1,058 research outputs found
Higher order weakly over-penalized symmetric interior penalty methods
In this paper we study higher order weakly over-penalized symmetric interior penalty methods for second-order elliptic boundary value problems in two dimensions. We derive hp error estimates in both the energy norm and the norm and present numerical results that corroborate the theoretical results. © 2012 Elsevier B.V. All rights reserved. L
Convergence Analysis of the Lowest Order Weakly Penalized Adaptive Discontinuous Galerkin Methods
In this article, we prove convergence of the weakly penalized adaptive
discontinuous Galerkin methods. Unlike other works, we derive the contraction
property for various discontinuous Galerkin methods only assuming the
stabilizing parameters are large enough to stabilize the method. A central idea
in the analysis is to construct an auxiliary solution from the discontinuous
Galerkin solution by a simple post processing. Based on the auxiliary solution,
we define the adaptive algorithm which guides to the convergence of adaptive
discontinuous Galerkin methods
Robust Estimation of High-Dimensional Mean Regression
Data subject to heavy-tailed errors are commonly encountered in various
scientific fields, especially in the modern era with explosion of massive data.
To address this problem, procedures based on quantile regression and Least
Absolute Deviation (LAD) regression have been devel- oped in recent years.
These methods essentially estimate the conditional median (or quantile)
function. They can be very different from the conditional mean functions when
distributions are asymmetric and heteroscedastic. How can we efficiently
estimate the mean regression functions in ultra-high dimensional setting with
existence of only the second moment? To solve this problem, we propose a
penalized Huber loss with diverging parameter to reduce biases created by the
traditional Huber loss. Such a penalized robust approximate quadratic
(RA-quadratic) loss will be called RA-Lasso. In the ultra-high dimensional
setting, where the dimensionality can grow exponentially with the sample size,
our results reveal that the RA-lasso estimator produces a consistent estimator
at the same rate as the optimal rate under the light-tail situation. We further
study the computational convergence of RA-Lasso and show that the composite
gradient descent algorithm indeed produces a solution that admits the same
optimal rate after sufficient iterations. As a byproduct, we also establish the
concentration inequality for estimat- ing population mean when there exists
only the second moment. We compare RA-Lasso with other regularized robust
estimators based on quantile regression and LAD regression. Extensive
simulation studies demonstrate the satisfactory finite-sample performance of
RA-Lasso
An adaptive discontinuous finite volume method for elliptic problems
AbstractAn adaptive discontinuous finite volume method is developed and analyzed in this paper. We prove that the adaptive procedure achieves guaranteed error reduction in a mesh-dependent energy norm and has a linear convergence rate. Numerical results are also presented to illustrate the theoretical analysis
Discontinuous Galerkin method for the spherically reduced BSSN system with second-order operators
We present a high-order accurate discontinuous Galerkin method for evolving
the spherically-reduced Baumgarte-Shapiro-Shibata-Nakamura (BSSN) system
expressed in terms of second-order spatial operators. Our multi-domain method
achieves global spectral accuracy and long-time stability on short
computational domains. We discuss in detail both our scheme for the BSSN system
and its implementation. After a theoretical and computational verification of
the proposed scheme, we conclude with a brief discussion of issues likely to
arise when one considers the full BSSN system.Comment: 35 pages, 6 figures, 1 table, uses revtex4. Revised in response to
referee's repor
Multiplicative Schwarz methods for discontinuous Galerkin approximations of elliptic problems
In this paper we introduce and analyze some non-overlapping multiplicative Schwarz methods for discontinuous Galerkin (DG) approximations of elliptic problems. The construction of the Schwarz preconditioners is presented in a unified framework for a wide class of DG methods. For symmetric DG approximations we provide optimal convergence bounds for the corresponding error propagation operator, and we show that the resulting methods can be accelerated by using suitable Krylov space solvers. A discussion on the issue of preconditioning non-symmetric DG approximations of elliptic problems is also included. Extensive numerical experiments to confirm the theoretical results and to assess the robustness and the efficiency of the proposed preconditioners are provided. © 2008 EDP Sciences SMAI
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