570 research outputs found
On some symmetric multidimensional continued fraction algorithms
We compute explicitly the density of the invariant measure for the Reverse
algorithm which is absolutely continuous with respect to Lebesgue measure,
using a method proposed by Arnoux and Nogueira. We also apply the same method
on the unsorted version of Brun algorithm and Cassaigne algorithm. We
illustrate some experimentations on the domain of the natural extension of
those algorithms. For some other algorithms, which are known to have a unique
invariant measure absolutely continuous with respect to Lebesgue measure, the
invariant domain found by this method seems to have a fractal boundary, and it
is unclear that it is of positive measure.Comment: Version 1: 22 pages, 12 figures. Version 2: new section on Cassaigne
algorithm, 25 pages, 15 figures. Version 3: corrections during review proces
Efficient 1-Laplacian Solvers for Well-Shaped Simplicial Complexes: Beyond Betti Numbers and Collapsing Sequences
We present efficient algorithms for solving systems of linear equations in 1-Laplacians of well-shaped simplicial complexes. 1-Laplacians, or higher-dimensional Laplacians, generalize graph Laplacians to higher-dimensional simplicial complexes and play a key role in computational topology and topological data analysis. Previously, nearly-linear time solvers were developed for simplicial complexes with known collapsing sequences and bounded Betti numbers, such as those triangulating a three-ball in R3 (Cohen, Fasy, Miller, Nayyeri, Peng, and Walkington [SODA’2014], Black, Maxwell, Nayyeri, and Winkelman [SODA’2022], Black and Nayyeri [ICALP’2022]). Furthermore, Nested Dissection provides quadratic time solvers for more general systems with nonzero structures representing well-shaped simplicial complexes embedded in R3. We generalize the specialized solvers for 1-Laplacians to simplicial complexes with additional geometric structures but without collapsing sequences and bounded Betti numbers, and we improve the runtime of Nested Dissection. We focus on simplicial complexes that meet two conditions: (1) each individual simplex has a bounded aspect ratio, and (2) they can be divided into “disjoint” and balanced regions with well-shaped interiors and boundaries. Our solvers draw inspiration from the Incomplete Nested Dissection for stiffness matrices of well-shaped trusses (Kyng, Peng, Schwieterman, and Zhang [STOC’2018]).ISSN:1868-896
Efficient -Laplacian Solvers for Well-Shaped Simplicial Complexes: Beyond Betti Numbers and Collapsing Sequences
We present efficient algorithms for approximately solving systems of linear
equations in -Laplacians of well-shaped simplicial complexes up to high
precision. -Laplacians, or higher-dimensional Laplacians, generalize graph
Laplacians to higher-dimensional simplicial complexes and play a key role in
computational topology and topological data analysis. Previously, nearly-linear
time approximate solvers were developed for simplicial complexes with known
collapsing sequences and bounded Betti numbers, such as those triangulating a
three-ball in (Cohen, Fasy, Miller, Nayyeri, Peng, and
Walkington [SODA'2014], Black, Maxwell, Nayyeri, and Winkelman [SODA'2022],
Black and Nayyeri [ICALP'2022]). Furthermore, Nested Dissection provides
quadratic time exact solvers for more general systems with nonzero structures
representing well-shaped simplicial complexes embedded in .
We generalize the specialized solvers for -Laplacians to simplicial
complexes with additional geometric structures but without collapsing sequences
and bounded Betti numbers, and we improve the runtime of Nested Dissection. We
focus on simplicial complexes that meet two conditions: (1) each individual
simplex has a bounded aspect ratio, and (2) they can be divided into "disjoint"
and balanced regions with well-shaped interiors and boundaries. Our solvers
draw inspiration from the Incomplete Nested Dissection for stiffness matrices
of well-shaped trusses (Kyng, Peng, Schwieterman, and Zhang [STOC'2018]).Comment: 45 pages, 3 figures, ESA 202
Preconditioning of weighted H(div)-norm and applications to numerical simulation of highly heterogeneous media
In this paper we propose and analyze a preconditioner for a system arising
from a finite element approximation of second order elliptic problems
describing processes in highly het- erogeneous media. Our approach uses the
technique of multilevel methods and the recently proposed preconditioner based
on additive Schur complement approximation by J. Kraus (see [8]). The main
results are the design and a theoretical and numerical justification of an
iterative method for such problems that is robust with respect to the contrast
of the media, defined as the ratio between the maximum and minimum values of
the coefficient (related to the permeability/conductivity).Comment: 28 page
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