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.
domingo, 29 de noviembre de 2009
A new efficient technique for finding the solution of initial-value problems using He's variational iteration method
The well-known He's variational iteration method has been implemented by researchers for solving various kinds of problems. Although it is a very efficient method for solving problems, the approximation is not accurate in large regions. In this work a new technique based on the successive use of the He's variational iteration method in smaller regions has been presented. The new method has been named as local He's variational iteration method. By applying the new technique, an accurate approximation provided in large domains in a shorter time and less computations. Copyright © 2009 John Wiley & Sons, Ltd.
domingo, 22 de noviembre de 2009
Coloring plane graphs with independent crossings
We show that every plane graph with maximum face size four in which all faces of size four are vertex-disjoint is cyclically 5-colorable. This answers a question of Albertson whether graphs drawn in the plane with all crossings independent are 5-colorable. © 2009 Wiley Periodicals, Inc. J Graph Theory
Balloons, cut-edges, matchings, and total domination in regular graphs of odd degree
A balloon in a graph G is a maximal 2-edge-connected subgraph incident to exactly one cut-edge of G. Let b(G) be the number of balloons, let c(G) be the number of cut-edges, and let [alpha][prime](G) be the maximum size of a matching. Let be the family of connected (2r+1)-regular graphs with n vertices, and let . For , we prove the sharp inequalities c(G)[les][r(n-2)-2]/(2r2+2r-1)-1 and [alpha][prime](G)[ges]n/2-rb/(2r+1). Using b[les][(2r-1)n+2]/(4r2+4r-2), we obtain a simple proof of the bound proved by Henning and Yeo. For each of these bounds and each r, the approach using balloons allows us to determine the infinite family where equality holds. For the total domination number [gamma]t(G) of a cubic graph, we prove [gamma]t(G)[les]n/2-b(G)/2 (except that [gamma]t(G) may be n/2-1 when b(G)=3 and the balloons cover all but one vertex). With [alpha][prime](G)[ges]n/2-b(G)/3 for cubic graphs, this improves the known inequality [gamma]t(G)[les][alpha][prime](G). © 2009 Wiley Periodicals, Inc. J Graph Theory
Choosability of toroidal graphs without short cycles
Let G be a toroidal graph without cycles of a fixed length k, and [chi]l(G) the list chromatic number of G. We establish tight upper bounds of [chi]l(G) for the following values of k: © 2009 Wiley Periodicals, Inc. J Graph Theory
The geometry of t-spreads in k-walk-regular graphs
A graph is walk-regular if the number of closed walks of length [ell] rooted at a given vertex is a constant through all the vertices for all [ell]. For a walk-regular graph G with d+1 different eigenvalues and spectrally maximum diameter D=d, we study the geometry of its d-spreads, that is, the sets of vertices which are mutually at distance d. When these vertices are projected onto an eigenspace of its adjacency matrix, we show that they form a simplex (or tetrahedron in a three-dimensional case) and we compute its parameters. Moreover, the results are generalized to the case of k-walk-regular graphs, a family which includes both walk-regular and distance-regular graphs, and their t-spreads or vertices at distance t from each other. © 2009 Wiley Periodicals, Inc. J Graph Theory
On Ramsey-type positional games
Beck introduced the concept of Ramsey games by studying the game versions of Ramsey and van der Waerden theorems. We contribute to this topic by investigating games corresponding to structural extensions of Ramsey and van der Waerden theorems - the theorem of Brauer, structural and restricted Ramsey theorems. © 2009 Wiley Periodicals, Inc. J Graph Theory
On recognizing graphs by numbers of homomorphisms
Let hom (G, H) be the number of homomorphisms from a graph G to a graph H. A well-known result of Lovász states that the function hom (·, H) from all graphs uniquely determines the graph H up to isomorphism. We study this function restricted to smaller classes of graphs. We show that several natural classes (2-degenerate graphs and graphs homomorphic to an arbitrary non-bipartite graph) are sufficient to recognize all graphs, and provide a description of graph properties that are recognizable by graphs with bounded tree-width. © 2009 Wiley Periodicals, Inc. J Graph Theory
Brown, IBM Unveil Multimillion-Dollar Supercomputer
Brown University and IBM today announced the opening of a multimillion-dollar supercomputer at Brown's Center for Computation and Visualization.
