solve this issue, we introduce an Augmented version of the RIIS method (ARIIS), extending a previous method proposed in the literature to the case of a mesh not conforming to the valve. Second, we present an original multiscale Fluid-Structure Interaction model for valve dynamics, coupling a lumped-parameter model for the structural mechanics of the valve leaflets with the three-dimensional blood flow, via the RIIS method.
Applications of the abovementioned models to the cases of native and prosthetic valves will be presented.
This is joint work with Alberto Zingaro, Michele Bucelli, Luca Dede', Alfio Quarteroni.
This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No 740132, IHEART 2017-2022, P.I. A. Quarteroni).
A journey through stochastic trace estimation
When: Tue, November 1, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=cef53583-e1ab-4cee-bb30-af40015787c3
Speaker: Daniel Kressner (EPFL, Lausanne, Switzerland) -
Abstract: The development and analysis of randomized techniques for estimating the
trace of a large matrix through matrix-vector products is a topic with
a long history that has recently seen increased attention and
new exciting developments. This talk aims at providing a survey of these
developments, with an emphasis on estimating the trace of a matrix function
f(A) for some function f and a symmetric matrix A. Such problems feature
prominently in a variety of applications from, e.g., scientific computing,
statistical learning, and computational physics. One significant recent
innovation is the combination of classical stochastic estimators, like
the Hutchinson estimator, with randomized low-rank approximation. We
will discuss novel error bounds that lead to insight and improvement of
such techniques. This talk is based on joint work with Alice Cortinovis
and David Persson.
CANCELLED
When: Tue, November 15, 2022 - 3:30pmWhere:
Speaker: -
Abstract:
A nonlinear bending theory for nematic LCE plates
When: Tue, November 29, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=db95c918-98cb-4196-91b1-af5c016c2aea
Speaker: Stefan Neukamm (TU Dresden, Germany) -
Abstract: In this talk we consider a bending theory for an elastic bilayer plate composed of a nematic liquid crystal elastomer in the top layer and a nonlinearly elastic material in the bottom layer.
The state of the plate is descibed by two kinematic variables: an isometric deformation of the plate and a director field with unit-length that describes the liquid crystal orientation on the deformed surface.
The model comes in the form of an energy functional that consists of an elastic energy that is quadratic in curvature and a surface Oseen-Frank energy for the liquid crystal.
Both energies are nonlinearly coupled in form of spontaneous curvature tensor that depends on the director field.
In the talk I shall explain how to derive the model via Gamma-convergence from a three-dimensional nonlinear elasticity model. In view of the spontaneous curvature term, the plate has typically non-flat equilibrium states with a geometry that non-trivially depends on the relative thickness and shape of the plate, material parameters, boundary conditions for the deformation, and anchorings of the liquid crystal orientation.
We explore this rich mechanical behavior with help of numerical experiments that rely on a discrete gradient flow approach that treats the isometry- and unit-length constraint in its linearized form.
The talk is based on joint work with M. Griehl & D.Padilla-Garza (Dresden), and S.Bartels & C. Pauls (Freiburg).
A Measure Perspective on Uncertainty Quantification
When: Tue, December 6, 2022 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=e75655ec-6543-4094-8aa1-af630164555a
Speaker: Amir Sagiv (Columbia University) -
Abstract: In many scientific areas, deterministic models (e.g., differential equations) use numerical parameters. Often, such parameters might be uncertain or noisy. A more honest model should therefore provide a statistical description of the quantity of interest. Underlying this numerical analysis problem is a fundamental question - if two "similar" functions push-forward the same measure, would the new resulting measures be close, and if so, in what sense? We will first show how the probability density function (PDF) of the quantity of interest can be approximated. We will then discuss how, through the lense of the Wasserstein-distance, our problem yields a simpler and more robust theoretical framework.
Finally, we will take a steep turn to a seemingly unrelated topic: the computational sampling problem. In particular, we will discuss the emerging class of sampling-by-transport algorithms, which to-date lacks rigorous theoretical guarantees. As it turns out, the mathematical machinery developed in the first half of the talk provides a clear avenue to understand this latter class of algorithms.
Accurate error bounds in finite element approximation and a posteriori error analysis
When: Tue, January 24, 2023 - 3:30pmWhere: Video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=6b69469b-dd63-4729-9242-af940174cb78
Speaker: Andreas Veeser (Università degli Studi di Milano) -
Abstract: Error bounds for numerical approximations may be viewed as a comparison of two seminorms. An accurate bound then corresponds to the equivalence of the seminorms. Taking this viewpoint, we survey several results on finite element approximation and a posteriori error estimates.
