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Title: Meshfree Approximation of Integral Operators

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  1. Los Alamos National Laboratory
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA), Office of Defense Programs (DP) (NA-10)
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Report Number(s):
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Resource Relation:
Conference: UCSD-Sandia Workshop on Meshfree and Related Methods ; 2017-11-14 - 2017-11-15 ; San Diego, California, United States
Country of Publication:
United States

Citation Formats

Dilts, Gary Allen. Meshfree Approximation of Integral Operators. United States: N. p., 2017. Web.
Dilts, Gary Allen. Meshfree Approximation of Integral Operators. United States.
Dilts, Gary Allen. 2017. "Meshfree Approximation of Integral Operators". United States. doi:.
title = {Meshfree Approximation of Integral Operators},
author = {Dilts, Gary Allen},
abstractNote = {},
doi = {},
journal = {},
number = ,
volume = ,
place = {United States},
year = 2017,
month =

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  • Many discretizations of integral equations and compact fixed point problems are collectively compact and strongly convergent in spaces of continuous functions. These properties not only lead to stable and convergent approximations but also can be used in the construction of fast multilevel algorithms. Recently the GMRES algorithm has become a standard coarse mesh solver. The purpose of this paper is to show how the special properties of integral operators and their approximations are reflected in the performance of the GMRES iteration and how these properties can be used to strengthen the norm in which convergence takes place. The authors illustratemore » these ideas with composite Gauss rules for integral equations on the unit interval.« less
  • The Integral Transport Matrix Method (ITMM) has been shown to be an effective method for solving the neutron transport equation in large domains on massively parallel architectures. In the limit of very large number of processors, the speed of the algorithm, and its suitability for unstructured meshes, i.e. other than an ordered Cartesian grid, is limited by the construction of four matrix operators required for obtaining the solution in each sub-domain. The existing algorithm used for construction of these matrix operators, termed the differential mesh sweep, is computationally expensive and was developed for a structured grid. This work proposes themore » use of a new algorithm for construction of these operators based on the construction of a single, fundamental matrix representing the transport of a particle along every possible path throughout the sub-domain mesh. Each of the operators is constructed by multiplying an element of this fundamental matrix by two factors dependent only upon the operator being constructed and on properties of the emitting and incident cells. The ITMM matrix operator construction time for the new algorithm is demonstrated to be shorter than the existing algorithm in all tested cases with both isotropic and anisotropic scattering considered. While also being a more efficient algorithm on a structured Cartesian grid, the new algorithm is promising in its geometric robustness and potential for being applied to an unstructured mesh, with the ultimate goal of application to an unstructured tetrahedral mesh on a massively parallel architecture. (authors)« less
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