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Multilevel acceleration of scattering-source iterations with application to electron transport
Clif Drumm,Wesley Fan 한국원자력학회 2017 Nuclear Engineering and Technology Vol.49 No.6
Acceleration/preconditioning strategies available in the SCEPTRE radiation transport code are described. A flexible transport synthetic acceleration (TSA) algorithm that uses a low-order discrete-ordinates (SN) or spherical-harmonics (PN) solve to accelerate convergence of a high-order SN source-iteration (SI) solve is described. Convergence of the low-order solves can be further accelerated by applying off-the-shelf incomplete-factorization or algebraic-multigrid methods. Also available is an algorithm that uses a generalized minimum residual (GMRES) iterative method rather than SI for convergence, using a parallel sweep-based solver to build up a Krylov subspace. TSA has been applied as a preconditioner to accelerate the convergence of the GMRES iterations. The methods are applied to several problems involving electron transport and problems with artificial cross sections with large scattering ratios. These methods were compared and evaluated by considering material discontinuities and scattering anisotropy. Observed accelerations obtained are highly problem dependent, but speedup factors around 10 have been observed in typical applications.
Bisectors determining unique pairs of points in the bidisk
Charette, Virginie,Drumm, Todd A.,Kim, Youngju World Scientific 2018 International Journal of Mathematics Vol.29 No.3
<P>Bisectors are equidistant hypersurfaces between two points and are basic objects in a metric geometry. They play an important part in understanding the action of subgroups of isometries on a metric space. In many metric geometries (spherical, Euclidean, hyperbolic, complex hyperbolic, to name a few) bisectors do not uniquely determine a pair of points, in the following sense: completely different sets of points share a common bisector. The above examples of this non-uniqueness are all rank <TEX>$ 1$</TEX> symmetric spaces. However, generically, bisectors in the usual <TEX>$ L^{2}$</TEX> metric are such for a unique pair of points in the rank <TEX>$ 2$</TEX> geometry <TEX>$ \mathbf{H}^{2} \times \mathbf{H}^{2}$</TEX>. This result indicates the striking assertion that non-uniqueness of bisectors holds for “most” geometries.</P>