Preservation and destruction of periodic orbits by symplectic integrators
Abstract We investigate what happens to periodic orbits and lower-dimensional tori of Hamiltonian systems under discretisation by a
symplectic one-step method where the system may have more than one degree of freedom. We use an embedding of a symplectic
map in a quasi-periodic non-autonomous flow and a KAM result of Jorba and Villaneuva (J Nonlinear Sci 7:427–473, 1997) to show that periodic orbits persist in the new flow, but with slightly perturbed period and an additional degree of freedom
when the map is non-resonant with the periodic orbit. The same result holds for lower-dimensional tori with more degrees of
freedom. Numerical experiments with the two degree of freedom Hénon–Heiles system are used to show that in the case where
the method is resonant with the periodic orbit, the orbit is destroyed and replaced by two invariant sets of periodic points—analogous
to what is understood for one degree of freedom systems.
symplectic one-step method where the system may have more than one degree of freedom. We use an embedding of a symplectic
map in a quasi-periodic non-autonomous flow and a KAM result of Jorba and Villaneuva (J Nonlinear Sci 7:427–473, 1997) to show that periodic orbits persist in the new flow, but with slightly perturbed period and an additional degree of freedom
when the map is non-resonant with the periodic orbit. The same result holds for lower-dimensional tori with more degrees of
freedom. Numerical experiments with the two degree of freedom Hénon–Heiles system are used to show that in the case where
the method is resonant with the periodic orbit, the orbit is destroyed and replaced by two invariant sets of periodic points—analogous
to what is understood for one degree of freedom systems.
- Content Type Journal Article
- Category Original Paper
- DOI 10.1007/s11075-009-9352-6
- Authors
- Robert I. McLachlan, Massey University I.F.S. Palmerston North New Zealand
- Dion R. J. O’Neale, Massey University I.F.S. Palmerston North New Zealand
- Journal Numerical Algorithms
- Online ISSN 1572-9265
- Print ISSN 1017-1398
sábado, 21 de noviembre de 2009
Erratum to: MIR closures of polyhedral sets
Erratum to: MIR closures of polyhedral sets
- Content Type Journal Article
- Category Erratum
- DOI 10.1007/s10107-009-0328-z
- Authors
- Sanjeeb Dash, IBM, T.J. Watson Research Center P.O. Box 218 Yorktown Heights NY 10598 USA
- Oktay Günlük, IBM, T.J. Watson Research Center P.O. Box 218 Yorktown Heights NY 10598 USA
- Andrea Lodi, DEIS, University of Bologna viale Risorgimento 2 40136 Bologna Italy
- Journal Mathematical Programming
- Online ISSN 1436-4646
- Print ISSN 0025-5610
Numerical Studies of Adaptive Finite Element Methods for Two Dimensional Convection-Dominated Problems
Abstract In this paper, we study the stability and accuracy of adaptive finite element methods for the convection-dominated convection-diffusion-reaction
problem in the two-dimension space. Through various numerical examples on a type of layer-adapted grids (Shishkin grids),
we show that the mesh adaptivity driven by accuracy alone cannot stabilize the scheme in all cases. Furthermore the numerical
approximation is sensitive to the symmetry of the grid in the region where the solution is smooth. On the basis of these two
observations, we develop a multilevel-homotopic-adaptive finite element method (MHAFEM) by combining streamline diffusion
finite element method, anisotropic mesh adaptation, and the homotopy of the diffusion coefficient. We use numerical experiments
to demonstrate that MHAFEM can efficiently capture boundary or interior layers and produce accurate solutions.
problem in the two-dimension space. Through various numerical examples on a type of layer-adapted grids (Shishkin grids),
we show that the mesh adaptivity driven by accuracy alone cannot stabilize the scheme in all cases. Furthermore the numerical
approximation is sensitive to the symmetry of the grid in the region where the solution is smooth. On the basis of these two
observations, we develop a multilevel-homotopic-adaptive finite element method (MHAFEM) by combining streamline diffusion
finite element method, anisotropic mesh adaptation, and the homotopy of the diffusion coefficient. We use numerical experiments
to demonstrate that MHAFEM can efficiently capture boundary or interior layers and produce accurate solutions.
