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Articles 1 - 7 of 7
Full-Text Articles in Physical Sciences and Mathematics
Nonlinear Multigrid Based On Local Spectral Coarsening For Heterogeneous Diffusion Problems, Chak Shing Lee, Francois Hamon, Nicola Castelletto, Panayot S. Vassilevski, Joshua A. White
Nonlinear Multigrid Based On Local Spectral Coarsening For Heterogeneous Diffusion Problems, Chak Shing Lee, Francois Hamon, Nicola Castelletto, Panayot S. Vassilevski, Joshua A. White
Mathematics and Statistics Faculty Publications and Presentations
This work develops a nonlinear multigrid method for diffusion problems discretized by cell-centered finite volume methods on general unstructured grids. The multigrid hierarchy is constructed algebraically using aggregation of degrees of freedom and spectral decomposition of reference linear operators associated with the aggregates. For rapid convergence, it is important that the resulting coarse spaces have good approximation properties. In our approach, the approximation quality can be directly improved by including more spectral degrees of freedom in the coarsening process. Further, by exploiting local coarsening and a piecewise-constant approximation when evaluating the nonlinear component, the coarse level problems are assembled and …
A Posteriori Error Estimates For Elliptic Eigenvalue Problems Using Auxiliary Subspace Techniques, Stefano Giani, Luka Grubišić, Harri Hakula, Jeffrey S. Ovall
A Posteriori Error Estimates For Elliptic Eigenvalue Problems Using Auxiliary Subspace Techniques, Stefano Giani, Luka Grubišić, Harri Hakula, Jeffrey S. Ovall
Mathematics and Statistics Faculty Publications and Presentations
We propose an a posteriori error estimator for high-order p- or hp-finite element discretizations of selfadjoint linear elliptic eigenvalue problems that is appropriate for estimating the error in the approximation of an eigenvalue cluster and the corresponding invariant subspace. The estimator is based on the computation of approximate error functions in a space that complements the one in which the approximate eigenvectors were computed. These error functions are used to construct estimates of collective measures of error, such as the Hausdorff distance between the true and approximate clusters of eigenvalues, and the subspace gap between the corresponding true and approximate …
Convex Analysis Of Minimal Time And Signed Minimal Time Functions, D. V. Cuong, B. S. Mordukhovich, Mau Nam Nguyen, M. L. Wells
Convex Analysis Of Minimal Time And Signed Minimal Time Functions, D. V. Cuong, B. S. Mordukhovich, Mau Nam Nguyen, M. L. Wells
Mathematics and Statistics Faculty Publications and Presentations
In this paper we first consider the class of minimal time functions in the general setting of locally convex topological vector (LCTV) spaces. The results obtained in this framework are based on a novel notion of closedness of target sets with respect to constant dynamics. Then we introduce and investigate a new class of signed minimal time functions, which are generalizations of the signed distance functions. Subdifferential formulas for the signed minimal time and distance functions are obtained under the convexity assumptions on the given data.
Structure Aware Runge–Kutta Time Stepping For Spacetime Tents, Jay Gopalakrishnan, Joachim Schöberl, Christoph Wintersteiger
Structure Aware Runge–Kutta Time Stepping For Spacetime Tents, Jay Gopalakrishnan, Joachim Schöberl, Christoph Wintersteiger
Mathematics and Statistics Faculty Publications and Presentations
We introduce a new class of Runge–Kutta type methods suitable for time stepping to propagate hyperbolic solutions within tent-shaped spacetime regions. Unlike standard Runge–Kutta methods, the new methods yield expected convergence properties when standard high order spatial (discontinuous Galerkin) discretizations are used. After presenting a derivation of nonstandard order conditions for these methods, we show numerical examples of nonlinear hyperbolic systems to demonstrate the optimal convergence rates. We also report on the discrete stability properties of these methods applied to linear hyperbolic equations.
A Tutorial Of Bland Altman Analysis In A Bayesian Framework, Krissina M. Alari, Steven B. Kim, Jeffrey O. Wand
A Tutorial Of Bland Altman Analysis In A Bayesian Framework, Krissina M. Alari, Steven B. Kim, Jeffrey O. Wand
Mathematics and Statistics Faculty Publications and Presentations
There are two schools of thought in statistical analysis, frequentist, and Bayesian. Though the two approaches produce similar estimations and predictions in large-sample studies, their interpretations are different. Bland Altman analysis is a statistical method that is widely used for comparing two methods of measurement. It was originally proposed under a frequentist framework, and it has not been used under a Bayesian framework despite the growing popularity of Bayesian analysis. It seems that the mathematical and computational complexity narrows access to Bayesian Bland Altman analysis. In this article, we provide a tutorial of Bayesian Bland Altman analysis. One approach we …
Numerical Results For Adaptive (Negative Norm) Constrained First Order System Least Squares Formulations, Andreas Schafelner, Panayot S. Vassilevski
Numerical Results For Adaptive (Negative Norm) Constrained First Order System Least Squares Formulations, Andreas Schafelner, Panayot S. Vassilevski
Mathematics and Statistics Faculty Publications and Presentations
We perform a follow-up computational study of the recently proposed space–time first order system least squares ( FOSLS ) method subject to constraints referred to as CFOSLS where we now combine it with the new capability we have developed, namely, parallel adaptive mesh refinement (AMR) in 4D. The AMR is needed to alleviate the high memory demand in the combined space time domain and also allows general (4D) meshes that better follow the physics in space–time. With an extensive set of computational experiments, performed in parallel, we demonstrate the feasibility of the combined space–time AMR approach in both two space …
A Mass Conserving Mixed Stress Formulation For Stokes Flow With Weakly Imposed Stress Symmetry, Jay Gopalakrishnan, Philip L. Lederer, Joachim Schoeberl
A Mass Conserving Mixed Stress Formulation For Stokes Flow With Weakly Imposed Stress Symmetry, Jay Gopalakrishnan, Philip L. Lederer, Joachim Schoeberl
Mathematics and Statistics Faculty Publications and Presentations
We introduce a new discretization of a mixed formulation of the incompressible Stokes equations that includes symmetric viscous stresses. The method is built upon a mass conserving mixed formulation that we recently studied. The improvement in this work is a new method that directly approximates the viscous fluid stress $\sigma$, enforcing its symmetry weakly. The finite element space in which the stress is approximated consists of matrix-valued functions having continuous “normal-tangential” components across element interfaces. Stability is achieved by adding certain matrix bubbles that were introduced earlier in the literature on finite elements for linear elasticity. Like the earlier work, …