We use Newton divided differences for calculation of Greene sums -- the
rational functions determined by linear extensions of partially ordered sets.
Identities for Greene sums generate relations for Newton divided differences
and Arnold differential forms. Also generalizations of the Newton interpolation
series which are indexed by sequences of partially ordered sets are received.
lunes, 30 de noviembre de 2009
Calculus of linear extensions and Newton interpolation. (arXiv:0911.5620v1 [math.CO])
Coarse space over the ages. (arXiv:0911.5725v1 [math.NA])
The objective of this paper is to explain the principles of the design of a
coarse space in a simplified way and by pictures. The focus is on ideas rather
than on a more historically complete presentation. Also, space limitation does
not allow even to mention many important methods and papers that should be
rightfully included.
On Adaptive-Multilevel BDDC. (arXiv:0911.5730v1 [math.NA])
We combine the advantages of the adaptive and multilevel approaches, proposed
previously by the authors, to propose a new method that preserves both,
parallel scalability with increasing number of subdomains and excellent
convergence properties. Performance of the method is illustrated on a
two-dimensional problem of linear elasticity.
Numerical Study of Liquid Crystal Elastomer Using Mixed Finite Element Method. (arXiv:0911.5415v1 [math.NA])
In this paper, we tried to model the elastic behavior of liquid crystal
elastomer using mixed finite element method. We start from an energy functional
which includes Blandon's stored energy of LCE, penalization of change of
directors, and two Lagrangian terms enforcing incompressibility and the unity
of the directors. The resulting Euler-Lagrange equation is a nonlinear equation
of the displacement ${mathbf u}$, the director ${mathbf n}$, the pressure $p$
and the Lagrange multiplier $lambda$. Inf-sup conditions for the
well-posedness of the linearized system were proposed and some are proved. For
those inf-sup conditions that are not easy to prove, we suggest ways to do the
numerical verification (the inf-sup tests). Finally, some numerical results are
presented.
Quasi-Lp norm orthogonal Galerkin expansions in sums of Jacobi polynomials
Abstract In the study of differential equations on [ − 1,1] subject to linear homogeneous boundary conditions of finite order, it is
often expedient to represent the solution in a Galerkin expansion, that is, as a sum of basis functions, each of which satisfies
the given boundary conditions. In order that the functions be maximally distinct, one can use the Gram-Schmidt method to generate
a set orthogonal with respect to a particular weight function. Here we consider all such sets associated with the Jacobi weight
function, w(x) = (1 − x)
α
(1 + x)
β
. However, this procedure is not only cumbersome for sets of large degree, but does not provide any intrinsic means to characterize
the functions that result. We show here that each basis function can be written as the sum of a small number of Jacobi polynomials,
whose coefficients are found by imposing the boundary conditions and orthogonality to the first few basis functions only.
That orthogonality of the entire set follows—a property we term “auto-orthogonality”—is remarkable. Additionally, these basis
functions are shown to behave asymptotically like individual Jacobi polynomials and share many of the latter’s useful properties.
Of particular note is that these basis sets retain the exponential convergence characteristic of Jacobi expansions for expansion
of an arbitrary function satisfying the boundary conditions imposed. Further, the associated error is asymptotically minimized
in an L
p(α) norm given the appropriate choice of α = β. The rich algebraic structure underlying these properties remains partially obscured by the rather difficult form of the
non-standard weighted integrals of Jacobi polynomials upon which our analysis rests. Nevertheless, we are able to prove most
of these results in specific cases and certain of the results in the general case. However a proof that such expansions can
satisfy linear boundary conditions of arbitrary order and form appears extremely difficult.
often expedient to represent the solution in a Galerkin expansion, that is, as a sum of basis functions, each of which satisfies
the given boundary conditions. In order that the functions be maximally distinct, one can use the Gram-Schmidt method to generate
a set orthogonal with respect to a particular weight function. Here we consider all such sets associated with the Jacobi weight
function, w(x) = (1 − x)
α
(1 + x)
β
. However, this procedure is not only cumbersome for sets of large degree, but does not provide any intrinsic means to characterize
the functions that result. We show here that each basis function can be written as the sum of a small number of Jacobi polynomials,
whose coefficients are found by imposing the boundary conditions and orthogonality to the first few basis functions only.
That orthogonality of the entire set follows—a property we term “auto-orthogonality”—is remarkable. Additionally, these basis
functions are shown to behave asymptotically like individual Jacobi polynomials and share many of the latter’s useful properties.
Of particular note is that these basis sets retain the exponential convergence characteristic of Jacobi expansions for expansion
of an arbitrary function satisfying the boundary conditions imposed. Further, the associated error is asymptotically minimized
in an L
p(α) norm given the appropriate choice of α = β. The rich algebraic structure underlying these properties remains partially obscured by the rather difficult form of the
non-standard weighted integrals of Jacobi polynomials upon which our analysis rests. Nevertheless, we are able to prove most
of these results in specific cases and certain of the results in the general case. However a proof that such expansions can
satisfy linear boundary conditions of arbitrary order and form appears extremely difficult.
