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On the Hierarchical Preconditioning of the PMCHWT Integral Equation on Simply and Multiply Connected Geometries
We present a hierarchical basis preconditioning strategy for the
Poggio-Miller-Chang-Harrington-Wu-Tsai (PMCHWT) integral equation considering
both simply and multiply connected geometries.To this end, we first consider
the direct application of hierarchical basis preconditioners, developed for the
Electric Field Integral Equation (EFIE), to the PMCHWT. It is notably found
that, whereas for the EFIE a diagonal preconditioner can be used for obtaining
the hierarchical basis scaling factors, this strategy is catastrophic in the
case of the PMCHWT since it leads to a severly ill-conditioned PMCHWT system in
the case of multiply connected geometries. We then proceed to a theoretical
analysis of the effect of hierarchical bases on the PMCHWT operator for which
we obtain the correct scaling factors and a provably effective preconditioner
for both low frequencies and mesh refinements. Numerical results will
corroborate the theory and show the effectiveness of our approach
The dissimilarity map and representation theory of
We give another proof that -dissimilarity vectors of weighted trees are
points on the tropical Grassmanian, as conjectured by Cools, and proved by
Giraldo in response to a question of Sturmfels and Pachter. We accomplish this
by relating -dissimilarity vectors to the representation theory of Comment: 11 pages, 8 figure
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