Applied/ACMS/absS21: Difference between revisions
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Abstract: In this talk, I will first introduce the variational formulation for the numerical spectral approximation of the second-order elliptic operators, followed by the introduction of the particular methods: softFEM, isogeometric analysis (IGA), and the hybrid high-order (HHO) method. The main idea of softFEM is to reduce the stiffness of the variational problem by subtracting to the standard stiffness bilinear form a least-squares penalty on the gradient jumps across the mesh interfaces. I will discuss briefly the motivation and why one wants to soften the stiffness of the resulting systems arising from the classical FEM. I will present a sharp upper bound on the softness parameter weighting the stabilization bilinear form so as to maintain coercivity for the softFEM bilinear form and then prove that softFEM delivers the optimal convergence rates as the standard Galerkin FEM approximation for the eigenvalues and the eigenvectors. The main idea of IGA is to apply highly-smooth basis functions within the Galerkin FEM framework. For this method, I will present dispersion analysis and develop analytical eigenpairs for the resulting generalized matrix eigenvalue problems. Lastly, the HHO method is formulated using cell and face unknowns which are polynomials of some degree. The key idea for the discrete eigenvalue problem is to introduce a discrete operator where the face unknowns have been eliminated. Using the abstract theory of spectral approximation of compact operators in Hilbert spaces, we prove that the eigenvalues converge as $h^{2t}$ and the eigenfunctions as $h^{t}$ in the $H^1$-seminorm, where $h$ is the mesh-size, $t\in [s,k+1]$ depends on the smoothness of the eigenfunctions, and $s>\frac12$ results from the elliptic regularity theory. The convergence rates for smooth eigenfunctions are thus $h^{2k+2}$ for the eigenvalues and $h^{k+1}$ for the eigenfunctions. The first two methods are continuous Galerkin (CG) methods while the third one is a discontinuous Galerkin (DG) method. I will make comparisons by showing some numerical examples. | Abstract: In this talk, I will first introduce the variational formulation for the numerical spectral approximation of the second-order elliptic operators, followed by the introduction of the particular methods: softFEM, isogeometric analysis (IGA), and the hybrid high-order (HHO) method. The main idea of softFEM is to reduce the stiffness of the variational problem by subtracting to the standard stiffness bilinear form a least-squares penalty on the gradient jumps across the mesh interfaces. I will discuss briefly the motivation and why one wants to soften the stiffness of the resulting systems arising from the classical FEM. I will present a sharp upper bound on the softness parameter weighting the stabilization bilinear form so as to maintain coercivity for the softFEM bilinear form and then prove that softFEM delivers the optimal convergence rates as the standard Galerkin FEM approximation for the eigenvalues and the eigenvectors. The main idea of IGA is to apply highly-smooth basis functions within the Galerkin FEM framework. For this method, I will present dispersion analysis and develop analytical eigenpairs for the resulting generalized matrix eigenvalue problems. Lastly, the HHO method is formulated using cell and face unknowns which are polynomials of some degree. The key idea for the discrete eigenvalue problem is to introduce a discrete operator where the face unknowns have been eliminated. Using the abstract theory of spectral approximation of compact operators in Hilbert spaces, we prove that the eigenvalues converge as $h^{2t}$ and the eigenfunctions as $h^{t}$ in the $H^1$-seminorm, where $h$ is the mesh-size, $t\in [s,k+1]$ depends on the smoothness of the eigenfunctions, and $s>\frac12$ results from the elliptic regularity theory. The convergence rates for smooth eigenfunctions are thus $h^{2k+2}$ for the eigenvalues and $h^{k+1}$ for the eigenfunctions. The first two methods are continuous Galerkin (CG) methods while the third one is a discontinuous Galerkin (DG) method. I will make comparisons by showing some numerical examples. | ||
=== Yulong Lu (UMass) == | |||
Theoretical guarantees of machine learning methods for statistical sampling and PDEs in high dimensions | |||
Abstract: Neural network-based machine learning methods, including the most notably deep learning have achieved extraordinary successes in numerous fields. In spite of the rapid development of learning algorithms based on neural networks, their mathematical analysis are far from understood. In particular, it has been a big mystery that neural network-based machine learning methods work extremely well for solving high dimensional problems. | |||
In this talk, I will demonstrate the power of neural network methods for solving two classes of high dimensional problems: statistical sampling and PDEs. In the first part of the talk, I will present a universal approximation theorem of deep neural networks for representing high dimensional probability distributions. In the second part of the talk, I will discuss the generalization error analysis of the Deep Ritz Method for solving high dimensional elliptic PDEs. For both problems, our theoretical results show that neural networks-based methods can overcome the curse of dimensionality. | |||
Revision as of 18:21, 1 March 2021
ACMS Abstracts: Spring 2021
Christina Kurzthaler (Princeton)
Complex Transport Phenomena
Abstract: Self-propelled agents are intrinsically out of equilibrium and exhibit a variety of unusual transport features. In this talk, I will discuss the spatiotemporal dynamics of catalytic Janus colloids characterized in terms of the intermediate scattering function. Our findings show quantitative agreement of our analytic theory for the active Brownian particle model with experimental observations from the smallest length scales, where translational diffusion and self-propulsion dominate, up to the larges ones, which probe the rotational diffusion of the active agents. In the second part of this talk, I will address the hydrodynamic interactions between sedimenting particles and surfaces with corrugated topographies, omnipresent in biological and microfluidic environments. I will present an analytic theory for the roughness-induced mobility and discuss the sedimentation behavior of a sphere next to periodic and randomly structured surfaces.