- Content Type Journal Article
- Category Original Paper
- DOI 10.1007/s11075-009-9353-5
- Authors
- Philip W. Livermore, University of California, San Diego Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography La Jolla CA 92093-0225 USA
- Glenn R. Ierley, University of California, San Diego Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography La Jolla CA 92093-0225 USA
- Journal Numerical Algorithms
- Online ISSN 1572-9265
- Print ISSN 1017-1398
Matroid Polytopes and their Volumes
Abstract We express the matroid polytope P
M
of a matroid M as a signed Minkowski sum of simplices, and obtain a formula for the volume of P
M
. This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian Gr
k,n
. We then derive analogous results for the independent set polytope and the underlying flag matroid polytope of M. Our proofs are based on a natural extension of Postnikov’s theory of generalized permutohedra.
M
of a matroid M as a signed Minkowski sum of simplices, and obtain a formula for the volume of P
M
. This gives a combinatorial expression for the degree of an arbitrary torus orbit closure in the Grassmannian Gr
k,n
. We then derive analogous results for the independent set polytope and the underlying flag matroid polytope of M. Our proofs are based on a natural extension of Postnikov’s theory of generalized permutohedra.
- Content Type Journal Article
- DOI 10.1007/s00454-009-9232-9
- Authors
- Federico Ardila, San Francisco State University San Francisco CA USA
- Carolina Benedetti, York University Toronto Canada
- Jeffrey Doker, University of California, Berkeley Berkeley CA USA
- Journal Discrete and Computational Geometry
- Online ISSN 1432-0444
- Print ISSN 0179-5376
Fast Multiresolution Algorithms and Their Related Variational Problems for Image Denoising
Abstract When considering fast multiresolution techniques for image denoising problems, there are three important aspects. The first
one is the choice of the specific multiresolution, the second one the choice of a proper filter function and the third one
the choice of the thresholding parameter. Starting from the classical one, namely, linear wavelet algorithms with Donoho and
Johnstone’s Soft-thresholding with the universal shrinkage parameter, the first aim of this paper is to improve it in the
three mentioned directions. Thus, a new nonlinear approach is proposed and analyzed. On the other hand, the linear approach
of Donoho and Johnstone is related with a well known variational problem. Our second aim is to find a related variational
problem, more adapted to the denoising problem, for the new approach. We would like to mention that the analysis of theoretical
properties in a nonlinear setting are usually notoriously more difficult. Finally, a comparison with other approaches, including
linear and nonlinear multiresolution schemes, SVD-based schemes and filters with a non-multiresolution nature, is presented.
one is the choice of the specific multiresolution, the second one the choice of a proper filter function and the third one
the choice of the thresholding parameter. Starting from the classical one, namely, linear wavelet algorithms with Donoho and
Johnstone’s Soft-thresholding with the universal shrinkage parameter, the first aim of this paper is to improve it in the
three mentioned directions. Thus, a new nonlinear approach is proposed and analyzed. On the other hand, the linear approach
of Donoho and Johnstone is related with a well known variational problem. Our second aim is to find a related variational
problem, more adapted to the denoising problem, for the new approach. We would like to mention that the analysis of theoretical
properties in a nonlinear setting are usually notoriously more difficult. Finally, a comparison with other approaches, including
linear and nonlinear multiresolution schemes, SVD-based schemes and filters with a non-multiresolution nature, is presented.
- Content Type Journal Article
- DOI 10.1007/s10915-009-9336-7
- Authors
- Sergio Amat, Universidad Politécnica de Cartagena Departamento de Matemática Aplicada y Estadística Cartagena Spain
- Juan Ruiz, Centro de Investigaciones Energéticas Medioambientales y Tecnológicas Madrid Spain
- J. Carlos Trillo, Universidad Politécnica de Cartagena Departamento de Matemática Aplicada y Estadística Cartagena Spain
- Journal Journal of Scientific Computing
- Online ISSN 1573-7691
- Print ISSN 0885-7474
An Iterative Method for Solving Non-Linear Hydromagnetic Equations. (arXiv:0911.5214v1 [math.NA])
We propose an iterative finite element method for solving non-linear
hydromagnetic and steady Euler's equations. Some three-dimensional
computational tests are given to confirm the convergence and the high
efficiency of the method.
Non-convexly constrained linear inverse problems. (arXiv:0911.5098v1 [math.NA])
This paper considers the inversion of ill-posed linear operators. To
regularise the problem the solution is enforced to lie in a non-convex subset.
Theoretical properties for the stable inversion are derived and an iterative
algorithm akin to the projected Landweber algorithm is studied. This work
extends recent progress made on the efficient inversion of finite dimensional
linear systems under a sparsity constraint to the Hilbert space setting and to
more general non-convex constraints.
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