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| Structure-preserving machine learning moment closures for the radiative transfer equation Juntao Huang Department of Mathematics and Statistics, Texas Tech University |
Please attend this week's Applied Math seminar at 4 PM Wednesday via this Zoom link.
Meeting ID: 976 3095 1027
Passcode: applied
| Recent Progress on Q^k Spectral Element Method: Accuracy, Monotonicity and Applications Xiangxiong Zhang Department of Mathematics, Purdue University |
1) Accuracy: when the least accurate (k+1)-point Gauss-Lobatto quadrature is used, the spectral element method is also a finite difference (FD) scheme, and this FD scheme can sometimes be (k+2)-th order accurate for k>=2. This has been observed in practice but never proven before as rigorous a priori error estimates. We are able to prove it for linear elliptic, wave, parabolic and Schrödinger equations for Dirichlet boundary conditions. For Neumann boundary conditions, (k+2)-th order can be proven if there is no mixed second order derivative. Otherwise, only (k+3/2)-th order can be proven and some order loss is indeed observed in numerical tests. The accuracy result also applies to spectral element method on any curvilinear mesh that can be smoothly mapped to a rectangular mesh, e.g., solving a wave equation on an annulus region with a curvilinear mesh generated by polar coordinates.
2) Monotonicity: consider solving the Poisson equation, then a scheme is called monotone if the inverse of the stiffness matrix is entrywise non-negative. It is well known that second order centered difference or P1 finite element method can form an M-matrix thus they are monotone, and high order accurate schemes in general are not monotone. But on structured meshes, high order accurate schemes can be monotone, though they do not form M-matrices. In particular, we have proven that the fourth order accurate FD scheme (Q^2 spectral element method) is a product of two M-matrices thus monotone for a variable coefficient diffusion operator: this is the first time that a high order accurate scheme is proven monotone for a variable coefficient operator. We have also proven the fifth order accurate FD scheme (Q^3 spectral element method) is a product of three M-matrices thus monotone for the Poisson equation: this is the first time that a fifth order accurate discrete Laplacian is proven monotone in two dimensions (all previously known high order monotone discrete Laplacian in 2D are fourth order accurate).
Please attend this week's Applied Math seminar at 4 PM Wednesday via this Zoom link.
Meeting ID: 976 3095 1027
Passcode: applied
| Global regularity issue of the two-and-a-half dimensional Hall-magnetohydrodynamics system Mohammad Mahabubur Rahman Department of Mathematics and Statistics, Texas Tech University |
Please attend this week's Applied Math seminar at 4 PM Wednesday via this Zoom link.
Meeting ID: 976 3095 1027
Passcode: applied
| Model Reduction using Moment Models for Kinetic Equations and Shallow Flows Julian Koellermeier Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, University of Groningen |
In this talk, I will introduce hierarchical moment models as a flexible way to derive hierarchies of models in fluid dynamics and other applications. The general derivation procedure of the reduced models results in structural similarities of the models, which facilitate physical insight, model adaptivity, and the development of suitable numerical methods. Based on kinetic equations and shallow flows, I will exemplify the hierarchical moment approach and highlight runtime and accuracy improvements.
Please attend this week's Applied Math seminar at 4 PM Wednesday via this Zoom link.
Meeting ID: 976 3095 1027
Passcode: applied
| Eigenvalue Analysis for Rigorous Error Bounds in a Reduced Basis Method Jehanzeb H. Chaudhry Department of Mathematics and Statistics, University of New Mexico |
In the first part of this talk, we present a reduced basis method with a sharp error estimate with respect to the exact solution of the PDE. A crucial element in developing such a method is the least-squares finite element method (LSFEM). LSFEMs are widely used for the solution of PDEs arising in many applications in science and engineering. LSFEMs are based on minimizing the residual of the PDE in an appropriate norm, and have a number of attractive properties. In particular, the property relevant to this work is that these methods provide a robust and inexpensive a posteriori error estimate with respect to the true solution. This estimate is utilized in developing the Least-Squares Reduced Basis Method presented in this talk.
The second part of the talk concerns a key ingredient in error estimates for variational problems and reduced basis methods: the so-called coercivity or inf-sup constant of the continuous problem. We characterize the coercivity constant as a spectral value of a self-adjoint linear operator; for several differential equations, we show that the coercivity constant is related to the eigenvalue of a compact operator. For these applications, convergence rates are derived and verified with numerical examples.
Please attend this week's Applied Math seminar at 4 PM Wednesday via this Zoom link.
Meeting ID: 976 3095 1027
Passcode: applied
| Modeling Calcium Dynamics in Neurons with Endoplasmic Reticulum: Well-Posedness and Numerical Methods Qingguang Guan School of Mathematics and Natural Sciences, University of Southern Mississippi |
Please attend this week's Applied Math seminar at 4 PM Wednesday via this Zoom link.
Meeting ID: 976 3095 1027
Passcode: applied
| Learning operators using deep neural networks for multiphysics, multiscale, & multifidelity problems Lu Lu Department of Chemical and Biomolecular Engineering, University of Pennsylvania |
Please attend this week's Applied Math seminar at 4 PM Wednesday via this Zoom link.
Meeting ID: 976 3095 1027
Passcode: applied
| A Flavor of Electron Transport: Localized Edge Modes in 2D Materials Dionisios Margetis Department of Mathematics & Institute for Physical Science and Technology, University of Maryland |
In many applications of photonics at the nanoscale, 2D materials such as graphene may behave as conductors, and allow for the excitation and propagation of electromagnetic waves with surprisingly small length scales. These surface waves are tightly confined to the material. They can possibly beat the optical diffraction limit, in the sense that the wavelength of the excited surface waves can be much smaller than that of the incident wave in a frequency range of practical interest. A broad goal in mathematical modeling is to understand how distinct kinetic regimes of 2D electron transport can be probed, and even controlled, by electromagnetic signals.
In this talk, I will discuss recent work in describing the dispersion of electromagnetic modes that may propagate along edges of flat, anisotropic conducting sheets. Some emphasis will be placed on the role of the fractional Laplacian in this context. An emergent concept of topological character pertaining to the existence of such modes will be presented. The starting point is a boundary value problem for Maxwell’s equations coupled with the physics of the moving electrons in 2D structures. This problem will be converted to singular integral equations of the Wiener-Hopf type, and be solved explicitly. Some extensions of the analysis will be discussed, if time permits.
Please attend this week's Applied Math seminar at 4 PM Wednesday via this Zoom link.
Meeting ID: 976 3095 1027
Passcode: applied