Optimization Problems Constrained by PDEs and Augmented Lagrangian Methods
When: Tue, January 31, 2023 - 3:30pmWhere: MTH3206 and ONLINE (ter.ps/joinNAS to get Zoom link)
Speaker: Harbir Antil (George Mason University) -
Abstract: In the first part of the talk, we analyze an optimization problem constrained by Darcy’s law, to design permeability that achieve uniform flow properties despite having nonuniform geometries. We establish well-posedness of the problem, as well as differentiability, which enables the use of rapidly converging, derivate-based optimization methods.
The second part of the talk will focus on ALESQP, which is a general purpose augmented Lagrangian based optimization algorithm that can handle generic constraints such as PDEs. Extensions of ALESQP to risk-averse optimization problems will also be considered.
The talk will end with a few realistic interdisciplinary applications. Examples include, optimal HVAC outlay to minimize pathogen propagation and numeromorphic imaging.
Spatial Manifestations of Order Reduction, and Remedies via Weak Stage Order
When: Tue, February 7, 2023 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=b4b6dd35-439f-4fe3-ab76-afa201761057
Speaker: Benjamin Seibold (Temple University) -
Abstract: Order reduction, i.e., the convergence of the solution at a lower rate than the formal order of the chosen time-stepping scheme, is a fundamental challenge in stiff problems. Runge-Kutta schemes with high stage order provide a remedy, but unfortunately high stage order is incompatible with DIRK schemes. We first highlight the spatial manifestations of order reduction in PDE IBVPs. Then we introduce the concept of weak stage order, and (a) demonstrate how it overcomes order reduction in important linear PDE problems; and (b) how high-order DIRK schemes can be constructed that are devoid of order reduction.
Internal waves in 2D aquaria and homeomorphisms of the circle (Avron Douglis Lecture)
When: Tue, February 14, 2023 - 3:30pmWhere: MTH3206 and ONLINE (ter.ps/joinNAS to get Zoom link)
Speaker: Maciej Zworski (UC Berkeley) -
Abstract: The connections between the formation of internal waves in fluids, spectral theory, and homeomorphisms of the circle were investigated by oceanographers in the 90s and resulted in novel experimental observations (Leo Maas et al, 1997). The specific homeomorphism is given by a "chess billiard" and has been considered by many authors (Fritz John 1941, Vladimir Arnold 1957, Jim Ralston 1973... ).
The relation between the nonlinear dynamics of this homeomorphism and linearized internal waves provides a striking example of classical/quantum correspondence (in a classical and surprising setting of fluids!). I will illustrate the results with numerical and experimental examples and explain how classical concepts such as rotation numbers of homeomorphisms (introduced by Henri Poincare) are related to solutions of the Poincare evolution problem (so named by Elie Cartan). The talk is based on joint work with Semyon Dyatlov and Jian Wang. I will also mention recent progress by Zhenhao Li on the case of irrational rotation numbers.
Learning Low Bending and Low Distortion Manifold Embeddings
When: Tue, February 21, 2023 - 3:30pmWhere: ONLINE (ter.ps/joinNAS to get Zoom link)
Speaker: Martin Rumpf (University of Bonn, Germany) -
Abstract: Autoencoders, which consist of an encoder and a decoder, are widely used in machine learning for dimension reduction of high-dimensional data. The encoder embeds the input data manifold into a lower-dimensional latent space, while the decoder represents the inverse map, providing a parametrization of the data manifold by the manifold in latent space. A good regularity and structure of the embedded manifold may substantially simplify further data processing tasks such as cluster analysis or data interpolation.
In this talk we propose and analyze a novel regularization for learning the encoder component of an autoencoder: a loss functional that prefers isometric, extrinsically flat embeddings and allows to train the encoder on its own. To perform the training it is assumed that for pairs of nearby points on the input manifold their local Riemannian distance and their local Riemannian average can be evaluated.
The loss functional is computed via Monte Carlo integration with different sampling strategies for pairs of points on the input manifold.
The main theorem identifies a geometric loss functional of the embedding map as the \(\Gamma\)-limit of the sampling-dependent loss functionals.
Numerical tests, using image data that encodes different explicitly given data manifolds, show that smooth manifold embeddings into latent space are obtained. Due to the promotion of extrinsic flatness, these embeddings are regular enough such that interpolation between not too distant points on the manifold is well approximated by linear interpolation in latent space as one possible postprocessing.
