25,231 research outputs found
Efficient and Robust Weighted Least-Squares Cell-Average Gradient Construction Methods for the Simulation of Scramjet Flows
The ability to solve the equations governing the hypersonic turbulent flow of a real gas on unstructured grids using a spatially-elliptic, 2nd-order accurate, cell-centered, finite-volume method has been recently implemented in the VULCAN-CFD code. The construction of cell-average gradients using a weighted linear least-squares method and the use of these gradients in the construction of the inviscid fluxes is the focus of this paper. A comparison of least-squares stencil construction methodologies is presented and approaches designed to minimize the number of cells used to augment/stabilize the least-squares stencil while preserving accuracy are explored. Due to our interest in hypersonic flow, a robust multidimensional cell-average gradient limiter procedure that is consistent with the stencil used to construct the cellaverage gradients is described. Canonical problems are computed to illustrate the challenges and investigate the accuracy, robustness and convergence behavior of the cell-average gradient methods on unstructured cell-centered finite-volume grids. Finally, thermally perfect, chemically frozen, Mach 7.8 turbulent flow of air through a scramjet engine flowpath is computed and compared with experimental data to demonstrate the robustness, accuracy and convergence behavior of the preferred gradient method for a realistic 3-D geometry on a non-hex-dominant grid
Rational minimax approximation via adaptive barycentric representations
Computing rational minimax approximations can be very challenging when there
are singularities on or near the interval of approximation - precisely the case
where rational functions outperform polynomials by a landslide. We show that
far more robust algorithms than previously available can be developed by making
use of rational barycentric representations whose support points are chosen in
an adaptive fashion as the approximant is computed. Three variants of this
barycentric strategy are all shown to be powerful: (1) a classical Remez
algorithm, (2) a "AAA-Lawson" method of iteratively reweighted least-squares,
and (3) a differential correction algorithm. Our preferred combination,
implemented in the Chebfun MINIMAX code, is to use (2) in an initial phase and
then switch to (1) for generically quadratic convergence. By such methods we
can calculate approximations up to type (80, 80) of on in
standard 16-digit floating point arithmetic, a problem for which Varga, Ruttan,
and Carpenter required 200-digit extended precision.Comment: 29 pages, 11 figure
Proximal Multitask Learning over Networks with Sparsity-inducing Coregularization
In this work, we consider multitask learning problems where clusters of nodes
are interested in estimating their own parameter vector. Cooperation among
clusters is beneficial when the optimal models of adjacent clusters have a good
number of similar entries. We propose a fully distributed algorithm for solving
this problem. The approach relies on minimizing a global mean-square error
criterion regularized by non-differentiable terms to promote cooperation among
neighboring clusters. A general diffusion forward-backward splitting strategy
is introduced. Then, it is specialized to the case of sparsity promoting
regularizers. A closed-form expression for the proximal operator of a weighted
sum of -norms is derived to achieve higher efficiency. We also provide
conditions on the step-sizes that ensure convergence of the algorithm in the
mean and mean-square error sense. Simulations are conducted to illustrate the
effectiveness of the strategy
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