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Yanlai Chen, University of Massachusetts at DartmouthYanlai Chen, University of Massachusetts at DartmouthEwing 336Title: <br> Reduced Basis Method: Applications, recent improvements on efficiency and robustness, and a new iterative scheme Abstract:<br> Models of reduced computational complexity is indispensable in scenarios where a large number of numerical solutions to a parametrized problem are desired in a fast/real-time fashion. Thanks to an offline-online procedure and the recognition that the parameter-induced solution manifolds can be well approximated by finite-dimensional spaces, reduced basis method (RBM) and reduced collocation method (RCM) can improve efficiency by several orders of magnitudes. The accuracy of the RBM/RCM solutions is maintained through a rigorous a posteriori error estimator whose efficient and accurate evaluation is critical. After a brief introduction of the RBM, this talk will show our recent work on novel approaches for speeding up the offline portion of the approaches by around 6-fold, and new residual-based and residual-free strategies for circumventing error stagnation that is traditional of the classical RBM. If time permits, we will introduce a new iterative scheme inspired by RBM that outperforms conjugate gradient method with multigrid as a preconditioner for linear systems discretizing Poisson's equation. 4/12/2018 3:00:00 PM4/12/2018 4:00:00 PMFalse
Benjamin Seibold, Temple University Benjamin Seibold, Temple UniversityEWG 336Title: <br>Spatial Manifestations of Order Reduction and Weak State Order</br> Abstract: <br>When advancing a time-dependent PDE forward via Runge-Kutta methods, the numerical solution may exhibit a convergence order that is less than the actual order of the scheme. We demonstrate that this order reduction phenomenon arises rather generically via spatial boundary layers that are paradoxically caused by the fact that stage values are too accurate near the boundary. We show via modal analysis under which circumstances boundary layers persist globally in time, and present a new property of Runge-Kutta schemes, called weak stage order, which can remedy order reduction and is compatible with DIRK schemes.3/15/2018 3:00:00 PM3/15/2018 4:00:00 PMFalse
Harbir Antil, George Mason UniversityHarbir Antil, George Mason UniversityEWG 336Title: Fractional linear and semilinear PDEs: analysis, control and discretization <br></br> Abstract: Fractional differential operators provide an attractive mathematical tool to model effects with limited regularity properties. Particular examples are image processing and phase field models in which jumps across lower dimensional subsets and sharp transitions across interfaces are of interest. In this talk we will provide an introduction to fractional derivatives. We will discuss how to incorporate nonzero boundary conditions in fractional PDEs and study the boundary optimal control problems. We will conclude this talk with analysis and error estimates for fractional semilinear problems.12/7/2017 4:00:00 PM12/7/2017 5:00:00 PMFalse
Juan Pablo Borthagaray, University of Maryland College ParkJuan Pablo Borthagaray, University of Maryland College ParkEWG 336Title: Finite element approximations of the nonhomogeneous fractional Dirichlet problem <br></br> Abstract: We study finite element approximations of the following non-homogeneous Dirichlet problem for the integral fractional Laplacian on a bounded domain in n-dimensional space. We analyze two different approaches: the first one consists on the direct imposition of the Dirichlet condition; the second one is based on weak imposition of the Dirichlet condition and incorporating a nonlocal analogous of the normal derivative as a Lagrange multiplier in the formulation of the problem. In order to obtain convergence orders for our scheme, regularity estimates are developed, both for the solution and its nonlocal derivative. The method we propose requires that, as meshes are refined, the discrete problems be solved in a family of domains of growing diameter. This is joint work with Gabriel Acosta (UBA, Argentina) and Norbert Heuer (PUC, Chile).11/30/2017 4:00:00 PM11/30/2017 5:00:00 PMFalse

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