This is joint work with Juliane Braunsmann, Marko Rajkovic, and Benedikt Wirth.
The discrete maximum principle and positivity-preservation in finite element methods
When: Tue, February 28, 2023 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=f4445913-2bfa-4e4e-9fe3-afb70184d401
Speaker: Gabriel Barrenechea (University of Strathclyde, Scotland) -
Abstract: The quest for physical consistency in the discretisation of PDEs started as soon as the numerical methods started being proposed. By physical consistency we mean a discretisation that, by design, satisfies a property also satisfied by the continuous PDE. This property might be positivity of the discrete solution, or preservation of some bounds (e.g., concentrations should belong to the interval [0,1]), or can also be energy preservation, or exactly divergence-free velocities for incompressible fluids, etc.
Regarding positivity preservation, this topic has been around since the pioneering work by Ph. Ciarlet in the late 1960s and early 1970s. In the context of finite element methods, it was shown in those early works that in order for a finite element method to preserve positivity the mesh needs to satisfy certain geometrical restrictions, e.g., in two space dimensions with simplicial elements the triangulation needs to be of Delaunay type (in higher dimensions or quadrilateral meshes the restrictions are more involved). Throughout the years several conclusions have been reached in this topic, but in the context of finite element methods the discretisations tend to be of first order in space, and tend to be nonlinear. So, many important problems still remain open. In particular, one open problem is how to build a discretisation that will lead to a positive solution regardless of the geometry of the mesh and the order of the finite element method.
This talk will be divided in two parts. In the first one I will give a very quick summary of some nonlinear discretisations that respect the discrete maximum principle. I will focus mostly in the convection-diffusion equation, and in the discretisation known as Algebraic Flux Correction (AFC) scheme. I will review some of the results that I have obtained in collaboration with several collaborators (E. Burman, V. John, and P. Knobloch, mainly), results that include stability, and convergence (or lack of convergence) of the method depending on the geometry of the mesh, and the precise definition of the method. In the second part of the talk I will present a method that enforces bound-preservation (at the degrees of freedom) of the discrete solution. 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, then the outcome is a finite element function that satisfies the bounds at the degrees of freedom. Another important observation to make is that this approach is related to variational inequalities, and this fact will be exploited in the error analysis. The core of the second part 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.
Monotone discretization of the Monge-Ampère equation of optimal transport
When: Tue, March 14, 2023 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=99cb0889-ae8b-49ab-af97-afc5015af66d
Speaker: Guillaume Bonnet (University of Maryland College Park)
Abstract: The Monge-Ampère equation is a nonlinear second-order equation whose one main application is the modelization of optimal transport problems. The relevant boundary value problem in that setting, called the second boundary value problem for the Monge-Ampère equation, involves the Monge-Ampère equation, a convexity constraint, and a boundary condition often referred to as the optimal transport boundary condition. In the first part of the talk, I will discuss the discretization of the Monge-Ampère operator, together with the convexity constraint. The Monge-Ampère operator belongs to the class of degenerate elliptic operators. It is often suitable for discretizations of such operators to satisfy a property of monotonicity in order for the convergence of the resulting numerical schemes to be guaranteed. I will show how Selling's algorithm, a tool originating from the field of low-dimensional lattice geometry, may be used to build a monotone finite difference discretization of the Monge-Ampère operator whose evaluation is particularly efficient numerically. In the second part of the talk, I will present a strategy for handling the optimal transport boundary condition in numerical schemes, with theoretical guarantees in the setting of quadratic optimal transport. This involves studying the equivalence at the continuous level between the standard formulation of the second boundary value problem and a reformulation more amenable to discretization. I will conclude by showing some numerical results, including an application to the far-field refractor problem in nonimaging optics.
From blood flow to airplanes: efficient models for fluid flows around structures
When: Tue, March 28, 2023 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=cbd525dd-06bc-4535-8b0e-afd3015f994c
Speaker: Rebecca Durst (University of Pittsburgh) -
Abstract: In this talk, we explore models designed to efficiently model fluid flows past elastic and inelastic structures. We will first focus on the fluid-structure interaction problem modeling flow past an elastic solid, a primary example of which is blood flow through a blood vessel. We present significant results in the development of a loosely coupled time-splitting method for this system based on Robin-type interface conditions. Assuming negligible deformations of the interface, this method has been proven to be unconditionally stable with quasi-optimal error estimates in time. Critically, the values of the physical parameters do not impact the stability, so the splitting method does not suffer from the added-mass instability that is common to other loosely coupled methods for blood flow.
