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Archive : Numerical Analysis and PDE

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Dr. Qingguo Hong, Pennsylvania State UniversityDr. Qingguo Hong, Pennsylvania State UniversityEWG336Title: Parameter-robust iterative methods for Biot and multiple-permeability poroelasticity systems. <br><br> Abstract:In this talk, we consider the flux-based Biot and multiple-permeability poroelasticity systems describing multiple-network flow and deformation in a poro-elastic medium, also referred to as MPET models. The focus of the talk is on the convergence analysis of some iterative methods for the MPET models. These iterative methods include a commonly used fixed-stress split methods and the novelly designed Uzawa type methods. We prove the linear convergence of these fixed-point iteration and show that the contraction rates of these iterative methods do not depend on any of the physical parameters appearing in the models. These results are confirmed by numerical results. 3/13/2020 2:00:00 PM3/13/2020 3:00:00 PMFalse
Dr. Yuwen Li, Pennsylvania State university Dr. Yuwen Li, Pennsylvania State universityEWG336Title:A posteriori error and convergence analysis of adaptive mixed methods <br><br> Abstract: Many physical models are naturally derived as critical points of variational principles, which could be numerically solved by mixed finite element methods. On the other hand, adaptivity is indispensable for achieving the optimal computational complexity of numerical methods in general. <br> In this talk, first I will illustrate several basic ingredients in adaptive mixed finite element methods (AMFEMs) using the simple Poisson's equation in the mixed form. Then I will discuss AMFEMs for more complicated problems including the Stokes, Maxwell, Hodge Laplace equations, linear elasticity, heat/Darcy's flow, Biot's consolidation model, etc. 3/6/2020 3:00:00 PM3/6/2020 4:00:00 PMFalse
TALK CANCELLED TO BE RESCHEDULED Dr. Jennifer Ryan, Colorado School of MinesTALK CANCELLED TO BE RESCHEDULED Dr. Jennifer Ryan, Colorado School of MinesEWG336TALK CANCELLED TO BE RESCHEDULED 2/28/2020 3:00:00 PM2/28/2020 4:00:00 PMFalse
Dr. Chris Rackauckas, MIT and University of Maryland-BaltimoreDr. Chris Rackauckas, MIT and University of Maryland-BaltimoreEWG336Title:Universal Differential Equations for Scientific Machine Learning <br><br> <a href="https://www.mathsci.udel.edu/content-sub-site/Documents/universal_diffeq_full.pptx"> To download a copy of the presentation slides click here</a> <br><br> Abstract: In the context of science, the well-known adage "a picture is worth a thousand words" might well be "a model is worth a thousand datasets." Scientific models, such as Newtonian physics or biological gene regulatory networks, are human-driven simplifications of complex phenomena that serve as surrogates for the countless experiments that validated the models. Recently, machine learning has been able to overcome the inaccuracies of approximate modeling by directly learning the entire set of nonlinear interactions from data. However, without any predetermined structure from the scientific basis behind the problem, machine learning approaches are flexible but data-expensive, requiring large databases of homogeneous labeled training data. A central challenge is reconciling data that is at odds with simplified models without requiring "big data". In this talk we discuss a new methodology, universal differential equations (UDEs), which augments scientific models with machine-learnable structures for scientifically-based learning. We show how UDEs can be utilized to discover previously unknown governing equations, accurately extrapolate beyond the original data, and accelerate model simulation, all in a time and data-efficient manner. This advance is coupled with open-source software that allows for training UDEs which incorporate physical constraints, delayed interactions, implicitly-defined events, and intrinsic stochasticity in the model. Our examples show how a diverse set of computationally-difficult modeling issues across scientific disciplines, from automatically discovering biological mechanisms to accelerating climate simulations by 15,000x, can be handled by training UDEs. <br><br> <a href="https://arxiv.org/abs/2001.04385">https://arxiv.org/abs/2001.04385</a> 2/21/2020 3:00:00 PM2/21/2020 4:00:00 PMFalse

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