- Content Type Journal Article
- DOI 10.1007/s10915-009-9337-6
- Authors
- Pengtao Sun, University of Nevada, Las Vegas Department of Mathematical Sciences 4505 Maryland Parkway Las Vegas NV 89154 USA
- Long Chen, University of California, Irvine Department of Mathematics Irvine CA 92697 USA
- Jinchao Xu, Pennsylvania State University Department of Mathematics University Park PA 16802 USA
- Journal Journal of Scientific Computing
- Online ISSN 1573-7691
- Print ISSN 0885-7474
Gaussian quadrature rules using function derivatives
For finite positive Borel measures supported on the real line we consider a new type of quadrature rule with maximal algebraic degree of exactness that involves function derivatives. We prove the existence of such quadrature rules and describe their basic properties. We also give an application of these quadrature rules to the solution of a Cauchy problem without solving it directly. Numerical examples are included as well.
viernes, 20 de noviembre de 2009
Numerical integration of an elastic-viscoplastic constitutive model for dry metamorphosed snow
A constitutive model for dry metamorphosed snow is proposed, within the framework of elasto-viscoplasticity, which is able to reproduce the most relevant features of the macroscopic behaviour of snow, particularly its time and rate dependency. The basic ideas for modelling stem from the conceptual forms proposed for bonded geomaterials, such as cemented soils or soft rocks. The high viscosity of snow is accounted for by adopting an overstress approach, suitably modified. An evolution law for the curvature-driven process of sintering, by which intergranular ice necks form and grow, is established. The system of constitutive equations is then numerically integrated via a fully implicit time stepping scheme. Selected results from finite element simulations of laboratory tests, available in the literature, are presented. Copyright © 2009 John Wiley & Sons, Ltd.
On explicit two-derivative Runge-Kutta methods
Abstract The theory of Runge-Kutta methods for problems of the form y′ = f(y) is extended to include the second derivative y′′ = g(y): = f′(y)f(y). We present an approach to the order conditions based on Butcher’s algebraic theory of trees (Butcher, Math Comp 26:79–106,
1972), and derive methods that take advantage of cheap computations of the second derivatives. Only explicit methods are considered
here where attention is given to the construction of methods that involve one evaluation of f and many evaluations of g per step. Methods with stages up to five and of order up to seven including some embedded pairs are presented. The first
part of the paper discusses a theoretical formulation used for the derivation of these methods which are also of wider applicability.
The second part presents experimental results for non-stiff and mildly stiff problems. The methods include those with the
computation of one second derivative (plus many first derivatives) per step, and embedded methods for changing stepsize as
well as those involving one first derivative (plus many second derivatives) per step. The experiments have been performed
on standard problems and comparisons made with some standard explicit Runge-Kutta methods.
1972), and derive methods that take advantage of cheap computations of the second derivatives. Only explicit methods are considered
here where attention is given to the construction of methods that involve one evaluation of f and many evaluations of g per step. Methods with stages up to five and of order up to seven including some embedded pairs are presented. The first
part of the paper discusses a theoretical formulation used for the derivation of these methods which are also of wider applicability.
The second part presents experimental results for non-stiff and mildly stiff problems. The methods include those with the
computation of one second derivative (plus many first derivatives) per step, and embedded methods for changing stepsize as
well as those involving one first derivative (plus many second derivatives) per step. The experiments have been performed
on standard problems and comparisons made with some standard explicit Runge-Kutta methods.