We will then shift our focus to discuss the challenges of modeling fluid flows around bluff bodies at higher Reynolds numbers. We will discuss numerical computations of the drag and lift for the classical benchmark problem of 2D flow around a cylinder. In particular, we introduce a dynamics perspective that we hope to develop as a useful tool for studying the behavior of numerical models at increasing Reynolds numbers. We will present results for a well-established Scott-Vogelius IMEX method and discuss our plans to use these dynamics tools to further investigate and validate a breakthrough methodology developed for computational aerodynamics based on the highly efficient Direct FEM Simulation (DFS) method.
This talk is based on joint work with Johnny Guzman, Erik Berman, Miguel Fernandez, Ridgway Scott, and Johan Jansson.
Efficient Numerical Methods for Weak Solutions of Partial Differential Equations
When: Tue, April 4, 2023 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=9c553077-d355-4756-a6db-afda0177126a
Speaker: Chunmei Wang (University of Florida) -
Abstract: Approximating weak solutions of partial differential equations (PDEs) is known to be important and extremely challenging in scientific computing and data science. In this talk, the speaker will discuss two kinds of numerical methods for weak solutions: (1) Primal-Dual Weak Galerkin (PDWG) finite element methods for low-dimensional PDEs; and (2) Deep Learning methods (Friedrichs Learning) for high-dimensional PDEs. The essential idea of PDWG is to interpret the numerical solutions as a constrained minimization of some functionals with constraints that mimic the weak formulation of the PDEs by using weak derivatives. The resulting Euler-Lagrange formulation results in a symmetric scheme involving both the primal variable and the dual variable (Lagrangian multiplier). Friedrichs Learning is a novel deep learning methodology that could learn the weak solutions of PDEs via a mini-max optimization characterization of the original problem. The speaker will explain what Friedrichs Learning is and how it can be used for solving PDEs with discontinuous solutions without any prior knowledge of the solution discontinuity.
Three stories about quantum scientific computing (APPLIED MATH COLLOQUIUM)
When: Tue, April 11, 2023 - 3:30pmWhere: ONLINE (ter.ps/joinNAS to get Zoom link)
Speaker: Lexing Ying (Stanford University) -
Abstract: The recent development in quantum computing has inspired rapid progress in developing quantum algorithms for scientific computing. This includes examples in numerical linear algebra, partial differential equations, and machine learning. In this talk, I will share three recent stories in this direction: (1) a new algorithm for forming rather general functions of quantum operators, (2) a new scheme for representing pseudodifferential operators with quantum circuits, and (3) a new low-depth algorithm for quantum phase estimation for early fault-tolerant quantum devices.
Some Mathematical Aspects of Deep Learning and Stochastic Gradient Descent (AZIZ LECTURE)
When: Wed, April 12, 2023 - 3:15pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=cbd3733f-b247-4984-84b0-afe2015dab5a
Speaker: Lexing Ying (Stanford University) -
Abstract: This talk concerns several mathematical aspects of deep learning and stochastic gradient descent. The first aspect is why deep neural networks trained with stochastic gradient descent often generalize. We will make a connection between the generalization and the stochastic stability of the stochastic gradient descent dynamics. The second aspect is to understand the training process of stochastic gradient descent. Here, we use several simple mathematical examples to explain several key empirical observations, including the edge of stability, exploration of flat minimum, and learning rate decay. Based on joint work with Chao Ma.
Some numerical topics in stellarator optimization
When: Tue, April 18, 2023 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=47d251d2-0c7c-4fc2-8ebf-afe8015d5c03
Speaker: Matt Landreman (Institute for Research in Electronics & Applied Physics, University of Maryland, College Park) -
Abstract: A stellarator is a configuration of magnets, without continuous rotation symmetry, for confining a plasma. Stellarators have applications for fusion energy and basic physics, and their design requires numerical shape optimization of surfaces and curves. In this seminar, some of the numerical problems in this application area will be discussed, emphasizing ongoing research at UMD. One problem of interest is optimization of the plasma shape to maximize confinement of the charged particles, using direct numerical simulation or physics-based surrogates. Another very challenging problem is optimizing the plasma shape to reduce the plasma turbulence that causes loss of heat. A new method of topology optimization for the electromagnetic coils will also be discussed.
