121,364 research outputs found
Analysis of the convergence rate for the cyclic projection algorithm applied to basic semi-algebraic convex sets
In this paper, we study the rate of convergence of the cyclic projection
algorithm applied to finitely many basic semi-algebraic convex sets. We
establish an explicit convergence rate estimate which relies on the maximum
degree of the polynomials that generate the basic semi-algebraic convex sets
and the dimension of the underlying space. We achieve our results by exploiting
the algebraic structure of the basic semi-algebraic convex sets.Comment: 35 pages, revision incorporating referees' comment
An efficient algorithm for computing the Baker-Campbell-Hausdorff series and some of its applications
We provide a new algorithm for generating the Baker--Campbell--Hausdorff
(BCH) series Z = \log(\e^X \e^Y) in an arbitrary generalized Hall basis of
the free Lie algebra generated by and . It is based
on the close relationship of with a Lie algebraic structure
of labeled rooted trees. With this algorithm, the computation of the BCH series
up to degree 20 (111013 independent elements in ) takes less
than 15 minutes on a personal computer and requires 1.5 GBytes of memory. We
also address the issue of the convergence of the series, providing an optimal
convergence domain when and are real or complex matrices.Comment: 30 page
Algebraic chromatic homotopy theory for -comodules
In this paper, we study the global structure of an algebraic avatar of the
derived category of ind-coherent sheaves on the moduli stack of formal groups.
In analogy with the stable homotopy category, we prove a version of the
nilpotence theorem as well as the chromatic convergence theorem, and construct
a generalized chromatic spectral sequence. Furthermore, we discuss analogs of
the telescope conjecture and chromatic splitting conjecture in this setting,
using the local duality techniques established earlier in joint work with
Valenzuela.Comment: All comments welcom
Convergence of homogeneous manifolds
We study in this paper three natural notions of convergence of homogeneous
manifolds, namely infinitesimal, local and pointed, and their relationship with
a fourth one, which only takes into account the underlying algebraic structure
of the homogeneous manifold and is indeed much more tractable. Along the way,
we introduce a subset of the variety of Lie algebras which parameterizes the
space of all n-dimensional simply connected homogeneous spaces with
q-dimensional isotropy, providing a framework which is very advantageous to
approach variational problems for curvature functionals as well as geometric
evolution equations on homogeneous manifolds.Comment: 26 pages, final version to appear in J. London Math. So
Performance of algebraic multigrid methods for non-symmetric matrices arising in particle methods
Large linear systems with sparse, non-symmetric matrices arise in the
modeling of Markov chains or in the discretization of convection-diffusion
problems. Due to their potential to solve sparse linear systems with an effort
that is linear in the number of unknowns, algebraic multigrid (AMG) methods are
of fundamental interest for such systems. For symmetric positive definite
matrices, fundamental theoretical convergence results are established, and
efficient AMG solvers have been developed. In contrast, for non-symmetric
matrices, theoretical convergence results have been provided only recently. A
property that is sufficient for convergence is that the matrix be an M-matrix.
In this paper, we present how the simulation of incompressible fluid flows with
particle methods leads to large linear systems with sparse, non-symmetric
matrices. In each time step, the Poisson equation is approximated by meshfree
finite differences. While traditional least squares approaches do not guarantee
an M-matrix structure, an approach based on linear optimization yields
optimally sparse M-matrices. For both types of discretization approaches, we
investigate the performance of a classical AMG method, as well as an AMLI type
method. While in the considered test problems, the M-matrix structure turns out
not to be necessary for the convergence of AMG, problems can occur when it is
violated. In addition, the matrices obtained by the linear optimization
approach result in fast solution times due to their optimal sparsity.Comment: 16 pages, 7 figure
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