- Content Type Journal Article
- Category Original Paper
- DOI 10.1007/s11075-009-9349-1
- Authors
- Robert P. K. Chan, University of Auckland Department of Mathematics Private Bag 92019 Auckland 1142 New Zealand
- Angela Y. J. Tsai, University of Auckland Department of Mathematics Private Bag 92019 Auckland 1142 New Zealand
- Journal Numerical Algorithms
- Online ISSN 1572-9265
- Print ISSN 1017-1398
A stabilized Lagrange multiplier method for the finite element approximation of contact problems in elastostatics
Abstract In this work we consider a stabilized Lagrange (or Kuhn–Tucker) multiplier method in order to approximate the unilateral contact
model in linear elastostatics. The particularity of the method is that no discrete inf-sup condition is needed in the convergence
analysis. We propose three approximations of the contact conditions well adapted to this method and we study the convergence
of the discrete solutions. Several numerical examples in two and three space dimensions illustrate the theoretical results
and show the capabilities of the method.
model in linear elastostatics. The particularity of the method is that no discrete inf-sup condition is needed in the convergence
analysis. We propose three approximations of the contact conditions well adapted to this method and we study the convergence
of the discrete solutions. Several numerical examples in two and three space dimensions illustrate the theoretical results
and show the capabilities of the method.
- Content Type Journal Article
- DOI 10.1007/s00211-009-0273-z
- Authors
- Patrick Hild, Université de Franche-Comté Laboratoire de Mathématiques de Besançon, CNRS UMR 6623 16 route de Gray 25030 Besançon Cedex France
- Yves Renard, Université de Lyon, CNRS, INSA-Lyon, ICJ UMR 5208, LaMCoS UMR 5259 69621 Villeurbanne France
- Journal Numerische Mathematik
- Online ISSN 0945-3245
- Print ISSN 0029-599X
Geometry of Configuration Spaces of Tensegrities
Abstract Consider a graph G with n vertices. In this paper we study geometric conditions for an n-tuple of points in ℝ
d
to admit a nonzero self-stress with underlying graph G. We introduce and investigate a natural stratification, depending on G, of the configuration space of all n-tuples in ℝ
d
. In particular we find surgeries on graphs that give relations between different strata. Further we discuss questions related
to geometric conditions defining the strata for plane tensegrities. We conclude the paper with particular examples of strata
for tensegrities in the plane with a small number of vertices.
d
to admit a nonzero self-stress with underlying graph G. We introduce and investigate a natural stratification, depending on G, of the configuration space of all n-tuples in ℝ
d
. In particular we find surgeries on graphs that give relations between different strata. Further we discuss questions related
to geometric conditions defining the strata for plane tensegrities. We conclude the paper with particular examples of strata
for tensegrities in the plane with a small number of vertices.
- Content Type Journal Article
- DOI 10.1007/s00454-009-9229-4
- Authors
- Franck Doray, Universiteit Leiden Mathematisch Instituut P.O. Box 9512 2300 RA Leiden The Netherlands
- Oleg Karpenkov, Universiteit Leiden Mathematisch Instituut P.O. Box 9512 2300 RA Leiden The Netherlands
- Jan Schepers, Universiteit Leiden Mathematisch Instituut P.O. Box 9512 2300 RA Leiden The Netherlands
- Journal Discrete and Computational Geometry
- Online ISSN 1432-0444
- Print ISSN 0179-5376
lunes, 16 de noviembre de 2009
Optimal Quadrature Formulas with Positive Coefficients in $L_2^{(m)}(0,1)$ Space. (arXiv:0911.2896v1 [math.NA])
In the Sobolev space $L_2^{(m)}(0,1)$ optimal quadrature formulas with the
nodes (1.5) are investigated. For optimal coefficients explicit form are
obtained and norm of the error functional is calculated. In particular, by
choosing parameter $eta_0$ in (1.5) the optimal quadrature formulas with
positive coefficients are obtained and compared with well known optimal
formulas.
miércoles, 11 de noviembre de 2009
The Penalized Lebesgue Constant for Surface Spline Interpolation. (arXiv:0911.1815v1 [math.CA])
Problems involving approximation from scattered data where data is arranged
quasi-uniformly have been treated by RBF methods for decades. Treating data
with spatially varying density has not been investigated with the same
intensity, and is far less well understood. In this article we consider the
stability of surface spline interpolation (a popular type of RBF interpolation)
for data with nonuniform arrangements. Using techniques similar to those
recently employed by Hangelbroek, Narcowich and Ward to demonstrate the
stability of interpolation from quasi-uniform data on manifolds, we show that
surface spline interpolation on R^d is stable, but in a stronger, local sense.