Analytical quadrature for (Fast Multipole Accelerated) Boundary Element Methods
When: Tue, April 25, 2023 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=7ad99494-9d66-4f69-9e15-afef015bfccb
Speaker: Shoken Kaneko (Department of Computer Science, University of Maryland, College Park) -
Abstract: In a set of works with my advisors Prof. Ramani Duraiswami and Dr. Nail Gumerov (before his tragic passing), methods for addressing accuracy and efficiency issues in quadrature for FMM-BEM for solving the Laplace and Helmholtz equation in R^3 have been developed. In this talk I will introduce Fast Multipole Accelerated Boundary Element Methods for solving the Laplace and Helmholtz equations, and discuss how formation of the linear system via element quadrature, fast matrix vector products via the FMM, and iterative solution all must be balanced to produce fast and accurate FMM-BEM. I then will discuss the three papers on analytical integral primitives for (FMM-)BEM which we have recently submitted.
Quadrature to Expansion (https://arxiv.org/abs/2107.10942): The far-field quadrature is needed to evaluate the multipole expansion coefficients, i.e. integrals of the spherical basis functions, in the FMM-BEM. In this work a method exploiting Euler’s homogeneous function theorem is developed for the analytical evaluation of these integrals over simplex elements in one, two, and three dimensions for the Laplace equation.
Recursive Integrals for Polynomial Elements (RIPE, https://arxiv.org/abs/2302.02196): A method for accurate and efficient analytical evaluation of layer potentials over flat boundary elements with polynomial densities (shape functions) is developed. This method addresses integral evaluation for nearly singular, singular, and hypersingular kernels arising in the collocation BEM for Laplace and Helmholtz equation.
Analytical Galerkin Integrals (https://arxiv.org/abs/2302.03247): in this work an analytical method is developed for evaluating the four-dimensional integrals arising in the Galerkin BEM for the Laplace equation in R^3. This method is based on recursive dimensionality reduction achieved by recursive application of Gauss’ divergence theorem, analytically reducing the four dimensional integrals to point evaluations.
I will conclude with a discussion of my current research in this area, on generalizing these results.
A fully discrete adaptive scheme for parabolic equations
When: Tue, May 2, 2023 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=7f361bb0-7363-4752-b15f-aff601603578
Speaker: Pedro Morin (Universidad Nacional del Litoral, Argentina) -
Abstract: We present an adaptive algorithm for solving linear parabolic equations using hierarchical B-splines and the implicit Euler method for the spatial and time discretizations, respectively. Our development improves upon one from 2018 from Gaspoz and collaborators, where fully discrete adaptive schemes have been analyzed within the framework of classical finite elements. Our approach is based on an a posteriori error estimation that essentially consists of four indicators: a time and a consistency error indicator that dictate the time-step size adaptation, and coarsening and a space error indicator that are used to obtain suitably adapted hierarchical meshes (at different time-steps). Even though we use hierarchical B-splines for the space discretization, a straightforward generalization to other methods, such as FEM, is possible. The algorithm is guaranteed to reach the final time within a finite number of operations, and keep the space-time error below a prescribed tolerance. Some numerical tests document the practical performance of the proposed adaptive algorithm.
This is joint work with F. Gaspoz, E. Garau and R. Vazquez
The energy technique for BDF methods
When: Tue, May 9, 2023 - 3:30pmWhere: video https://umd.hosted.panopto.com/Panopto/Pages/Viewer.aspx?id=fe363aac-8151-4326-8938-affd015b887d
Speaker: Georgios Akrivis (University of Ioannina and Institute of Applied and Computational Mathematics FORTH, Crete, Greece ) -
Abstract: The application of the energy technique to numerical methods with very good stability properties for parabolic equations, such as algebraically stable Runge–Kutta methods or A-stable multistep methods, is straightforward. The extension to high order multistep methods requires some effort; the main difficulty concerns suitable choices of test functions. We discuss the energy technique for all six backward difference formula (BDF) methods. In the cases of the A-stable one- and two-step BDF methods, the application is trivial. The energy technique is applicable also to the three-, four- and five-step BDF methods via Nevanlinna–Odeh multipliers. The main new results are: i) No Nevanlinna–Odeh multipliers exist for the six-step BDF method. ii) The energy technique is applicable under a relaxed condition on the multipliers. iii) We present multipliers that make the energy technique applicable also to the six-step BDF method. Besides its simplicity, the energy technique for BDF methods is powerful, it leads to several stability estimates, and flexible, it can be easily combined with other stability techniques.