Antoine Remond-Tiedrez (UW)
Instability of an Anisotropic Micropolar Fluid
Abstract: Many aerosols and suspensions, or more broadly fluids containing a non-trivial structure at a microscopic scale, can be described by the theory of micropolar fluids. The resulting equations couple the Navier-Stokes equations which describe the macroscopic motion of the fluid to evolution equations for the angular momentum and the moment of inertia associated with the microcopic structure. In this talk we will discuss the case of viscous incompressible three-dimensional micropolar fluids. We will discuss how, when subject to a fixed torque acting at the microscopic scale, the nonlinear stability of the unique equilibrium of this system depends on the shape of the microstructure.
Hugo Touchette (Stellenbosch University)
Large deviation theory: From physics to mathematics and back
Abstract: I will give a basic overview of the theory of large deviations, developed by Varadhan (Abel Prize 2007) in the 1970s, and of its applications in statistical physics. In the first part of the talk, I will discuss the basics of this theory and its historical sources, which can be traced back in mathematics to Cramer (1938) and Sanov (1960) and, on the physics side, to Einstein (1910) and Boltzmann (1877). In the second part, I will show how the theory can be applied to study equilibrium and nonequilibrium systems and to express many key concepts of statistical physics in a clear mathematical way.
Tijana Pfander (Ludwig-Maximilians-University of Munich)
Towards next generation data assimilation algorithms for convective scale applications
Abstract: The initial state for a geophysical numerical model is produced by combining observational data with a short-range model simulation using a data assimilation algorithm. Particularly challenging is the application of these algorithms in weather forecasting at the convective scale. For convective scale applications, high resolution nonlinear numerical models are used. In addition, intermittent convection is present in the simulations and observations, often leading to errors in locations and intensity of convective storms. In addition, the state vector has a large size, one third of which contains variables whose non-negativity needs to be preserved, and the estimation of the state vector has to be done frequently in order to catch fast changing convection. Finally, often, not only one, but rather an ensemble of predictions is needed in order to correctly specify, for example, the uncertainty of rain at a particular location, even further increasing the computational considerations. In current practice, many data assimilation methods do not preserve the non-negativity of variables and rely on Gaussian assumptions. We present an algorithm that could be used for weather forecasting at the convective scale, that is based on the ensemble Kalman filter (EnKF) and quadratic programming. This algorithm outperforms the EnKF as well as the EnKF with the lognormal change of variables for all ensemble sizes. For a model that was designed to mimic the important characteristics of convective motion, preserving non-negativity of rain and conserving mass reduce the error in all fields; they prevent the data assimilation algorithm from producing artificial mass or artificial rain. Finally, important reduction in the computational costs has been recently achieved, making it possible to apply this algorithm in high dimensional weather forecasting problems in the future.
Quanling Deng (UW)
Spectral approximation of elliptic operators by softFEM, isogeometric analysis, and the hybrid high-order method
Abstract: In this talk, I will first introduce the variational formulation for the numerical spectral approximation of the second-order elliptic operators, followed by the introduction of the particular methods: softFEM, isogeometric analysis (IGA), and the hybrid high-order (HHO) method. The main idea of softFEM is to reduce the stiffness of the variational problem by subtracting to the standard stiffness bilinear form a least-squares penalty on the gradient jumps across the mesh interfaces. I will discuss briefly the motivation and why one wants to soften the stiffness of the resulting systems arising from the classical FEM. I will present a sharp upper bound on the softness parameter weighting the stabilization bilinear form so as to maintain coercivity for the softFEM bilinear form and then prove that softFEM delivers the optimal convergence rates as the standard Galerkin FEM approximation for the eigenvalues and the eigenvectors. The main idea of IGA is to apply highly-smooth basis functions within the Galerkin FEM framework. For this method, I will present dispersion analysis and develop analytical eigenpairs for the resulting generalized matrix eigenvalue problems. Lastly, the HHO method is formulated using cell and face unknowns which are polynomials of some degree. The key idea for the discrete eigenvalue problem is to introduce a discrete operator where the face unknowns have been eliminated. Using the abstract theory of spectral approximation of compact operators in Hilbert spaces, we prove that the eigenvalues converge as $h^{2t}$ and the eigenfunctions as $h^{t}$ in the $H^1$-seminorm, where $h$ is the mesh-size, $t\in [s,k+1]$ depends on the smoothness of the eigenfunctions, and $s>\frac12$ results from the elliptic regularity theory. The convergence rates for smooth eigenfunctions are thus $h^{2k+2}$ for the eigenvalues and $h^{k+1}$ for the eigenfunctions. The first two methods are continuous Galerkin (CG) methods while the third one is a discontinuous Galerkin (DG) method. I will make comparisons by showing some numerical examples.
= Yulong Lu (UMass)
Theoretical guarantees of machine learning methods for statistical sampling and PDEs in high dimensions
Abstract: Neural network-based machine learning methods, including the most notably deep learning have achieved extraordinary successes in numerous fields. In spite of the rapid development of learning algorithms based on neural networks, their mathematical analysis are far from understood. In particular, it has been a big mystery that neural network-based machine learning methods work extremely well for solving high dimensional problems.
In this talk, I will demonstrate the power of neural network methods for solving two classes of high dimensional problems: statistical sampling and PDEs. In the first part of the talk, I will present a universal approximation theorem of deep neural networks for representing high dimensional probability distributions. In the second part of the talk, I will discuss the generalization error analysis of the Deep Ritz Method for solving high dimensional elliptic PDEs. For both problems, our theoretical results show that neural networks-based methods can overcome the curse of dimensionality.