We also obtain pointwise estimates showing that the Lagrange function decays
very rapidly, and at a rate determined by the local spacing of datasites. These
results, in conjunction with a Lebesgue lemma, show that surface spline
interpolation enjoys the same rates of convergence as those of the local
approximation schemes recently developed by DeVore and Ron.
A convergent mixed method for the Stokes approximation of viscous compressible flow. (arXiv:0911.1870v1 [math.NA])
We propose a mixed finite element method for the motion of a strongly
viscous, ideal, and isentropic gas. At the boundary we impose a Navier-slip
condition such that the velocity equation can be posed in mixed form with the
vorticity as an auxiliary variable. In this formulation we design a finite
element method, where the velocity and vorticity is approximated with the div-
and curl- conforming Nedelec elements, respectively, of the first order and
first kind. The mixed scheme is coupled to a standard piecewise constant upwind
discontinuous Galerkin discretization of the continuity equation. For the time
discretization, implicit Euler time stepping is used. Our main result is that
the numerical solution converges to a weak solution as the discretization
parameters go to zero. The convergence analysis is inspired by the continuous
analysis of Feireisl and Lions for the compressible Navier-Stokes equations.
Tools used in the analysis include an equation for the effective viscous flux
and various renormalizations of the density scheme.
martes, 10 de noviembre de 2009
Symmetry-restrained molecular dynamics simulations improve homology models of potassium channels
Most crystallized homo-oligomeric ion channels are highly symmetric, which dramatically decreases conformational space and facilitates building homology models (HMs). However, in molecular dynamics (MD) simulations channels deviate from ideal symmetry and accumulate thermal defects, which complicate the refinement of HMs using MD. In this work we evaluate the ability of symmetry constrained MD simulations to improve HMs accuracy, using an approach conceptually similar to Critical Assessment of techniques for protein Structure Prediction (CASP) competition: build HMs of channels with known structure and evaluate the efficiency of proposed methods in improving HMs accuracy (measured as deviation from experimental structure). Results indicate that unrestrained MD does not improve the accuracy of HMs, instantaneous symmetrization improves accuracy but not stability of HMs during subsequent unrestrained MD, while gradually imposing symmetry constraints improves both accuracy (by 5-50%) and stability of HMs. Moreover, accuracy and stability are strongly correlated, making stability a reliable criterion in predicting the accuracy of new HMs. Proteins 2010. © 2009 Wiley-Liss, Inc.
lunes, 9 de noviembre de 2009
Waveform Transmission Method, a New Waveform-relaxation Based Algorithm to Solve Ordinary Differential Equations in Parallel. (arXiv:0911.1166v1 [math.NA])
Waveform Relaxation method (WR) is a beautiful algorithm to solve Ordinary
Differential Equations (ODEs). However, because of its poor convergence
capability, it was rarely used. Virtual Transmission Method (VTM) is a new
distributed algorithm to solve large sparse linear systems, with an appreciable
convergence speed. In this paper, by marrying WR with VTM, a new distributed
algorithm, named Waveform Transmission Method (WTM), was born. WTM has better
convergence capability than the traditional WR algorithms.
Selection of Corners for the BDDC Method. (arXiv:0911.1245v1 [math.NA])
The Balancing Domain Decomposition by Constraints (BDDC) method has evolved
quite fast since its introduction in 2003, as the primal counterpart to the
earlier FETI-DP method. Recent results have shown close connection of these
methods and theoretically supported equivalent rate of convergence. In both
methods, a fundamental role is played by the coarse space. Optimal choice of
constraints on continuity of the coarse space is still not a satisfactorily
solved problem. The usual basic choice is a 'minimal' set of coarse nodes
(sometimes called corners), that assures invertibility of local subdomain
problems and also of the global coarse problem. However, this set alone does
not suffice for optimal preconditioning in 3D. For this reason, continuity of
some generalized degrees of freedom, such as average values on edges or faces
of subdomains, have to be added. While theoretically correct, this approach
does not easily offer a flexible size of desired coarse problem. In our
contribution, we compare this approach with adding more coarse nodes into the
coarse problem, which is technically simpler and allows flexible setting of
desired approximation. Most attention is given to the basic set of corners, and
a new algorithm for its selection is proposed.
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