455 research outputs found
Absolute value preconditioning for symmetric indefinite linear systems
We introduce a novel strategy for constructing symmetric positive definite
(SPD) preconditioners for linear systems with symmetric indefinite matrices.
The strategy, called absolute value preconditioning, is motivated by the
observation that the preconditioned minimal residual method with the inverse of
the absolute value of the matrix as a preconditioner converges to the exact
solution of the system in at most two steps. Neither the exact absolute value
of the matrix nor its exact inverse are computationally feasible to construct
in general. However, we provide a practical example of an SPD preconditioner
that is based on the suggested approach. In this example we consider a model
problem with a shifted discrete negative Laplacian, and suggest a geometric
multigrid (MG) preconditioner, where the inverse of the matrix absolute value
appears only on the coarse grid, while operations on finer grids are based on
the Laplacian. Our numerical tests demonstrate practical effectiveness of the
new MG preconditioner, which leads to a robust iterative scheme with minimalist
memory requirements
High-temperature environments of human evolution in East Africa based on bond ordering in paleosol carbonates
Many important hominid-bearing fossil localities in East Africa are in regions that are extremely hot and dry. Although humans are well adapted to such conditions, it has been inferred that East African environments were cooler or more wooded during the Pliocene and Pleistocene when this region was a central stage of human evolution. Here we show that the Turkana Basin, Kenyaâtoday one of the hottest places on Earthâhas been continually hot during the past 4 million years. The distribution of ^(13)C-^(18)O bonds in paleosol carbonates indicates that soil temperatures during periods of carbonate formation were typically above 30 °C and often in excess of 35 °C. Similar soil temperatures are observed today in the Turkana Basin and reflect high air temperatures combined with solar heating of the soil surface. These results are specific to periods of soil carbonate formation, and we suggest that such periods composed a large fraction of integrated time in the Turkana Basin. If correct, this interpretation has implications for human thermophysiology and implies a long-standing human association with marginal environments
Galerkin FEM for fractional order parabolic equations with initial data in
We investigate semi-discrete numerical schemes based on the standard Galerkin
and lumped mass Galerkin finite element methods for an initial-boundary value
problem for homogeneous fractional diffusion problems with non-smooth initial
data. We assume that , is a convex
polygonal (polyhedral) domain. We theoretically justify optimal order error
estimates in - and -norms for initial data in . We confirm our theoretical findings with a number of numerical tests
that include initial data being a Dirac -function supported on a
-dimensional manifold.Comment: 13 pages, 3 figure
On thin plate spline interpolation
We present a simple, PDE-based proof of the result [M. Johnson, 2001] that
the error estimates of [J. Duchon, 1978] for thin plate spline interpolation
can be improved by . We illustrate that -matrix
techniques can successfully be employed to solve very large thin plate spline
interpolation problem
Nematic liquid crystal alignment on chemical patterns
Patterned Self-Assembled Monolayers (SAMs) promoting both homeotropic and planar degenerate alignment of 6CB and 9CB in their nematic phase, were created using microcontact printing of functionalised organothiols on gold films. The effects of a range of different pattern geometries and sizes were investigated, including stripes, circles and checkerboards. EvanescentWave Ellipsometry was used to study the orientation of the liquid crystal (LC) on these patterned surfaces during the isotropic-nematic phase transition. Pretransitional growth of a homeotropic layer was observed on 1 Âčm homeotropic aligning stripes, followed by a homeotropic mono-domain state prior to the
bulk phase transition. Accompanying Monte-Carlo simulations of LCs aligned on nano-patterned surfaces were also performed. These simulations also showed the presence of the homeotropic mono-domain state prior to the transition.</p
Smoothness-Increasing Accuracy-Conserving (SIAC) filtering and quasi interpolation: A unified view
Filtering plays a crucial role in postprocessing and analyzing data in scientific and engineering applications. Various application-specific filtering schemes have been proposed based on particular design criteria. In this paper, we focus on establishing the theoretical connection between quasi-interpolation and a class of kernels (based on B-splines) that are specifically designed for the postprocessing of the discontinuous Galerkin (DG) method called Smoothness-Increasing Accuracy-Conserving (SIAC) filtering. SIAC filtering, as the name suggests, aims to increase the smoothness of the DG approximation while conserving the inherent accuracy of the DG solution (superconvergence). Superconvergence properties of SIAC filtering has been studied in the literature. In this paper, we present the theoretical results that establish the connection between SIAC filtering to long-standing concepts in approximation theory such as quasi-interpolation and polynomial reproduction. This connection bridges the gap between the two related disciplines and provides a decisive advancement in designing new filters and mathematical analysis of their properties. In particular, we derive a closed formulation for convolution of SIAC kernels with polynomials. We also compare and contrast cardinal spline functions as an example of filters designed for image processing applications with SIAC filters of the same order, and study their properties
Hexagonal Smoothness-Increasing Accuracy-Conserving Filtering
Discontinuous Galerkin (DG) methods are a popular class of numerical techniques to solve partial differential equations due to their higher order of accuracy. However, the inter-element discontinuity of a DG solution hinders its utility in various applications, including visualization and feature extraction. This shortcoming can be alleviated by postprocessing of DG solutions to increase the inter-element smoothness. A class of postprocessing techniques proposed to increase the inter-element smoothness is SIAC filtering. In addition to increasing the inter-element continuity, SIAC filtering also raises the convergence rate from order k+1k+1 to order 2k+12k+1 . Since the introduction of SIAC filtering for univariate hyperbolic equations by Cockburn et al. (Math Comput 72(242):577â606, 2003), many generalizations of SIAC filtering have been proposed. Recently, the idea of dimensionality reduction through rotation has been the focus of studies in which a univariate SIAC kernel has been used to postprocess a two-dimensional DG solution (Docampo-SĂĄnchez et al. in Multi-dimensional filtering: reducing the dimension through rotation, 2016. arXiv preprint arXiv:1610.02317). However, the scope of theoretical development of multidimensional SIAC filters has never gone beyond the usage of tensor product multidimensional B-splines or the reduction of the filter dimension. In this paper, we define a new SIAC filter called hexagonal SIAC (HSIAC) that uses a nonseparable class of two-dimensional spline functions called hex splines. In addition to relaxing the separability assumption, the proposed HSIAC filter provides more symmetry to its tensor-product counterpart. We prove that the superconvergence property holds for a specific class of structured triangular meshes using HSIAC filtering and provide numerical results to demonstrate and validate our theoretical results
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