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Archive : Numerical Analysis and PDE

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Jake Jacavage, Lafayette CollegeJake Jacavage, Lafayette CollegeZoomTitle: A Least Squares Discretization for Parametric Mixed Variational Formulations <br> Abstract: Mixed variational formulations for partial differential equations arise naturally when modeling physical systems. When approximating the physical quantities from these systems, it is beneficial to obtain estimates that are robust with respect to the parameters associated with the system. In this talk, we will summarize some general ideas regarding an approach to approximating solutions of mixed variational problems using a saddle point reformulation. We will also discuss the concepts of optimal and almost optimal test norm as well as provide estimates for the continuity and stability constants. Applications of the method will include second order elliptic interface problems as well as reaction-diffusion problems. 12/11/2020 4:00:00 PM12/11/2020 5:00:00 PMFalse
Sigal Gottlieb, UMass DartmouthSigal Gottlieb, UMass DartmouthZoomTitle: Optimal time filtering methods as GLMs <br> Abstract: In this talk I will describe our work on time filtering methods for the Navier Stokes equations as well as other applications. Time filtering has been used to enhance the order of accuracy of given methods. This is particularly useful in the context of legacy codes, in which the time-stepping module is given and difficult to change. However, modifying the inputs and outputs is simple and allows for higher order. In this talk, we show how time filtering approaches can be seen as equivalent to generating a general linear methods. We use this GLM approach to develop an optimization routine that enabled us to find new time-filtering methods with high order and efficient linear stability properties. We will present our new methods and show their performance when tested on sample problems. Co-authors: Victor De Caria, Zachary Grant, William Layton.12/4/2020 4:00:00 PM12/4/2020 5:00:00 PMFalse
Jennifer Ryan, Colorado School of MinesJennifer Ryan, Colorado School of MinesZoomTitle: Extracting Extra Accuracy Through Post-Processing <br> Abstract: Many numerical simulations produce data that contains hidden information. This hidden information can be exploited to create even more accurate representations of the data by appropriately constructing convolution post-processors. In this presentation we address one particular form of data — that produced by discontinuous Galerkin finite element methods. Specifically, a discussion how the Smoothness-Increasing Accuracy-Conserving (SIAC) post-processing filter takes advantage of the information hidden in the numerical solution . Previous work focused on adapting the convolution kernel for boundaries, unstructured grids, and non-smooth solutions. This presentation will focus on identifying where this hidden accuracy comes from, why the hidden accuracy is important, and how to construct convolution post-processors to take advantage of this information in order to reduce the computational cost of multi-dimensional post-processing. <br>11/13/2020 4:00:00 PM11/13/2020 5:00:00 PMFalse
Benjamin Peherstorfer, Courant Institute, New York UniversityBenjamin Peherstorfer, Courant Institute, New York UniversityZoomTitle: Nonlinear model reduction for transport-dominated problems <br> Abstract: Solution manifolds induced by transport-dominated problems such as hyperbolic conservation laws typically exhibit nonlinear structures. This means that traditional model reduction methods based on linear approximations in subspaces are inefficient when applied to these problems. This presentation discusses model reduction methods for constructing nonlinear reduced models that seek approximations on manifolds, rather than in subspaces, and so lead to efficient dimensionality reduction even for transport-dominated problems. First, we will discuss an online adaptive approach that exploits locality in space and time to efficiently adapt piecewise linear approximations of the solution manifolds. Second, we present an approach that derives reduced approximations that are nonlinear by explicitly composing global transport dynamics with locally linear approximations of the solution manifolds. The compositions can be interpreted as one-hidden-layer neural networks. Numerical results demonstrate that the proposed approaches achieve speedups even for problems where traditional, linear reduced models are more expensive to solve than the high-dimensional, full model.10/30/2020 3:00:00 PM10/30/2020 4:00:00 PMFalse

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